3D ribbon woven structures and methods of making and using same

Abstract

Structures, space frames, lattice materials and articles of a great variety of shapes and method to make and use same. The structures feature interwoven ribbon components. The advantages are improved performance, low weight, simple fabrication of components, rapid assembly and less bonding relative to traditional frames and articles. The woven structures can be used in a great variety of different ways.

Description

3D RIBBON WOVEN STRUCTURES AND METHODS OF MAKING AND USING SAMERELATED PATENTS / PATENT APPLICATIONS

[0001] This application claims priority to U.S. Patent Appl. Serial No. 63 / 557,495, filed February 24, 2024, which is entitled “3D Ribbon Woven Structures And Methods of Making And Using Same.” This patent application is commonly owned by the owner of the present invention and is hereby incorporated herein by reference in its entirety for all purposes.TECHNICAL FIELD

[0002] The present invention relates to structures, space frames, lattices and articles of a great variety of shapes and methods to make and use the same.BACKGROUND

[0003] Structures of interconnected struts such as space frames and lattice materials are very useful because they are lighter than ordinary structures and materials while maintaining good strength and structural stiffness. Therefore, these types of structures often find use in applications where low weight and high strength and stiffness are important. Spaceframes and lattice materials are usually composed of repeating unit cells of tubular struts interconnected at nodes. However, space frames typically have nodes that are costly or complicated to manufacture and assemble, particularly when they need to be customized for shapes other than a plane. Stresses concentrate at nodes and so can constitute weak points in the structure.SUMMARY OF THE INVENTION

[0004] The present invention is directed towards structures, space frames, lattice materials, freeform articles, etc. in a great variety of shapes for a great variety of uses. The structures are formed by interweaving ribbon components.

[0005] The present invention is based on a connection between polygon meshes and ribbon topology and the realization that any structure that can be represented by a polygon mesh can be formed from interwoven ribbons. The ribbons are strips of material that have specific shapes that depend on the interconnections and shapes of polygons in the polygon mesh.

[0006] The present invention relates to structures that can have a great variety of shapes, sizes and uses. In accordance with embodiments, the present invention provides for a method of making a structure from a polygon mesh with a target shape. The polygon mesh includes polygons and vertices arranged to represent the target shape. One distinguishing feature is that an article whose shape can be represented by a polygon mesh can be fabricated into a structure using one or more ribbons as components that are interwoven. There are two ends to each ribbon. The ends of the one or more ribbons can be connected in to form a sequence of ribbons that is a closed loop or left open. The structure can have internal or external structure such that the polygon mesh cannot be flattened into a 2D sheet. This is topologically distinct from traditional weaving which can be flattend into a 2D sheet.

[0007] One of the distinguishing features is that unlike typical space frames, no nodes are needed to connect components. This reduces complexity, weight, cost, difficulty of assembly and difficulty of customization for free form shapes. Another distinguishing feature is that the ribbons in accordance with embodiments are fabricated from sheets of material. This can simplify manufacture, transportation and assembly as well as reduce cost. Material in sheet form is also typically less costly by weight than tubes or beams of the same material. Another distinguishing feature is that the ribbons are interwoven into a structure, also referred to as a 3D woven structure.A distinguishing feature from 3D woven fabric technologies is that the components of the present invention are three-dimensional ribbons instead of threads which are essentially one dimensional.A distinguishing feature of the present invention is that a 3D woven structure can have internal or external structure, wherein the structure is based on a polygon mesh that has more than two polygons sharing an edge or has two or more polygons sharing the same two or more edges.

[0008] A previous application, PCT Publication Number WO 2023 / 069785 (“WO 2023 / 069785”), relates to woven structures made of connected plates. The present application relates to woven structures of ribbons. A distinguishing feature of the present invention is that at least one of the ends of one of the ribbons is a wing in a connecting pair with the next wing in the ribbon. This is distinct from the previous application wherein each ribbon end is a matching pair. It has the unexpected advantage that the structure can be constructed without the need for connections, bonding, etc. of plates. In embodiments, the ribbons of the structure are held in place by mechanical interference. Another advantage is that assembly can be simplified because the structure can be formed out of modules that connect by means of ribbon ends of one module overlapping ribbons of another module.

[0009] Another distinguishing feature is that ribbon ends overlap. An advantage of this is that they are held in place by means of mechanical interference.

[0010] In embodiments, a ribbon end is locked in an interference fit by sliding a ribbon end under another ribbon. The ribbon end may be deformed to slide it under the other ribbon. Embodiments include placing an end wing of a ribbon between the wings of a wing pair, wherein one of the wings of the wing pair is a twin wing of the end wing. Embodiments include placing ribbon ends in a flap interlock closure.

[0011] In embodiments, a structure has an end wing of a ribbon in a pocket formed by a wing pair of wings. This forms a layered arrangement of three or more wings at the same strut tab surface. This has the advantage of locking the ribbon in place. In embodiments each ribbon is held in placeby fitting each end in a pocket formed by a wing pair of wings of the same or other ribbons. This has the advantage of holding each ribbon in place. This also has the advantage of creating an aesthetically pleasing look because each ribbon end is hidden from view.

[0012] In embodiments, a structure has three or more ribbon ends in a flap-interlock closure, wherein each end wing of the ribbon ends is under the connecting pair wing of the end wing of the end of the next ribbon. This has the advantage of locking the ribbons in place. In embodiments each ribbon end is held in place with a flap-interlock closure. This has the advantage of not needing bonding to hold it in place. This also has the advantage of creating an aesthetically pleasing look because each ribbon end is hidden from view.

[0013] Another distinguishing feature of embodiments is that ribbons are arranged in modules that have free ribbon ends extending from them. An advantage of this is that the modules can be connected by means of their free ribbon ends to form larger structures.

[0014] Another distinguishing feature of embodiments is that ribbons are arranged in modules that have side wings. An advantage of side wings is that modules can be connected by means of their side wings to form larger structures.

[0015] In general, in embodiments, the invention features a method that includes identifying a shape of a structure to be made. The method further includes generating a representation of the shape of the structure using a polygon mesh. The method further includes determining the shape and dimensions of components of the structure from the polygon mesh. The components are one or more ribbons that can be interwoven to form a structure represented by the polygon mesh. The invention features a method for predetermining the geometry, number and lengths of ribbons, dihedral angles, and geometric arrangement between ribbons for construction of the structure of any of the above-described structures. The method further includes fabricating the ribbons basedon the polygon mesh. The method further includes bending, placing and interweaving the ribbons according to the polygon mesh. Each end of each ribbon overlaps another end of the same or another ribbon at a shared surface wherein the interwoven ribbons form the structure. Each of the ribbons can connect to one or more other ribbons at surfaces of the one or more ribbons.

[0016] In general, in embodiments, the invention features a method to form a structure including one or more ribbons. Each ribbon includes a plurality of wings. The method includes identifying a shape of the structure. The method further includes generating a representation of the shape using a polygon mesh. The polygon mesh includes an assembly of polygons. Each polygon includes vertices and edges. Each polygon shares an edge with one or more adjacent polygons. Each edge shares a vertex with one or more adjacent edges. Each polygon has a first face and an opposite face, wherein each face has an outward direction. The method further includes determining a shape and dimensions of the wings based on the polygon mesh. Each particular wing has a particular edge of a particular face of a particular polygon of the polygon mesh. Each particular wing shares the particular edge with a second wing, wherein the second wing is associated with a second face of a second polygon. The outward direction of the particular wing and the outward direction of the second wing point into a same volume. When only the particular polygon has the particular edge, the second polygon is the particular polygon and the second face is the opposite face of the particular polygon. The particular wing and the second wing are a matching pair. Each particular wing shares a strut tab surface with a third wing, wherein the third wing is associated with the opposite face of the particular polygon. The particular wing and the third wing share the particular edge. When only the particular polygon has the particular edge, the third wing is the second wing. The particular wing and the third wing are a wing pair. Each particular wing shares a connecting tab surface with a fourth wing, wherein the fourth wing isassociated with the opposite face of the particular polygon. The fourth wing has an edge adjacent to the particular edge. The particular wing and the fourth wing point to a shared vertex of the particular edge and the adjacent edge. The particular wing and the fourth wing are a connecting pair. The method further includes determining a shape and dimensions of the one or more ribbons. Each ribbon includes a contiguous surface comprising a sequence of matching pairs and connecting pairs of wings. The method further includes fabricating the one or more ribbons based on the shape and dimensions of the ribbons. The method further includes forming the structure by bending and placing the ribbons according to the polygon mesh. Each wing pair shares a strut tab surface. Each end of each ribbon shares a connecting tab surface with another end of a ribbon.

[0017] Implementations of the invention can include one or more of the following features:

[0018] The one or more ribbons are connected to each other at their strut tab and / or connecting tab surfaces.

[0019] The polygon mesh can be a surface (2D) mesh or a volume (3D) mesh. There can be two or more polygons in the polygon mesh that share a same edge. There can be two or more edges that are shared by the same polygons.

[0020] Faces are subdivided into wings such that each wing has a particular edge of a face. The shapes of the wings are such that each wing overlaps part of the wing on the opposite face that has an adjacent edge to the particular edge. The same direction, counterclockwise or clockwise, is chosen for going around the face to determine the adjacent edge relative to the particular edge. The right-hand thumb rule is chosen to relate the outward direction of each face to the counterclockwise direction around the face. On the other hand, the left-hand thumb rule can also be chosen to relate the outward direction of each face to the clockwise direction around the face. No matter what ruleis chosen, it is applied consistently to each face in the polygon mesh. The right-hand thumb rule is used in all the examples below.

[0021] When the ribbons are assembled into the structure, the outward directions of the wings of each ribbon are used to place each ribbon relative to the same or other ribbons. The outward direction of each of the wings of the matching pair of wings points into the same volume. The outward direction of each wing is the same as the outward direction of the face with which it is associated. Each ribbon can extend from the surface of each wing into the volume indicated by the outward direction of each wing. This defines the thickness of each of the ribbons. The thickness can vary over the length of each ribbon or can vary from ribbon to ribbon.

[0022] The method can further include using one or both of guide pins or relief features to align or place the plurality of wings when placing the one or more ribbons to form the structure.

[0023] The contiguous surface or weaver of matching pairs and connecting pairs can be a closed loop or can be open. The method further includes opening a closed loop at a connecting pair or matching pair so that the contiguous surface has two ends. The method can further include opening a contiguous surface at several connecting pairs or matching pairs so that each of the resulting parts is a smaller contiguous surface with two ends and each end has a matching pair or connecting pair.

[0024] The method can further include flattening a contiguous surface of matching pairs and connecting pairs so that the dihedral angle for each matching pair becomes 180 degrees. The method can further include making a design for a ribbon based on the flattened contiguous surface, wherein the design includes the location of a bending axis at each edge of the polygon mesh shared by a matching pair of the ribbon. The value of the dihedral angle in the polygon mesh can be indicated at each bending axis. An index can identify a wing pair or a connecting pair.

[0025] The method can further include fabricating the ribbons with notches at the ends of the bending axes in the one or more ribbons.

[0026] The one or more ribbons can have substantially the same width and shape.

[0027] Each of the wing pairs can include wings of substantially the same width.

[0028] Polygons in the polygon mesh can be non-planar, that is a polygon can be a skew-polygon, wherein the vertices do not all lie in the same plane. The wings generated from the faces of the polygon mesh can be non-planar. A polygon can be made non-planar by making the surface bulge, have curvature or extend out to a point.

[0029] Polygons in the polygon mesh can be selected from the group consisting of triangles, quadrangles, pentagons, hexagons, heptagons, octagons, nonagons, and decagons, hendecagons, dodecagons, polygons with more than twelve edges, and combinations thereof.

[0030] Polygons of the polygon mesh can be divided into planar facets.

[0031] Generating a shape of a structure from a polygon mesh can include augmenting the polygon mesh with polyhedrons.

[0032] Generating of a shape of a structure from a polygon mesh can include subdividing a volume into polyhedrons.

[0033] The method can further include overlapping the interwoven ribbon structure with material such that there are no or minimal holes or gaps.

[0034] The method can further include adhering panels to the structure to close off holes in the structure. The method can further include adding a hole and removing the air such that the pressure inside the structure is less than the pressure outside the structure.

[0035] The structure can be selected from the group consisting of substantially plane shapes, substantially spherical shapes, substantially dome shapes, substantially cylindrical shapes, substantially shell-like shapes, and combinations thereof.

[0036] The structure can be a free-form shape.

[0037] The structure can have a plurality of branched volumes.

[0038] The structure can be chiral.

[0039] The structure can be a pentet block structure.

[0040] The polygon mesh can have a shape that is part of a lattice.

[0041] The polygon mesh can have a shape that is part of an octet lattice.

[0042] The polygon mesh can have a shape that is part of a gyroid lattice.

[0043] The polygon mesh can be in the form of adjacent polyhedral arrangements.

[0044] The polygon mesh can include polyhedrons selected from the group consisting of tetrahedrons, pentahedrons, pyramids, hexahedrons, cuboids, heptahedrons, octahedrons, nonahedrons, decahedrons, hendecahedrons, dodecahedrons, polyhedrons with more than twelve faces, and combinations thereof.

[0045] The fabricating the one or more ribbons can include a method of forming a sheet of material in the shape of the one or more ribbons selected from the group consisting of but not limited to (a) cutting by laser, mechanical means, plasma, waterjet, abrasives, etc., (b) molding, (c) casting, and (d) additive manufacturing methods, etc.

[0046] The fabricating the one or more ribbons can include the process of imposing a bend at each bending axis equal to the dihedral angle in the polygon mesh. The process of imposing a bend is selected from the group consisting of but not limited to (a) bending a sheet of material by applying a physical force, (b) applying heat and bending, (c) folding, (d) pinching, (e) molding, (f) casting,(g) by joining separate pieces at the dihedral angle (h) by using prepreg carbon fiber composite material curing in a mold, heat and / or vacuum, scoring, perforating etc., and combinations thereof.

[0047] The one or more ribbons can include a material selected from the group consisting of but not limited to metals, alloys, composites, carbon fiber composites, ceramics, polymers, copolymers, rubber, textile, paper, cardboard, wood, leather, stone, concrete, sandwich core materials, honeycomb core plates, natural materials, foams, elastomers, alloys of metals, graphene, carbon nanotubes, beta-protein sheets, glasses, and combinations thereof.

[0048] The one or more ribbons can include a material property selected from the group consisting of but not limited to rigid, flexible, elastic, electrically conducting, semiconducting, insulating, translucent, opaque, transparent, reflecting materials, and combinations thereof.

[0049] The one or more ribbons can include connectors to connect the one or more ribbons together. The connectors can be selected from the group consisting of but not limited to friction coupling with or without a retaining agent, adhesive bonds, welded materials, brazing materials, melted material, glue, sintered materials, fasteners, rivets, bolts, nails, screws, latches, buckles, catches, clasps, stitching material, sewing material, buttons, tying material, self-fasteners, hook and loop, hole and pin, closures, clamps, couplings, links, magnets, molecular bonds, and combinations thereof.

[0050] The ribbons feature a code or index for identifying wing pairs at strut tabs and connecting pairs at connecting tabs of the structure or any of the above-described structures. The method is selected from the group consisting of but not limited to using numbers, letters, symbols, RF tags, QR codes, color codes, bar codes.

[0051] The structure can be operable for use as a space frame.

[0052] The structure can be operable for use as a monocoque structure.

[0053] The structure can be operable for use as a lattice material.In general, in embodiments, the invention features a structure including one or more ribbons. Each ribbon includes a plurality of wing, Each particular wing has a particular edge of a particular face of a particular polygon of a polygon mesh. The polygon mesh includes an assembly of polygons. Each polygon includes vertices and edges. Each polygon shares an edge with one or more adjacent polygons. Each edge shares a vertex with one or more adjacent edges. Each polygon has a first face and an opposite face. Each face has an outward direction. The particular wing is joined to a second wing. The second wing is associated with a second face of a second polygon. The particular wing and the second wing share the particular edge. The outward direction of the particular wing and the outward direction of the second wing point into a same volume. When only the particular polygon has the particular edge, the second polygon is the particular polygon and the second face is the opposite face of the particular polygon. The particular wing and the second wing are a matching pair. The particular wing shares a strut tab surface with a third wing. The third wing is associated with the opposite face of the particular polygon. The particular wing and the third wing share the particular edge. When only the particular polygon has the particular edge, the third wing is the second wing. The particular wing and the third wing are a wing pair. The particular wing shares a connecting tab surface with a fourth wing. The fourth wing is associated with the opposite face of the particular polygon. The fourth wing has an edge adjacent to the particular edge. The particular wing and the fourth wing point to a shared vertex of the particular edge and the adjacent edge. The particular wing and the fourth wing are a connecting pair. Each ribbon includes a contiguous surface comprising a sequence of matching pairs and connecting pairs of wings. The ribbons form the structure. Each ribbon is bent and placed according to the polygon mesh. Eachwing pair shares a strut tab surface. Each end of each ribbon shares a connecting tab surface with another end of a ribbon.

[0054] In general, in an embodiment, the invention features a structure with the topology of a structure that includes one or more ribbons. Each ribbon includes a plurality of wings. Each particular wing has a particular edge of a particular face of a particular polygon of a polygon mesh. The polygon mesh includes an assembly of polygons. Each polygon includes vertices and edges. Each polygon shares an edge with one or more adjacent polygons. Each edge shares a vertex with one or more adjacent edges. Each polygon has a first face and an opposite face. Each face has an outward direction. The particular wing is joined to a second wing. The second wing is associated with a second face of a second polygon. The particular wing and the second wing share the particular edge. The outward direction of the particular wing and the outward direction of the second wing point into a same volume. When only the particular polygon has the particular edge, the second polygon is the particular polygon and the second face is the opposite face of the particular polygon. The particular wing and the second wing are a matching pair. The particular wing shares a strut tab surface with a third wing. The third wing is associated with the opposite face of the particular polygon. The particular wing and the third wing share the particular edge. When only the particular polygon has the particular edge, the third wing is the second wing. The particular wing and the third wing are a wing pair. The particular wing shares a connecting tab surface with a fourth wing. The fourth wing is associated with the opposite face of the particular polygon. The fourth wing has an edge adjacent to the particular edge. The particular wing and the fourth wing point to a shared vertex of the particular edge and the adjacent edge. The particular wing and the fourth wing are a connecting pair. Each ribbon includes a contiguous surface including a sequence of matching pairs and connecting pairs of wings. The ribbons form thestructure. Each ribbon is bent and placed according to the polygon mesh. Each wing pair shares a strut tab surface. Each end of each ribbon shares a connecting tab surface with another end of a ribbon.

[0055] The topology of a structure includes knot topology, link topology and ribbon topology.

[0056] In embodiments, the topology is locally locked or frozen by associations of ribbons due to wing pairs with shared strut tab surfaces at edges of the polygon mesh. Each weaver in a structure can be identified by a “weaver sequence” of wing indices, one index for each wing, ordered in a sequence of alternating adjacent matching pairs and connecting pairs of wings. Further, the topology of the structure can be represented by the interrelation of the weavers at their wing pairs which can be represented by replacing each wing index in the weaver sequences with its wing pair index.

[0057] The method includes fabricating one or more ribbons and interweaving the one or more ribbons by bending and placing. Each of the ribbons includes a contiguous sequence of matching pairs and connecting pairs of wings, wherein each matching pair includes a first wing and a second wing. The first wing is associated with a third wing. The first wing and the third wing are a wing pair of wings. The second wing and the third wing are the same or different wings. When the second wing and the third wing are different wings, the third wing is a wing of a different matching pair than the matching pair including the first wing and the second wing. The first wing and third wing can connect at a strut tab. The first wing is associated with a fourth wing. The first wing and the fourth wing are a connecting pair of wings. The fourth wing is different from each of the first wing, the second wing, and the third wing. The fourth wing is a wing of a second matching pair that is different from the matching pair or matching pairs including the first wing, the second wing, and the third wing. The first wing and the fourth wing can connect at a connecting tab. Themethod further includes forming one or more ribbons, wherein each ribbon comprises a contiguous surface, known as a weaver, comprising a linear sequence of a plurality of matching pairs and connecting pairs of wings that share connecting tab surfaces. The method further includes interweaving the ribbons to form the structure, wherein each particular wing at each end of each particular ribbon shares a connecting tab surface with a second wing at the end of a second ribbon, wherein the second wing is a connecting pair with the particular wing and the second ribbon is the particular ribbon or another ribbon.

[0058] In general, in an embodiment, the structure is selected from a group consisting of but not limited to space frames, lattice materials, geodesic domes, building structures, spans / decking of bridges, free form architecture, towers, stadiums, airports, pavilions, earthquake-resistant constructions, cladding systems for blast / explosion protection, light-weight panels for cars, trucks, buses, and trains, airframes, rocket frames and parts, space craft, hulls for ships, yachts, and submarines, frame structures for machines, robots or lifts, hyperloop tubes, space telescope supports, lightweight pressure vessels and tanks, spaceship construction systems, solar sail supports, tunnel cladding systems, parts of helmets / personal protective gear, parts of sports equipment, structures for a vacuum lift aircraft, heat exchangers, batteries, prosthetic implants, clothing, three-dimensional art, installations and sculptures, etc.

[0059] The structure can have the shape of a surface of a triangular mesh augmented by tetrahedrons and at least some of the struts of the surface and augmenting tetrahedrons are made up by a weaver in the shape of a figure ‘8’. The topology of the structure features ribbons in the shape of a figure ‘8’, wherein each loop forms a Borromean link with two other loops. Each Borromean link is associated with an augmenting tetrahedron.

[0060] In general, in another embodiment, the invention features a method for producing a structure. The method includes providing ribbons including a linear sequence of matching pairs and connecting pairs of wings. Each matching pair has a bend or is bendable along a bending axis. Each bending axis corresponds to an edge of the polygon mesh. The method can further include connecting one or more ribbons at strut tabs of wings in a wing pair such that the bending axes of the matching wings are parallel and coincident.

[0061] Implementations of the invention can include interweaving one or more ribbons and can include connecting wings of a connecting pair at one or more connecting tabs. The interweaving one or more of ribbons can include connecting wings of a wing pair at one or more strut tabs.

[0062] In general, in another embodiment, the invention features a structure with a topology of a structure made by the above-described methods.

[0063] In general, in another embodiment, the invention features a method to form a structure including one or more ribbons. The method includes identifying a shape of the structure. The method further includes generating a representation of the shape with a polygon mesh. The polygon mesh includes an assembly of polygons. Each polygon includes vertices, edges and a surface bounded by the edges. For each edge there is a first directed edge and an opposite directed edge. Each polygon has a first face and an opposite face. The method further includes subdividing the faces into a plurality of wings. Each particular wing is from a particular face and has a particular directed edge. Each particular wing is a matching pair with a second wing. The particular wing and the second wing have a shared edge and opposite directed edges. The particular wing and the second wing are from faces with outward directions pointing into a same volume. The particular wing and the second wing have a bending axis at the particular edge and a dihedral angle. Each particular wing is a connecting pair with a third wing. The particular wing and the third wing havedirected edges that point to a same vertex. The particular wing and the third wing are from opposite faces of the particular polygon. The method further includes determining a shape and dimensions of the one or more ribbons. Each ribbon of the one or more ribbons includes an alternating sequence of a matching pair and a connecting pair of wings. At least one of the ribbons has an end wing that is in a connecting pair of wings. The method further includes fabricating the one or more ribbons based on the shape and dimensions of the ribbons and the dihedral angles at the bending axes. The method further includes forming the structure by bending and placing the one or more ribbons according to the polygon mesh.

[0064] Implementations of the invention can include one or more of the following features:

[0065] The one or more ribbons can be a plurality of ribbons.

[0066] For at least some of the ribbons in the one or more ribbons, the alternating sequence of a matching pair and a connecting pair of wings can include an alternating sequence that includes a first matching pair of wings, a first connecting pair of wings, a second matching pair of wings, and a second connecting pair of wings.

[0067] For at least some of the ribbons in the one or more ribbons, the alternating sequence of a matching pair and a connecting pair of wings can include an alternating sequence that further includes a third matching pair of wings and a third connecting pair of wings, such that, for at least some of the ribbons in the one or more ribbons the alternating sequence includes the first matching pair of wings, the first connecting pair of wings, the second matching pair of wings, the second connecting pair of wings, the third matching pair of wings, and the third connecting pair of wings.

[0068] At least one of the ribbons can have a side wing connected to the connecting pair wing of an end wing of the ribbon.

[0069] Each of the ribbons of at least 50% of the ribbons in the one or more ribbons can have at least one end wing that is in a connecting pair of wings.

[0070] Each of the ribbons of the at least 50% of the ribbons in the one or more ribbons can have a side wing connected to the connecting pair wing of an end wing of the ribbon.

[0071] At least one of the ribbons in the one or more ribbons can have (a) a first end wing that is in a first connecting pair of wings and (b) a second end wing that is in a second connecting pair of wings.

[0072] The at least one of the ribbons can have (a) a first side wing connected to the connecting pair wing of the first end wing and (b) a second side wing connected to the connecting pair wing of the second end wing.

[0073] Each of the ribbons of at least 50% of the ribbons in the one or more ribbons can have a first end wing that is in a connecting pair of wings and a second end wing that is in a connecting pair of wings.

[0074] Each of the ribbons of the at least 50% of the ribbons in the one or more ribbons can have(a) a first side wing connected to the connecting pair wing of the first end wing of the ribbon and(b) a second side wing connected to the connecting pair wing of the second end wing of the ribbon.

[0075] The polygon mesh can include an edge shared by three or more polygons.

[0076] An end wing that is in a connecting pair of wings can be placed in a pocket formed by a wing pair of wings.

[0077] An end wing that is in a connecting pair of wings can be placed in flap-interlock closure.

[0078] The structure can be a module based on a degenerate polyhedron polygon mesh.

[0079] The ribbons can be connected at shared strut tab and / or connecting tab surfaces.

[0080] The step of forming the structure can include elastically deforming the one or more ribbons while bending and placing the one or more ribbons according to the polymer mesh.

[0081] After the ribbons are placed according to the polymer mesh, the elastic deformation can provide forces upon the structure that maintains placement of the one or more ribbons.

[0082] In general, in another embodiment, the invention features a structure including one or more ribbons. The shape of the structure is represented by a polygon mesh. The polygon mesh includes an assembly of polygons. Each polygon includes vertices, edges and a surface bounded by the edges. For each edge, there is a first directed edge and an opposite directed edge. Each polygon has a first face and an opposite face. The faces are subdivided into a plurality of wings. Each particular wing is from a particular face and has a particular directed edge. Each particular wing is a matching pair with a second wing. The particular wing and the second wing have a shared edge and opposite directed edges. The particular wing and the second wing are from faces with outward directions pointing into a same volume. The particular wing and the second wing have a bending axis at the particular edge and a dihedral angle. Each particular wing is a connecting pair with a third wing. The particular wing and the third wing have directed edges that point to a same vertex. The particular wing and the third wing are from opposite faces of the particular polygon. Each ribbon of the one or more ribbons includes an alternating sequence of a matching pair and a connecting pair of wings. At least one of the ribbons has an end wing that is in a connecting pair of wings. Each ribbon is bent and placed according to the polygon mesh.

[0083] Implementations of the invention can include one or more of the following features:

[0084] The one or more ribbons can be a plurality of ribbons.

[0085] For at least some of the ribbons in the one or more ribbons, the alternating sequence of a matching pair and a connecting pair of wings can include an alternating sequence that includes afirst matching pair of wings, a first connecting pair of wings, a second matching pair of wings, and a second connecting pair of wings.

[0086] For at least some of the ribbons in the one or more ribbons, the alternating sequence of a matching pair and a connecting pair of wings can include an alternating sequence that further includes a third matching pair of wings and a third connecting pair of wings, such that, for at least some of the ribbons in the one or more ribbons the alternating sequence includes the first matching pair of wings, the first connecting pair of wings, the second matching pair of wings, the second connecting pair of wings, the third matching pair of wings, and the third connecting pair of wings.

[0087] At least one of the ribbons can have a side wing connected to the connecting pair wing of an end wing of the ribbon.

[0088] Each of the ribbons of at least 50% of the ribbons in the one or more ribbons can have at least one end wing that is in a connecting pair of wings.

[0089] Each of the ribbons of the at least 50% of the ribbons in the one or more ribbons can have a side wing connected to the connecting pair wing of an end wing of the ribbon.

[0090] At least one of the ribbons in the one or more ribbons can have (a) a first end wing that is in a first connecting pair of wings and (b) a second end wing that is a in second connecting pair of wings.

[0091] The at least one of the ribbons can have (a) a first side wing connected to the connecting pair wing of the first end wing and (b) a second side wing connected to the connecting pair wing of the second end wing.

[0092] Each of the ribbons of at least 50% of the ribbons in the one or more ribbons can have a first end wing that is in a connecting pair of wings and a second end wing that is in a connecting pair of wings.

[0093] Each of the ribbons of the at least 50% of the ribbons in the one or more ribbons can have(a) a first side wing connected to the connecting pair wing of the first end wing of the ribbon and(b) a second side wing connected to the connecting pair wing of the second end wing of the ribbon.

[0094] The polygon mesh can be a 3D mesh comprising an edge shared by three or more polygons.

[0095] An end wing that is in a connecting pair of wings can be placed in a pocket formed by a wing pair of wings.

[0096] An end wing that is in a connecting pair of wings can be placed in flap-interlock closure.

[0097] The structure can be a module based on a degenerate polyhedron polygon mesh.

[0098] The ribbons can be connected to each other at shared strut tab and / or connecting tab surfaces.

[0099] A topology of the structure can be represented by one or more weaver sequences of wings in which each wing is indicated by its wing pair index.

[0100] The structure can be made by the above-described method.BRIEF DESCRIPTION OF THE DRAWINGS

[0101] FIG. 1A depicts a perspective view of a 3D polygon mesh with five polygons, showing the outward direction of five faces.

[0102] FIG. IB depicts a perspective view the polygon mesh in FIG. 1A showing the subdivision of five faces into wings.

[0103] FIG. 1C depicts a perspective view of the polygon mesh in FIG. 1A showing the outward direction of five opposite faces.

[0104] FIG. ID depicts a perspective view the polygon mesh in FIG. 1A showing the subdivision of five opposite faces into wings.

[0105] FIG. IE depicts a perspective view of the polygon mesh in FIG. 1A showing a first contiguous surface of wings with the outward directions of each constituent wing depicted by an arrow.

[0106] FIG. IF depicts a perspective view of the polygon mesh in FIG. 1A showing a second contiguous surface of wings with the outward directions of each constituent wing depicted by an arrow.

[0107] FIG. 1G depicts a perspective view of the polygon mesh in FIG. 1A showing the first and second contiguous surfaces of wings with the outward directions of each constituent wing depicted by an arrow.

[0108] FIG. 1H depicts a perspective view of the polygon mesh in FIG. 1A showing a first end of the second contiguous surface of wings overlapping the second end of the first contiguous surface.

[0109] FIG. II depicts a perspective view of the polygon mesh in FIG. 1A showing a third contiguous surface of wings completing the first loop.

[0110] FIG. 1J depicts a perspective view of the polygon mesh in FIG. 1A showing two contiguous surfaces of wings completing a second loop.

[0111] FIG. IK depicts designs for ribbons based on five contiguous surfaces for the polygon mesh in FIG. 1A.

[0112] FIG. IL depicts a perspective view of the polygon mesh in FIG. 1A showing a first ribbon bent at its bending axes and placed.

[0113] FIG. IM depicts a perspective view of the polygon mesh in FIG. 1A showing a first end of a second ribbon overlapping the second end of the first ribbon.

[0114] FIG. IN depicts a perspective view of the polygon mesh in FIG. 1A showing a first end of a third ribbon overlapping the second end of the second ribbon and completing a first loop.

[0115] FIG. IO depicts a perspective view of the polygon mesh in FIG. 1A showing a fourth and fifth ribbon completing a second loop.

[0116] FIG. IP depicts a perspective view of the polygon mesh in FIG. 1A showing the interwoven ribbons forming the structure.

[0117] FIG. IQ depicts another perspective view of the polygon mesh in FIG. 1A showing the interwoven ribbons forming the structure.

[0118] FIG. 1R depicts the knotted, twisted ribbon topology of the structure in FIG. IP, highlighting an association of three ribbons.

[0119] FIG. 2A depicts a view from above of a 2D polygon mesh with three polygons.

[0120] FIG. 2B depicts a view from below of the polygon mesh in FIG. 2A.

[0121] FIG. 2C depicts a perspective view of the polygon mesh in FIG. 2A showing the outward direction of the faces.

[0122] FIG. 2D depicts a view from above of the polygon mesh in FIG. 2A showing the subdivision of the faces on one side of the polygons into wings.

[0123] FIG. 2E depicts a perspective view from below of the polygon mesh in FIG. 2A showing the outward direction of the faces.

[0124] FIG. 2F depicts a view from below of the polygon mesh in FIG. 2A showing the subdivision of the faces on the other side of the polygons into wings.

[0125] FIG. 2G depicts a view from above of the polygon mesh in FIG. 2A showing the wings from the subdivision of the faces on both sides of the polygons.

[0126] FIG. 2H depicts a perspective view of the polygon mesh in FIG. 2A showing two wings in a connecting pair.

[0127] FIG. 21 depicts a view from above of the polygon mesh in FIG. 2A showing two wings in a connecting pair.

[0128] FIG. 2J depicts a perspective view of the polygon mesh in FIG. 2A showing two wings in a wing pair.

[0129] FIG. 2K depicts a view from above of the polygon mesh in FIG. 2A showing two wings in a wing pair.

[0130] FIG. 2L depicts a perspective view of the polygon mesh in FIG. 2A showing two wings in a matching pair.

[0131] FIG. 2M depicts a perspective view of the polygon mesh in FIG. 2A showing two wings in a matching pair.

[0132] FIG. 2N depicts the polygon mesh in FIG. 2A showing two matching pairs sharing strut tab surfaces.

[0133] FIG. 20 depicts the polygon mesh in FIG. 2A showing two matching pairs sharing connecting tab surfaces.

[0134] FIG. 2P depicts the polygon mesh in FIG. 2A showing a sequence of three matching pairs sharing connecting tab surfaces.

[0135] FIG. 2Q depicts a perspective view of the 3D woven ribbon structure for the polygon mesh in FIG. 2A

[0136] FIG. 2R depicts the ribbon topology of the 3D woven ribbon structure for the polygon mesh in FIG. 2A.

[0137] FIG. 2S depicts the pentagon in the polygon mesh in FIG. 2A showing the wings from the subdivision of both faces using the straight wing method.

[0138] FIG. 2T depicts the pentagon in the polygon mesh in FIG. 2A showing the wings from the subdivision of both faces using the coincident intersection method.

[0139] FIG. 2U depicts a perspective view of the 3D woven structure of ribbons for the polygon mesh in FIG. 2A with the ribbons designed using the straight wing method.

[0140] FIG. 2V depicts a perspective view of the 3D woven structure of ribbons for the polygon mesh in FIG. 2A with the ribbons designed using the coincident intersection method.

[0141] FIG. 2W depicts the pentagon in the polygon mesh in FIG. 2A showing the wings from the subdivision of both faces using the coincident intersection method with additional points.

[0142] FIG. 2X depicts a perspective view of the 3D woven structure of ribbons for the polygon mesh in FIG. 2A with curved wings designed using the method of coincident intersections with B-spline curves.

[0143] FIG. 2Y depicts a perspective view of a polygon mesh with non-planar polygons.

[0144] FIG. 2Z depicts a perspective view of the 3D woven structure of ribbons for the polygon mesh with non-planar polygons in FIG. 2Y with the wings designed using the method of coincident intersections with additional points.

[0145] FIG. 3A depicts a view of a polygon mesh for a pentet including a cube and an inner tetrahedron.

[0146] FIG. 3B depicts a view of wings for faces of the inner tetrahedron in FIG. 3A.

[0147] FIG. 3C depicts a view of wings for faces of the cube in FIG. 3A.

[0148] FIG. 3D depicts a view of a weaver for the polygon mesh in FIG. 3A.

[0149] FIG. 3E depicts a view of the weaver in FIG. 3D including the outward directions of the wings of the weaver.

[0150] FIG. 3F depicts a view of the weaver in FIG. 3D that has been opened and flattened.

[0151] FIG. 3G depicts designs for ribbons for the polygon mesh in FIG. 3A.

[0152] FIG. 3H depicts a view of a ribbon bent and placed for the weaver in FIG. 3E.

[0153] FIG. 31 depicts a view of two ribbons bent and placed for the two weavers in FIG. 3F.

[0154] FIG. 3J depicts a view of three ribbons bent and placed to complete three weavers for the polygon mesh in FIG. 3A.

[0155] FIG. 3K depicts a view of a 3D woven structure of six ribbons for the polygon mesh inFIG. 3A

[0156] FIG. 3L depicts the ribbon topology of the ribbon in FIG. 3H.

[0157] FIG. 3M depicts of the ribbon topology of the two ribbons in FIG. 31.

[0158] FIG. 3N depicts the ribbon topology of three ribbons for the structure in FIG. 3K.

[0159] FIG. 30 depicts the ribbon topology of six ribbons for the structure in FIG. 3K.

[0160] FIG. 4A depicts a view of a polygon mesh for a box with two compartments.

[0161] FIG. 4B depicts a view of wings for the polygon mesh in FIG. 4A.

[0162] FIG. 4C depicts designs for ribbons for the polygon mesh in FIG. 4A.

[0163] FIG. 4D depicts a view of a ribbon bent and placed for the polygon mesh in FIG. 4A.

[0164] FIG. 4E depicts a view of a weaver of two ribbons bent and placed for the polygon mesh in FIG. 4A

[0165] FIG. 4F depicts a view of a weaver of four ribbons bent and placed for the polygon mesh in FIG. 4A

[0166] FIG. 4G depicts a view of a 3D woven structure of two weavers for the polygon mesh inFIG. 4A

[0167] FIG. 4H depicts the ribbon topology of the 3D woven structure in FIG. 4G.

[0168] FIG. 5A depicts a view of a polygon mesh for a quadcopter drone frame.

[0169] FIG. 5B depicts two views of polygon faces subdivided into wings for the polygon mesh in FIG. 5A

[0170] FIG. 5C depicts designs for ribbons for the polygon mesh in FIG. 5A.

[0171] FIG. 5D depicts a view of a ribbon bent and placed for the polygon mesh in FIG. 5A.

[0172] FIG. 5E depicts a view of two ribbons bent and placed for the polygon mesh in FIG. 5A.

[0173] FIG. 5F depicts a view of a 3D woven structure for the polygon mesh in FIG. 5A.

[0174] FIG. 5G depicts a view of a 3D woven structure for the polygon mesh in FIG. 5A.

[0175] FIG. 5H depicts a view of a 3D woven structure for the polygon mesh in FIG. 5A.

[0176] FIG. 51 depicts the ribbon topology of the 3D woven structure in FIG. 5H.

[0177] FIG. 6A depicts a view of a polygon mesh for a unit cell for a gyroid lattice.

[0178] FIG. 6B depicts designs for ribbons for the polygon mesh in FIG. 6A.

[0179] FIG. 6C depicts a view of a ribbon bent and placed for the polygon mesh in FIG. 6A.

[0180] FIG. 6D depicts a view of two ribbons bent and placed for the polygon mesh in FIG. 6A.

[0181] FIG. 6E depicts a view of a weaver of three ribbons bent and placed for the polygon mesh in FIG. 6A

[0182] FIG. 6E depicts a view of a 3D woven structure of one weaver of three ribbons for the polygon mesh in FIG. 6A.

[0183] FIG. 7A depicts a view of a polygon mesh for a table.

[0184] FIG. 7B depicts designs for ribbons for the polygon mesh in FIG. 7A.

[0185] FIG. 7C depicts a view of a ribbon bent and placed for the polygon mesh in FIG. 7A.

[0186] FIG. 7D depicts a view of a 3D woven structure of four weavers for the polygon mesh inFIG. 7A

[0187] FIG. 7E depicts a view from below of the 3D woven structure in FIG. 7D.

[0188] FIG. 8A depicts a view of a polygon mesh for a rocket.

[0189] FIG. 8B depicts a view of polygon faces subdivided into wings for the polygon mesh inFIG. 8A

[0190] FIG. 8C depicts designs for ribbons for the polygon mesh in FIG. 8A.

[0191] FIG. 8D depicts a view of ribbons bent and placed for the polygon mesh in FIG. 8A.

[0192] FIG. 8E depicts a view of a 3D woven structure for the polygon mesh in FIG. 8A.

[0193] FIG. 8F depicts the ribbon topology of the 3D woven structure in FIG. 8H.

[0194] FIG. 9A depicts a view of a polygon mesh for a space frame.

[0195] FIG. 9B depicts a view of polygon faces subdivided into wings for the polygon mesh inFIG. 9A

[0196] FIG. 9C depicts designs for four ribbons for the polygon mesh in FIG. 9A.

[0197] FIG. 9D depicts a view of the four ribbons in FIG. 9C bent at their bending axes for the polygon mesh in FIG. 9A.

[0198] FIG. 9E depicts a view of two ribbons bent and placed for a unit cell for a space frame.

[0199] FIG. 9F depicts a view of two ribbons bent and placed for a unit cell for a space frame.

[0200] FIG. 9G depicts another view of the unit cell in FIG. 9D.

[0201] FIG. 9H depicts a view of a 3D woven structure for the polygon mesh in FIG. 9A.

[0202] FIG. 91 depicts a view of the 3D woven structure in FIG. 9H.

[0203] FIG. 9J depicts a view of the 3D woven structure in FIG. 9H.

[0204] FIG. 9K depicts the ribbon topology of the 3D woven structure in FIG. 9J.

[0205] FIG. 9L depicts three ribbons needed to form a tetrahedron module.

[0206] FIG. 9M depicts the first ribbon in FIG. 9L bent according to the dihedral angles.

[0207] FIG. 9N depicts the second ribbon in FIG. 9L bent according to the dihedral angles.

[0208] FIG. 90 depicts a tetrahedron module consisting of three interwoven ribbons with four free ribbon ends.

[0209] FIG. 9P depicts two tetrahedron modules aligned with each other.

[0210] FIG. 9Q depicts two tetrahedron modules connected by means of ribbon ends.

[0211] FIG. 9R depicts two views of many tetrahedrons connected in the form of a planar space frame.

[0212] FIG. 10A depicts two views of a polygon mesh for a sports car.

[0213] FIG. 10B depicts two views of a 3D woven structure for the polygon mesh in FIG. 10A.

[0214] FIG. 10C depicts two views for a ribbon topology of the 3D woven structure in FIG. 10B.

[0215] FIG. 11A depicts a target structure in the shape of a tetrahedron sharing vertices with a cube.

[0216] FIG. 11B depicts a polygon mesh in the form of a tetradihedron with planar faces in the shape of a tetrahedron and nonplanar faces in the shape of a cube.

[0217] FIG. 11C depicts a polygon mesh in the form of a tetradihedron with subdivision vertices for the nonplanar faces in the shape of a cube.

[0218] FIG. 11D depicts a polygon mesh in the form of a tetradihedron with subdivision vertices for the planar faces.

[0219] FIG. HE depicts four wings of a weaver based on the tetradihedron polygon mesh.

[0220] FIG. HF depicts eight wings of a weaver based on the tetradihedron polygon mesh.

[0221] FIG. 11G depicts a weaver comprising an alternating sequence of eight matching pairs and eight connecting pairs of wings.

[0222] FIG. 11H depicts the weaver in FIG. 11G opened and flattened out.

[0223] FIG. Ill depicts a design for a ribbon comprising an alternating sequence of four connecting pairs and three matching pairs of wings based on the weaver in FIG. 11H.

[0224] FIG. 11 J depicts the ribbon in FIG. Ill of bent and placed according to the polygon mesh.

[0225] FIG. UK depicts another ribbon with the same shape as the one in FIG. Ill bent and placed according to the tetradihedron polygon mesh.

[0226] FIG. 11L depicts the ribbons in FIG. 11 J and FIG. 11K bent and placed according to the tetradihedron polygon mesh.

[0227] FIG. 11M depicts three ribbons overlapping each other at one of the planar triangular faces prior to bending and imposing a three-flap interlock closure to form a corner of the cube.

[0228] FIG. UN depicts three ribbons overlapping each other at one of the planar triangular faces with a three-flap interlock closure to form a corner of the cube in FIG. 11O.

[0229] FIG. 11O depicts a structure formed by six interwoven ribbons placed according to the tetradihedron polygon mesh, with ribbon ends placed in flap-interlock closures.

[0230] FIG. IIP depicts the 3D woven ribbon topology for the tetradihedron polygon mesh.

[0231] FIG. 11Q depicts the design for a ribbon with a side wing connected to each ribbon end.

[0232] FIG. HR depicts a module formed with two ribbons with side wings and four ribbons with no side wings.

[0233] FIG. 11S depicts two modules with side wings rotated and aligned for connection.

[0234] FIG. 11T depicts two modules connected by means of side wings.

[0235] FIG. 11U depicts a module formed with four ribbons with side wings and two ribbons with no side wings.

[0236] FIG. 11V depicts four modules like the one in FIG. 11U connected to each other by means of side wings.

[0237] FIG. 11W depicts a module formed with six ribbons with side wings.

[0238] FIG. 12A depicts a tetradihedron polygon mesh for a module to cover an icosahedron.

[0239] FIG. 12B depicts six ribbons designed from three ribbons based on the tetradihedron polygon mesh in FIG. 12A.

[0240] FIG. 12C depicts a module formed from the six ribbons in FIG. 12B.

[0241] FIG. 12D depicts the module in FIG. 12C placed at one edge of the icosahedron in FIG. 12A.

[0242] FIG. 12E depicts modules at several adjacent edges of the icosahedron.

[0243] FIG. 12F depicts modules at more adjacent edges of the icosahedron.

[0244] FIG. 12G depicts 30 modules, one for each edge of the icosahedron, forming a structure in the shape of a spherical shell.DETAILED DESCRIPTION OF THE INVENTION

[0245] The present invention relates to structures, space frames, building trusses, lattice materials and articles and the like that can have a great variety of shapes, sizes and uses. In accordance with embodiments, the present invention provides for a method of making a structure from a polygon mesh with a target shape. One distinguishing feature of the method is that any article whose shape can be represented by a polygon mesh can be fabricated into a structure. This includes structures of a great many shapes and sizes including free form structures, space frames, articles with internal compartments or bulkheads and lattice materials. One of the distinguishing features from previousart is that no nodes connect struts. Instead, the structure comprises interwoven ribbons. This reduces many problems including weight, cost, difficulty of assembly and difficulty of customization for free form shapes. Another distinguishing feature is that structures of interwoven ribbons can be held together by mechanical interference without the need for physical connections such as by adhesives, bonding, welding, riveting, etc. Another distinguishing feature is that the ribbons can be fabricated from flat sheets of material. This can simplify manufacture, transportation and assembly. Material in sheet form is also typically less expensive by weight than tubes or beams of the same material. Another distinguishing feature is that one or more ribbons can be coupled to each other in contiguous sequences that define a structure with a 3D woven pattern in a process referred to as 3D weaving.

[0246] The structure can have a myriad of different shapes and uses. The structure can have a cladding layer including polygonal plates to cover any holes and can have barrier as well as structural capabilities. The range of purposes include but are not limited to space-frames, lattice material, geodesic domes, decking for floors in buildings, spans / decking of bridges, towers, free form parametric architecture, high rise buildings, warehouses, mobile homes, stadiums, public buildings, airports, pavilions, earthquake-resistant constructions, cladding systems for blast / explosion protection, vehicle frames, light-weight panels for cars, trucks, buses, and trains, airplane fuselages and wings, drones, rocket bodies, frames for rocket sections, space craft, hulls for a ship, yacht, and submarine, frame structures for a machine, robots or lifts, hyperloop tube walls, space telescope supports, lightweight pressure vessels and tanks, spaceship construction systems, solar sail supports, tunnel cladding, parts of helmets / personal protective gear, parts of sports equipment, structures for a vacuum lift aircraft, batteries, heat exchangers with fluid flowingthrough the structure, prosthetic implants, clothing, bullet proof vests, shoes, toys, puzzles, jewelry, furniture, models, art, installations and sculpture.Method

[0247] In accordance with an embodiment, what follows in this section are steps for a method to design and fabricate ribbons and to interweave or assemble the ribbons into a structure with a shape represented by a polygon mesh. There are many terms used herein that are defined at the end of the Detailed Description. In short, the method to form a structure is:• Identify a target shape of a structure.• Represent the shape with a polygon mesh.• Subdivide the faces of the polygons into wings.• Connect the wings in a prescribed way to form ribbons.• Interweave the ribbons to form the structure.In greater detail, the method is:(a) Identify a target shape for the structure to be constructed. The shape can for example follow the surface of an object. It can also include the internal structure of an object or represent a lattice. It can also have external structure such as fins.(b) Represent the shape of the object by means of a polygon mesh. A polygon mesh is an assembly of polygons and can also be referred to as a polyhedron. It is used to define or approximate the target shape of a structure. A polygon mesh has (1) vertices or points in space, (2) edges that each connect two vertices and (3) polygons having adjacent edges. Polygons can be adjacent to other polygons by means of shared edges. Two or more polygons can share an edge and two or more edges can be shared by the same polygons. This means that three polygons can share one edge and two polygons can share the sametwo or more edges. For example, the polygon mesh can include degenerate polygons such as a dihedron, made of two polygons which share the same set of edges, or the polygons can be cycles in a lattice. A cycle is a sequence of distinct edges which joins a sequence of vertices, with the first and last vertices being the same. Polygons of the polygon mesh can be open or closed, planar or non-planar. Several methods can be used to generate polygon meshes which are explained below. There are different types of polygon meshes: surface or 2D meshes and 3D meshes: a. A Surface or 2D mesh is a collection of adjacent polygons such as triangles, quadrilaterals, pentagons, hexagons and / or other polygons and is used to model objects in 2D space or to model a surface in 3D space. In a 2D polygon mesh, an edge can be shared by one or two polygons, but not by three or more polygons. A surface mesh in 3D space has the property that it can be flattened into a 2D surface. Surface meshes are used in 3D computer graphics and modeling. Software tools from these fields can be applied to generate polygon meshes. A surface mesh is used for the target shape when the shape is a surface. The surface can be planar, non-planar, curved, open, closed, convex and / or concave. An example of a surface mesh is shown in FIG. 2A. The polygon mesh includes three polygons and twelve vertices. The edges are shared by one or two polygons, but not by three. b. A 3D polygon mesh is a collection of adjacent polygons that can include a polygonal representation of interior or external structure of an object or lattice as well as its surface. A 3D mesh can have an edge shared by any number including one, two, three or more polygons. Two or more edges can be shared by the same two or more polygons. Some of the polygons can be arranged into multipleconnected surfaces, including fins. Some of the polygons can be arranged into polyhedrons such as tetrahedra, pyramids, prisms, pentahedrons hexahedra, cubes, and / or higher polyhedrons. The polyhedrons can be connected to each other by means of shared edges, and / or faces. Three-dimensional meshes are used in finite element analysis and finite volume methods. 3D modeling software can be applied to generate 3D meshes that represent a target shape or lattice. A 3D mesh can be used when the target shape has internal structure. An example of a 3D mesh is shown in FIG. 1A, which has an edge shared by three polygons. A 3D mesh can also be obtained by augmenting a surface mesh with polyhedrons or building it up from a surface mesh. For example, a triangle surface mesh can be augmented by tetrahedrons, each tetrahedron sharing one of its faces with a triangle face of the surface mesh. An example of such a 3D mesh obtained by augmenting an inner tetrahedron is shown in FIG. 3A. In addition to augmenting tetrahedrons, further tetrahedrons or other polyhedrons can fill in the spaces between the augmenting tetrahedrons to form a layer of polyhedrons in a 3D mesh that follows the original surface mesh. A 3D mesh can be obtained by filling in the volume between surfaces with polyhedrons. A volume of a structure can be subdivided into tetrahedrons or a variety of polyhedrons. Also, additional vertices can be introduced in the interior volume of a 3D polygon mesh to further subdivide into polyhedrons. An example of this process is called tetrahedralization. Unit cells can be generated and repeated to generate a polygon mesh for a space frame or lattice material.(c) Generate a polygon mesh by defining a list of vertices in 3D space, edges and polygons that interconnect the vertices and so make up the mesh. To generate the mesh, define thex, y and z coordinates of each vertex identified in a notation as v(i) with coordinates (xi, yi, zi). Here i is a vertex index which is an integer between 1 and the total number of vertices of the polygon mesh. Each vertex index identifies each vertex by its position in the list of vertices. The vertices can be key points in the shape so that as a set they are a good approximation of the shape. For example, the vertices could include corners for planar shapes, places in a curved shape where changes in curvature occur or at edges of surfaces. Also, the polygon mesh can be coarsened or refined by decreasing or increasing the number of vertices and polygons, respectively. This can be done to change for example the mechanical or other properties of the material. If desired, select the vertices so that they are approximately evenly spaced or more closely spaced if there is much curvature or change in the shape. If desired, vertices can be placed so that the polygons have angles between adjacent edges not too different from each other, are concave or convex, planar or non-planar. The notation for a polygon is a list of adjacent vertices connected by edges. Each polygon has a surface bounded by the edges of the polygon. The surface of a polygon can be non-planar, curved or have holes. Each polygon has two faces, one for each side of the polygon. Therefore, each face has an opposite face. Each face has the same surface as the surface of the polygon with which it is associated. Each face of a polygon is defined by the same edges and vertices as those that define the polygon. Each face has directed edges and each face has the opposite directed edges of the opposite face. Each face has an outward direction. The outward direction is normal or approximately normal to the surface of the face. The outward directions of opposite faces of the same polygon point in opposite directions. Each pair of faces that share an edge and that have outward directions pointing into the same volume has a dihedral angle, which is the angle from one face around theshared edge to the other face in the volume that the outward directions point. The sum of the dihedral angles at any edge of the polygon mesh is 360 degrees. The set of faces is identified as f(j) where j is the face index which is an integer between 1 and the total number of faces. Each face is represented by an ordered list of adjacent vertex indices corresponding to the vertices going around the face in a counter-clockwise or clockwise direction when facing that face, which means facing in the opposite direction to the outward direction of the face. The right-hand thumb rule or left-hand thumb rule is used as convention for all the faces and corresponds to counter-clockwise or clockwise vertex ordering, respectively. By the right-hand thumb rule, point the right-hand thumb in the direction of the outward direction of a face, then curl the fingers to give the direction of vertices around the face. The direction around the face determines the direction and ordering of the list of vertices describing the face. The opposite face to a face of the same polygon has an opposite outward direction and is described by the list of the same vertices as the face but in reverse order. Let f(j 1) denote a face of face j 1 with N vertices. A face f(j 1) of facej l is represented by the ordered, directed list of vertex indices, {il, i2, i3, ..., iN} . Let j 1 * denote the face index for the opposite face. The notation for a directed, ordered vertex list such as for a face or a directed edge is a list of indices within braces (curly brackets). The face and its opposite face are described by, f(j 1) = {il, i2, ..., iN} and f(j 1 *) = {iN, iN-1, ..., il } where N is the number of vertices in polygon j 1, and iN is the index of the Nth vertex in the polygon vertex list. Each face contains directed edges connecting adjacent vertices in its vertex list. If a polygon is closed then there is also a directed edge from the last vertexwith index iN to the first vertex with index, il . A polygon can also be open and then il and iN indicate the vertices at the ends of the polygon.An example of a polygon mesh 100 depicting faces f(j), vertices v(i) and outward directions for each face is shown in FIG. 1A. The polygon mesh has ten vertices and five polygons. Polygon mesh 100 is a 3D mesh and includes 10 edges shared by only one polygon, two edges shared by two polygons and two edges shared by three polygons. The two edges shared by three polygons are (3,4) and (4,5). The notation uses round brackets for undirected lists such as for edges and polygons. FIG. 1C shows the opposite faces f(j*) for the polygon mesh in FIG. 1A. Each face has directed edges that point from one vertex in the face to the adj acent vertex in the ordered, directed vertex list of the face. For example, in FIG. 1A, face f(3) = {3, 4, 8, 7} and has directed edges, {3,4}, {4,8}, {8,7} and {7,3}. Each directed edge of each face points in the opposite direction to the directed edge of the opposite face. For example, in FIG. 1C, opposite face f(3*) = {7, 8, 4, 3} has directed edges {4,3}{8,4}, {7,8} and {3,7}. Each pair of adjacent faces has a dihedral angle between their surfaces. The dihedral angle 104 between faces f(3) and f(4) at edge (8,4) is 90 degrees. The dihedral angle between the surfaces of faces f(3) and f(3 *) at edge (7,8) is 360 degrees. Another example of a polygon mesh depicting faces, vertices and outward directions is shown in FIG. 2C for the faces of the polygon mesh in FIG. 2A and FIG. 2E for the opposite faces of the polygon mesh in FIG. 2B.Mesh generation means determining the coordinates for each vertex and identifying the connected vertices that make up the polygons. This can be done by using a great variety of methods depending on the type of shape, type of mesh, type of polygons, application, etc.These methods can include but are not limited to:• By hand for simple shapes, such as a cube.• By generating a lattice with finite element analysis software or modeling software.• By using a vertex generating equation or mathematical function for lattices or shapes with vertices at regular or predictable intervals in 3D space. For example, the lattice vertices can be described by the maxima of a periodic function in 3D space. Then the edges connect adjacent points together with the faces represented by a certain number or numbers of vertices joined by edges in a closed circuit or cycle. The number of vertices joined depends on the geometry of the lattice. For example, for a tetrahedral lattice the number is 3 because it has triangular cycles. For a gyroid lattice, each cycle for a polygon has 10 vertices.• By using an equation or mathematical function that describes the surface of a shape such as for spheres and other shapes. The surface defined by the equation can then be triangulated into a triangle mesh or filled with tetrahedrons or other polyhedrons using 3D graphics software.• By using a 3D scanner for complex shapes. A 3D scanner typically generates a vertex cloud that can be converted to a polygon mesh that has a great number of polygons. The polygon mesh can be simplified using software tools.• By creating a mesh using 3D graphics software to construct, model or sculpt a shape, including a lattice, a free form shape or one with internal structure. Polygon meshes can also be modified by subdividing, simplifying, morphing, shearing, extruding, augmenting, stellateing, etc.(d) Determine shapes and dimensions of wings based on the polygon mesh. A wing is associated with each edge of each face of each polygon in the polygon mesh. Each wingincludes part of the surface of the face and each wing has a directed edge of the face as an edge. Each wing has an outward direction that is the same as the outward direction of the face with which it is associated. The shape of each wing is such that each wing shares part of its surface with a wing on the opposite face, wherein the wing on the opposite face is chosen such that the directed edges of the two wings point towards the same vertex. For example, FIG. IB shows a subdivision of the faces of the polygon mesh 100 in FIG. 1A into wings such as 10, 14, 18, 20 and 26. The subdivision into wings need not include the entire surface of the face and can be designed so that there are holes in the final structure, as is the case for FIG. IB. There is one wing for each directed edge of each face. Wing 10 is associated with directed edge {4,5} of face f(4), where f(4) is the face given by the ordered, directed list {4, 5, 9, 8}. FIG. ID shows a subdivision of the opposite faces of the polygon mesh 100 in FIG. IB into wings. Wing 12 in FIG. ID is associated with directed edge {9,5} of face f(4*) in FIG. 1C, where f(4*) is the face given by the ordered, directed list {8, 9, 5, 4}. Part of wing 12 coincides with (or overlaps) part of the wing 10 as shown in FIG. IE. The part that overlaps is shared surface 32 and is called a connecting tab. Both wings 10 and 12 point to the same vertex, v(5). Wings 10 and 12 are called a connecting pair of wings and share a connecting tab surface.Each wing shares an edge with a matching pair wing, wherein the outward directions of the wing and the matching pair wing point into the same volume. For example, wing 10 has the same outward direction of face f(4) and shares an edge (4,5) 42 with wing 16 of face f(5*) in FIG. ID. Wing 16 has the outward direction of face f(5*), as shown by the straight arrows emanating from the wing surface. The outward directions of both wings point into the same volume, which is the volume to the left of face f(5*) and in front offace f(4). Wings 10 and 16 are called a matching pair. Each matching pair of wings has a dihedral angle, which is the angle between the faces sharing the edge. Each matching pair has opposite directed edges.Each wing shares part of its surface with another wing that is associated with the same edge of the opposite face. This means these wings have opposite directed edges. The shared surface is called a strut tab and the two wings are called a wing pair. An example of a wing pair is wing 14 in FIG. IB, associated with edge {4,5} of face f(5), and wing 16 in FIG. ID, associated with the same edge {5,4} of opposite face f(5*). Another example of a wing pair is wing 12 in FIG. ID, associated with directed edge {9,5} of face f(4*), and wing 20 in FIG. IB, associated with the same edge (5,9) and opposite directed edge {5,9} of face f(4). Part of wing 12 shares a surface with (coincides with) part of wing 20. The shared surface 52 is called a strut tab as shown in FIG. IE. Since edge (5,9) is shared by only 1 polygon, the wing pair including wings 12 and 20 is also a matching pair. The outward directions of wings 12 and 20 point in opposite directions, but since there is no other polygon to break up the volume between them, they still point into the same volume.(e) Determine shapes and dimensions of one or more ribbons. The one or more ribbons have the required shape and dimensions to interweave and form the structure in the shape of the polygon mesh representing the target shape. Each ribbon is based on a contiguous surface comprising a sequence of a plurality of wings in matching pairs and connecting pairs. The sequence also known as a weaver sequence is a sequence of alternating matching pairs and connecting pairs of wings. Whereas a ribbon is a physical article with volume that is a component of the structure being assembled, a face, a wing or a weaver is a surface in 3D space.An example of a contiguous surface 60 of an alternating sequence of matching pairs and connecting pairs of wings is shown in FIG. IE. The sequence has 5 connecting pairs and 4 matching pairs sharing connecting tab surfaces. The sequence begins with wings 18 and 16 on one end. End wing 18 is in a connecting pair with wing 16 with which it shares connecting tab surface 34. Wing 16 is in a matching pair with wing 10 at shared edge (4,5) 42. The dihedral angle 44 at edge 42 between wings 10 and 16 is 90°. Wing 10 is in a connecting pair with wing 12 with which it shares connecting tab surface 32. Wing 12 is in a matching pair at edge (5,9) with wing 20. The alternating sequence continues with matching pairs of wings at edge (8,9) and edge (4,8) and ends with connecting pair of wings 24 and 26 that share connecting tab surface 38. The dihedral angle between the wings in the matching pair at edge 43 indicated by (4,8) shown in FIG. IE is 270 degrees because of the orientation of the outward normal of the wings at (4,8) as shown by the arrows pointing from the wings. Locally, the contiguous surface has the outward direction of its constituent wings and therefore has not one but a set of outward directions, each outward direction associated with a wing. Therefore, the weaver surface is a multi -oriented surface. At each connecting tab surface, the outward directions of the contiguous surface point in opposite directions because wings with opposite outward directions associated with opposite faces share the connecting tab surface. At each strut tab surface, the outward direction of the contiguous surface points in one direction, the outward direction of the wing.A contiguous surface that includes a sequence of matching pairs and connecting pairs of wings is also known as a weaver. A weaver can be a closed-loop or be open. In the example for the polygon mesh in FIG. 1A, there are two closed-loop weavers. One or more ribbonsare designed from each weaver. Each ribbon is part designed from the whole or part of the weaver sequence of wings. The sequences for the ribbons can be chosen so that they overlap at their ends with the ends of the same or other sequences. Each end of each sequence includes one or more wings that share a connecting tab surface or strut tab surface with one or more wings of a second end, wherein the second end is an end of the same or another sequence. FIG. IF shows a second sequence 62 of 4 matching pairs. The sequence begins with wings 24 and 26 that share connecting tab 38. The sequences in FIG. IE and FIG. IF share wings 24 and 26 at their ends and form part of a longer sequence as shown in FIG. 1G FIG. 1H is another view showing the two contiguous surfaces sharing a surface at their ends. A third sequence of contiguous surfaces 64 of 4 matching pairs at edges (4,1), (1,2), (2,3) and (3,4) shares a surface with the end of contiguous surface 62 in FIG. II. The end of this third sequence shares a surface with the first end of first sequence 60 at wings 16 and 18 as shown in FIG. 11. The three sequences of contiguous surfaces of 4 matching pairs each are a closed loop weaver. The remaining wings in the polygon mesh not included in the first weaver form parts of two other overlapping sequences of four matching pairs each that form a second closed loop weaver as shown in FIG. 1 J.Each ribbon is a physical component that has a shape and dimensions based on a design based on all or part of a contiguous surface of a sequence of wings. Each ribbon comprises part of the surface of the contiguous surface and has a thickness. When the structure is assembled, each ribbon is placed at its associated contiguous surface location in the polygon mesh such that its thickness extends from the surface outwards in the outward direction of its constituent wings. At the connecting tab surfaces, the thickness of the ribbon can extend from the surface in both directions.Each sequence of contiguous surfaces is flattened by bending at the bending axis of each matching pair so that the matching pair lies flat. When the contiguous surface is flattened, the outward direction at the strut tab surface in a matching pair is opposite to the outward direction of the strut tab surface of the previous matching pair in the sequence. The design is adjusted to account for the thickness and placement of the ribbon as well as any bending allowance needed when forming the ribbon by bending. A design for each ribbon includes the shape of all or part of the flattened contiguous surface. The design includes a bending axis at each shared edge of each matching pair as well as its dihedral angle. Designs for three ribbons based on the first weaver and two ribbons based on the second weaver are shown in FIG. IK. Design 170 is based on flattened sequence 60 in FIG. IE, and has wings such as end wing 71 and next wing 72 in a connecting pair which includes a connecting tab and two strut tabs with wing pair indices 9 and B12, respectively. Design 172 is based on flattened sequence 62 in FIG. IF and has an end wing 73 and next wing 74 in a connecting pair which includes a connecting tab and two strut tabs also with indices B12 and 9. Each ribbon end has an end wing in a connecting pair with the next wing. The sequences used for designs 170 and 172 overlap at ends 71 and 73. When the structure is assembled, the ribbon ends with wings 73 and 71 are placed so that the wings in the wing pairs indicated as B12 and 9 are one on top of the other with the curved boundary aligned. Design 174 is based on sequence 64 and has end wing 75 and connecting pair wing 76 overlapped by wing end wing 70 and connecting pair wing 77 of ribbon 170 to complete a first weaver loop. The bending axis at edge 42 from FIG. IE is shown in the first ribbon design 170, and a dihedral angle 177 of 90 degrees is indicated next to edge 42. A dihedral angle of 270 degrees is indicated next to edge 43. This matches the dihedral angle of facesf(3*) and f(4*) from which the wings sharing edge 43 in FIG. IE are subdivided. The design includes wing pair indices 175 for placement of each strut tab surface of each ribbon against the strut tab surface of its wing pair wing. A code identifies the outward direction of the strut tab surfaces. Each ribbon includes a sequence of wings, each of which has two opposite sides, referred to as an “a-side” and a “b-side.” The outward direction of each wing points outwards from the a-side. The a-side is generally visible when the structure is assembled. The b-side is the side that has a surface that is shared with or is in the neighborhood of another b-side of the wing in the same wing pair. At strut tabs, ribbons have an a-side opposite to the b-side. At connecting tabs, two wings share a surface and have a-sides on both sides of the ribbon. In FIG. IK, a ‘B’ before a wing pair index indicates that the wing is shown on its b-side in the design. If there is no ‘B’ then the wing is shown on the a-side.(f) Fabricate the ribbons from the design. Select a material and select the thickness. The thickness, that can vary over the extent of the ribbon. With the shape and dimensions of each ribbon defined, fabricate each ribbon according to the design for its corresponding sequence. The fabricating can be done by a great variety of means including but not limited to by cutting a sheet of material with a mechanical cutter, laser, plasma or waterjet cutter, by additive manufacturing, by molding, by casting, and by forging. Identify abending axis at the shared edge of each matching pair of wings of the ribbon. Bend the ribbon along its bending axes by an angle equal to the dihedral angle between the faces from which the wings of that matching pairs originate. There are many methods to fabricate the ribbons with bends at the bending axes. For example, this can be accomplished by scoring or perforating at bending axes with a laser cutter or other cutting means, and then bending theribbon around the bending axis according to the dihedral angle of the matching pair of wings associated with the edge. A bend allowance can be considered in the design. If the material cannot bend without breaking, then other means can be employed. For example, by cutting the material at the bending axis to separate the two wings. Then the wings are reconnected with a flexible strip of material or a hinge. Alternatively, in embodiments, the bent ribbon is formed by some of many other means such as by casting, using a mold or form or by 3D printing or other additive manufacturing methods, whichever is suitable for the intended component. Identify or mark strut tabs and / or connecting tabs in each ribbon with indices that allow wings to be matched up and placed correctly in wing pairs and connecting pairs. This allows wing pairs and connecting pairs to be placed according to the polygon mesh. Optionally cut or form notches at the ends of the bending axes. FIG. IK shows notches 179.(g) Form the structure by placing and interweaving the bent ribbons, wherein each particular wing at each end of each particular ribbon shares a connecting tab surface with a second wing at an end of a second ribbon, wherein the second wing is in a connecting pair with the particular wing and the second ribbon is the particular ribbon or another ribbon. In embodiments, the ends of ribbons include a strut tab surface and a connecting tab surface and ribbons overlap by sharing strut tab surfaces and connecting tab surfaces. In embodiments, the wings at the end of a ribbon are in a connecting pair.For each edge in the polygon mesh, match up wing pairs by using the indices and align the surfaces of the wing pairs in ribbons that share an edge so that their bending axes are coincident. In embodiments there are notches at each end of the bending axes. The strut tab surfaces of a ribbon are part of the ‘b’ sides of the ribbon and are in the neighborhoodof or flush against the ‘b’ side of a ribbon with a wing from the opposite face. The two wings in such a ‘b’ side to ‘b’ side arrangement constitutes a wing pair. The method can further include connecting or joining wings at their strut tab surfaces into winged struts. The winged struts then interconnect the ribbons, wherein there is a winged strut for each edge in the polygon mesh.For example, ribbon 180 in FIG. IL is fabricated and bent from design 170 for sequence 60 in FIG. IE. A fold or bend is imposed at each bending axis corresponding to edges (4,5), (5,9), (9,8) and (8,4) according to the dihedral angles listed in design 170 for each edge. For the edges shared by only one polygon, the dihedral angle is 360°. A second ribbon 182 fabricated from design 172, is bent and placed, as shown in FIG. IM. The second ribbon is placed with its first end 73 over the last end 71 of the first ribbon 180 so that their boundaries align. End wing 71 overlaps twin wing 74 and end wing 73 overlaps twin wing 72, as shown in FIG. IM. In FIG. IN, a third ribbon 184 fabricated from design 174, is bent and placed according to the polygon mesh for sequence 64 in FIG. 1G. The third ribbon is placed with its first end over the last end of the second ribbon 182 so that the ends are obscured by other parts of a ribbon in the final structure. Further, the last end of the third ribbon is placed under the first end of the first ribbon 180, thus completing the first loop. The placement of the ribbons for the second loop using ribbons designed from 176 and 178 in FIG. IK is shown in FIG. IO. When the structure is assembled, the ribbons are placed such that parts of the strut tab surfaces of wings in each wing pair, such as identified by the wing pair indices 175 in the design of FIG. IK, are shared or are close to each other. The ‘B’ code in the designs for some of the wing pair indices identifies whetherthe surface showing face up on the design is a ‘b’ side. The order of placing ribbons or parts of ribbons can be changed or optimized for ease of assembly.The result of placing the ribbons according to the polygon mesh is a 3D woven structure of interwoven ribbons 190 as depicted in different views in FIG. IP and FIG. IQ. A feature of the assembled structure is that each end wing is sandwiched in a pocket between its twin wing and its wing pair wing. This creates a layered arrangement of three wings that holds the ribbons in place by mechanical interference. Wing 70 is an end wing of ribbon 180 based on design 170 and is in a connecting pair with wing 77. Wing 70 is placed in a pocket between wing 76 and its wing pair wing 78, with wing pair index “17” as shown in the design for ribbon 172 in FIG. IK. Wing 70 and wing 76 are twin wings. Wing 76 is in a connecting pair with wing 75 which is an end wing of ribbon 184 based on design 174. End wing 75 is placed in a pocket between wing 77 and its wing pair wing 79, with wing pair index “B20”. Wing 77 and wing 75 are twin wings. A feature of the resulting structure is two adjacent layered arrangements of three wings. A middle wing is in a pocket between two outer wings that are in a wing pair and the middle wing is a twin wing of one of the outer wings of the layered arrangement. The adjacent layered arrangements of three wings together with the bends in the ribbons at their bending axes lock the ribbons in place by means of mechanical interference. An advantage is that the structure is held in place without the need for bonding, adhesives, welding, etc. As can be seen, no ribbon ends are directly visible as these are obscured by other parts of ribbons passing over them in the woven pattern. This aesthetic advantage requires the ribbon ends to be connecting pairs of wings. If the wings of ribbon ends were a matching pair, the ends would be visible.(h) A ribbon topology for the 3D woven structure is defined by the weavers, their knot topology, links, the twist and writhe in their ribbon topology and the associations between wings of the same or other weavers at edges of the polygon mesh. A depiction of a ribbon topology 192 for structure 184 is shown in FIG. 1R. It shows representations for weavers 194 and 196, one for each loop. As an example of an association, association 198 is shown by the dashed circle around a bundle of weaver parts for the shared strut tab surfaces around edge (3,4). The effect of an association at a shared strut tab surface is that a twist in the ribbon cannot slip past it. The associations further define the ribbon topology of the structure by locking ribbon topology between two associations.(i) In embodiments, there are many ways the shapes of wings can be obtained. Below is but one method to generate designs for wings. a. Determine the shape, dimensions and angles between edges of each face of each polygon in the polygon mesh from the location of the vertices and faces. Determine the dihedral angles between faces. b. For example, polygon mesh 200 in FIG. 2 A has three adjacent polygons and represents a desired target shape as seen from above. It has faces f(l), f(2) and f(3) with outward directions as shown in FIG. 2C. A view of the polygon mesh 200 from below is shown in FIG. 2B, showing the opposite faces, f(l *), f(2*) and f(3*). The corresponding outward directions are shown in FIG. 2E. Each face is represented by the counterclockwise directed, ordered list of vertex indices. c. For the polygon mesh 200 shown in FIG. 2D, the indices are as follows, il=10, i2=7, i3= 11, j l=2, j2=3 andj3=l. This means that f(j l) is represented by ordered list {i 1, i2, i3, i4, i5 } . This generalized notation can then be used to refer to anypolygon with the number of indices adjusted for the number of vertices in the polygon. Set a new point partway between vertex il and second vertex, i2, on edge (il, i2). Set another new point partway from the second to the third vertex, i3, and repeat for all directed edges in all faces. Also determine a point partway from the last vertex i5 to the first i 1. d. Construct a line from the vertex il to a new point partway from the second to the third vertices, i2 and i3. Construct a line from vertex i2 to the new point partway from the third to the fourth vertex. Repeat for all the vertices in f(j 1) including a line from the last vertex to the new point that is partway from the first to the second vertex. Repeat for all faces including opposite faces. This results in a line constructed from each vertex in each face to the new point partway along the next edge around the face, as shown in FIG. 2D. e. Determine the points where the line constructed above from the first vertex il intersects with the line constructed from the second vertex i2. This intersection point is represented by k(j 1, il) to identify it as associated with face j 1 and vertex i 1. Repeat for all the lines in face f(j 1 ) to define new vertices associated with indices i2 through i5. Examples of intersection points 208 are shown in FIG. 2D with wings constructed for each directed edge of each face f(j 1), f(j2) and f(j3) of the polygon mesh 200. For the opposite faces, f(j 1 *), f(j2*) and f(j3*), the wings point in the opposite directions as shown in FIG. 2F for polygon mesh 200 with directed edges as shown in FIG. 2E. FIG. 2G shows of all the wings in the polygon mesh which is an overlay of wings in FIG. 2D and FIG. 2F.f. Define a triangle-shaped wing with vertices i 1 and i2 and the intersection point k(j 1 , il) as shown in FIG. 2D. This wing has ordered vertex list {il, i2, k(j l, il} in the counterclockwise direction. The first two vertex indices, il and i2 in the list define a directed edge in the direction from v(il) to v(i2). This directed edge defines the direction for the triangular wing with ordered list { li, 12, k(j 1, il)}. The outward direction of the wing is the same as the outward direction of the face. Repeat for all vertices in f(j 1). Then repeat for each face including on opposite sides of each polygon. As a result of the method of construction there is a wing for each directed edge between adjacent vertices in each face. Let the wing constructed from vertex il in face f(j l) be identified as w(j 1, il), given by the ordered list {il, i2, k(j 1, il)}. For each edge of each polygon there are two wings, one associated for each face of each polygon. The two wings have the same edge but opposite directed edges. Wings associated with the same edge but opposite faces of a polygon are called a wing pair. g. Match up a first wing from f(j 1) with the adjacent wing from the adjacent face f(j2) such that the outward directions of the wings point into the same volume as shown in FIG. 2L. In this example the same volume is the volume above the plane of the polygon mesh. The first wing and the adjacent wing are joined at a shared edge and are a matching pair of wings. Pointing into the same volume means that there is no other polygon between the two wings in the volume into which the outward directions of the wings point. The first wing having ordered vertex list {il, i2, k(j 1, il)} is matched up with the wing that shares the same edge and vertices il and i2 and has as third vertex an intersection point k(j2, i2) from the construction ofintersecting lines of face f(j 2). In addition to its outward direction, wing w(j 1, il) has a wing direction as determined by its directed edge {il, i2}. Referring to FIG. 2L, the matching pair wing is part of face f(j 2) and has a vertex list with the vertices in the ordered list {i2, il, k(j2, i2)} in the counterclockwise direction as determined by the right-hand thumb rule. Note that the vertices with indices il and i2 are reversed in this list. Here k(j2, i2) denotes the intersection point in the adjacent face, f(j2)) associated with the directed edge from vertex, with index i2. The position of the vertex point associated with index k(j 2, i2) is the intersection of the line from v(i2) to a point on the edge from v(i 1) to the next vertex in the counterclockwise direction of face f(j2) and the line from v(i 1) to a point in the adjacent edge when going in the counterclockwise direction from the first edge. The two wings in a matching pair share the same edge and have parallel but opposite wing directions, as defined by their directed edges. The combined surface of the matching pair defines part of contiguous surface of the ribbon. Each ribbon has a thickness. When placed in the structure each ribbon has a volume that extends from this surface in the direction of the outward direction of the wings. At the connecting tab the volume can extend in the outward direction of either or both wings in the connecting pair. The ordered vertex list associated with the matching pair of wings is {il, k(j2, i2), i2, k(j 1, il)}. In this way each wing is matched up to define matching pairs of wings. If the wings are triangular, the matching pairs have a quadrangular shape and span diagonally across the edge shared by the wings. In this step, with adjacent polygons at an edge, matched up pairs of wings are from the faces on the same side of the adjacent polygon. If there is no such adjacentpolygon as for example on the boundary of a surface mesh, then wings from the two opposite faces of the same polygon are a matching pair. If there are more than two polygons sharing an edge as for example where two polyhedrons share an edge in a 3D mesh or where three or more surfaces share an edge, then the wing is a matching pair with the wing of the face that faces the first face such that the outward directions are pointing into the same volume. In this way, each edge has associated with it the same number of matching pairs as there are polygons that share that edge in the polygon mesh. If there is only one polygon at an edge then the volume spanned by the faces of the matching wings is everywhere around that polygon. h. A second matching pair at edge (il, i2) is shown in FIG. 2M with wing outward directions pointing into the volume below the mesh. i. Match up wing w(j l,il) from f(j 1) with w(j l*,i2) from the opposite face f(j I * ) that is associated with the same edge as the first wing as shown in FIG. 2J. The two wings share a strut tab surface 230 and are a wing pair. The outward directions of the wings in a wing pair point in opposite directions. The directed edges {i l,i2) and {i2,i 1 } of the wings point in opposite directions as shown in FIG. 2K. j . Match up wing w(j 1 ,i 1 ) from f(j 1 ) with wing w(j 1 *,i3) from the opposite face f(j 1 *) that is associated with the edge adjacent to the edge associated with the first wing as shown in FIG. 2H. The two wings share a connecting tab surface 220 and are a connecting pair. The outward directions of the wings in a connecting pair point in opposite directions. The directed edges {il,i2} and {i3,i2} of the wings point towards a shared vertex i2 as shown in FIG. 21.k. A set of one or more matching pairs associated with the same edge share strut tab surfaces where the wing pairs of the set overlap. For multiple wing pairs, the order that the wings go around the strut is the same order in which they appear in the polygon mesh at that edge. For example, two matching pairs 240 and 242 share edge 250 as shown in FIG. 2N. The wings overlap at strut tabs 230 and 232. l. The connecting pair wings associated with adjacent edges overlap at their connecting tab surfaces. Each wing overlaps the wing from the next edge in the same polygon but opposite face. The overlap is at the connecting tab near each vertex in the polygon mesh. This connecting tab corresponds to the portion of each wing shared with the wing from the next edge and opposite face. For example, matching pairs 242 and 246 associated with adjacent edges 250 and 252 are shown in FIG. 20. The matching pairs 242 and 246 share connecting tab surface 220. As shown in FIG. 2P, matching pair 248 associated with next adjacent edge 254 overlaps a wing of matching pair 246 and shares connecting tab surface 222. The contiguous surface of the sequence of matching pairs 242, 246 and 248 forms a surface which determines part of the shape of a ribbon. Each surface of the sequence of matching pairs has outward directions matching the outward directions of the wings in the surface. m. Each of the wings shares a connecting tab with its connecting pair wing to define contiguous surfaces comprising sequences of matching pairs and connecting pairs of wings. The wings of the ribbon are placed by matching them up with their wing pairs. The ribbons are bent according to the dihedral angles of the matching pairs. Bending and placing the ribbons results in the woven structure 280 shown in FIG.2Q. The ribbon topology 290 is depicted in FIG. 2R for the woven structure 280. The topology includes a single ribbon in a trefoil knot. There are 11 twists in the ribbon, one for each of the 11 edges shared by a single polygon. There are also three crossings that correspond to associations of wing pairs at edges shared by two polygons.(j) Instead of using a counterclockwise convention with a right-hand thumb rule for the ordered vertex lists, a clockwise convention with a left-hand thumb rule can be used. Changing this convention also changes the chirality of the topology.(k) The method of subdivision used above to define the shapes of the wings is one of a great variety of methods that can be used. The wings can have a great variety of shapes, wherein wings overlap or share a boundary with the wing from the next edge but opposite face of the polygon to define a shared connecting tab. The wings can be constructed in many ways whereby they overlap the wing of the next edge of the opposite face. By changing the point of intersection in the above method, the wings can individually be made wider or narrower. This still results in triangles for wings. The wings can also be defined as quadrilaterals with a second subdivision point and vertex index list {i 1 , i2, k(j 1, il), k(j 1 ,iN(j 1))} . Any shape for the wing can be used that results in each wing overlapping or sharing a boundary with the wing that is of the opposite face of the same polygon and that points at the same shared vertex. The wings can have curved boundaries or be polygonal with straight edges whereby they overlap partially with the next wing on the other side so that part of the b-side of each wing is not shared by another wing and so is visible. In embodiments, the wings have different shapes, the wings have varying widths or thickness, or the wings comprise different materials.(l) In embodiments, the matching pairs of wings are strips that have the same width, as seen in FIG. 2S resulting in woven structure 284 in FIG. 2U. For the matching pairs to have the same width, the wings are constructed as triangles. The shapes of these triangles are determined from the width, shared edge length and angle C, where C satisfies the condition that sin(C) = W / L. Here C is the angle between the edge {il, i2}and the line from il to k(j, i 1), as depicted in FIG. 2D. W is the width set for the matching pair and L is the length of the edge (i l,i2) in face f(j). Repeat for all edges of all faces.(m)In embodiments, the wings can be shaped so that the intersection point of a wing matches the intersection point of the wing associated with the edge two edges further around the polygon and the opposite face. Then the wings can be shaped such that no portion of the b side is visible because the b side is either part of a connecting tab or a strut tab. In this case, the wings are quadrilaterals, such as wing w(j, il) given by ordered vertex list {il, i2, k(j, il), k(j, i5)}, where k(j, il) = k(j *, i4) and k(j, i5) = k(j*, i3), as depicted withj = j l in FIG. 2T. The other wing in the connecting pair with w(j, i 1) is w(j *, i3) which is given by {i3, i2, k(j, i5), k(j, il)}. Connecting tab 220 is the overlapping surface of the two wings in a connecting pair which is the surface with vertices i2, k(j, il), k(j, i5). The other wing in the wing pair with w(j, il) is w(j*, i2) which is given by {i2, il, k(j*, i2), k(j, i5)}. Strut tab 230 is the overlapping surface of the two wings in the wing pair which is the surface with vertices il, i2, k(j, i5). The resulting structure 286 with interwoven ribbons is shown in FIG. 2V. There are many ways to locate the intersection points. In embodiments, a center point of the polygon is determined. The center point can be at the mean of the vertex positions of the polygon but can also be some other determined point in or out of the plane of the polygon. This point is used for the design of wings for both faces of the polygon sothat wing surfaces of opposite faces share a surface. The intersection points are then located between the center point and the edge midpoints. For example, intersection point k(j 1, il) associated with a vertex il of a face j 1 is on the line between the midpoint of the edge with vertices i2 and i3 and the center point 272, as depicted in FIG. 2T. This method is referred to as the “coincident intersection” method.(n) In embodiments, the polygons of the polygon mesh are triangles. For example, if face f(j) is given by {il, i2, i3 }, then the indices cycle back to il after i3, so with i4 becoming il and i5 becoming i2. Therefore, k(j 1, il) = k(j 1*, i4) becomes k(j 1 , i 1) = k(j 1 *, i 1) because i4 cycles around the triangle to refer to il. Wing w(j 1, il) is given by {il, i2, k(j 1, il), k(j 1, i2)}.(o) In embodiments, the wings have a curved edge. There are a great variety of means to construct curved edges. In an embodiment, a B-spline function is used to construct a curve using intersection points as control points for the B-spline. An example is depicted in FIG. 2W where the B-spline curve is shown with the dashed line. A structure 288 thus generated with interwoven ribbons using curved wings is depicted in FIG. 2X.(p) In embodiments polygons in the polygon mesh are non-planar or skew. Non-planar polygons can be subdivided into planar facets such as triangles using a great variety of means. One of these means is to facet the wings such that there is an additional bend between the strut tab and connecting tab part of the surface of each wing. For example, the coincident intersection method above can be used. There can be a bend along the axis i2, k(j, i5) of wing w(j, il), and along axis i2, k(j, il) of connecting pair wing w(j*, i3). When constructing the line for the boundary of each wing from the vertices, it can be angled or bent to stay in the plane of each facet. There are other ways that a non-planar polygoncan be accommodated, for example by defining a curved B-spline surface. In embodiments, the coincident intersection method above is used wherein the wings include at least one bending axis. A center point of the non-planar polygon is determined. The center point can be at the mean of the vertex positions of the polygon but can also be some other determined point. The center point can then be the intersection points. In another embodiment, the intersection points are then located between the center point and the edge midpoints. The coincident intersection method is then used with wing w(j, il) having a bending axis along the line from k(j*, i3) to i2. In embodiments additional points 274 are determined between the intersection points, as shown in FIG. 2W and these are used to construct additional bending axes within each non-planar wing from these points to the closest vertex in the original polygon mesh. A structure 289 generated with ribbons from a non-planar polygon mesh depicted in FIG. 2Y is depicted in FIG. 2Z.(q) No matter the shape of the wings or whether the mesh has planar or non-planar polygons, the topology stays the same. The topology shown in FIG. 2R of the woven structures in FIG. 2U, FIG. 2V, FIG. 2X and FIG. 2Z is the same as the topology of the structure in FIG. 2Q because they were derived from polygon meshes with the same topology.(r) In embodiments, whether the polygons are planar or non-planar, the material of the ribbons can be flexible or rigid. In embodiments, the material can accommodate sufficient curvature to approximately follow non-planar surfaces.(s) The optional connecting or joining of ribbon surfaces involves bonding or attaching part of a surface from one ribbon to part of a surface of the same or another ribbon. The connections can be made by many different means, including by friction coupling with orwithout a retaining compound, by interference coupling, by bonding with an adhesive or epoxy, by bolting, by riveting, by brazing, by gluing, by welding, by stitching, etc.Examples

[0248] As an example of an embodiment, a target shape is a unit cell for a cube-octet lattice. The unit cell for this lattice is referred to as a pentet because it has five tetrahedrons. A polygon mesh 300 for a pentet is shown in FIG. 3A and includes 16 polygons (triangles) and 32 faces. The polygon mesh includes an inner tetrahedron with vertex list (1,2, 3, 4) augmented by four outer tetrahedrons, one for each of the four polygon sides of the inner tetrahedron. This augmented arrangement means that each face of the inner polyhedron is also shared by an outer polyhedron. Each edge of the inner tetrahedron is shared by four polygons, two from the inner polygons and one each from two adjacent outer polygons.

[0249] Referring to the Figures, FIG. 3B shows a view of faces of the inner tetrahedron subdivided into wings 10, one wing for each edge of a face. The wings of the opposite faces are not shown in this depiction. FIG. 3C depicts a view of wings for faces and opposite faces of wings for the outer tetrahedrons of the polygon mesh in FIG. 3A. Indicated are a strut tab 352 and a connecting tab 332 that make up wing 312. Wings are matched up in connecting pairs and wing pairs. Contiguous sequences of connecting pairs of wings and matching pairs of wings form weavers. For this example, there are 6 weavers, each including 8 matching pairs of wings sharing connecting tab surfaces in a sequence. The weavers are sequences of alternating matching and connecting pairs of wings. FIG. 3D depicts a view of a weaver 360 for the polygon mesh in FIG. 3A. Further, FIG. 3E depicts a view of the weaver in FIG. 3D including the outward directions of the wings of the weaver as indicated by the arrows emanating from the strut tabs of each wing. During assembly, the outward directions determine positioning of the ribbons in relation to each other. The arrowspoint from the b-side of the positioned ribbon through the a-side. As can be seen from FIG. 3D, the weaver resembles a figure 8, with the crossing in the figure 8 corresponding to edge (1,3) of the inner tetrahedron with each loop of the figure 8 corresponding to the edges of two adjacent augmenting tetrahedrons. Each loop is opened at a connecting pair of wings and flattened at each bending axis. FIG. 3F depicts a view of opened and flattened weaver 360. The ends of the weaver have connecting tab surfaces. 362 and 364. FIG. 3G depicts a design 370 for a ribbon for opened, flattened weaver 360 in FIG. 3F. There are six ribbon designs, one for each weaver. Each design is based on the contiguous surface of a weaver that is opened at a connecting pair and the weaver flattened. A dihedral angle 372 is indicated for each matching pair of wings at each edge. Bending axes 374 are indicated where the ribbon is bent when positioned in the structure. If the dihedral angle is 180 degrees, then no line is indicated. Wing pair indices 376 are used in positioning parts of ribbons relative to each other according to their wing pairs. Each end of the ribbon has a connecting tab 378. Each end wing is in a connecting pair with the next wing in the ribbon.

[0250] In the same way that the inner tetrahedron is augmented by outer tetrahedrons in FIG. 3A, in general, triangular meshes can be augmented by outer tetrahedrons. The weavers of such augmented triangular meshes are figure 8’ s wherein the crossings are edges of the triangular mesh and wherein the loops of each figure 8’s form a Borromean link with loops of two other figure 8’s from adjacent edges. There is one Borromean link for each augmenting tetrahedron.

[0251] The fabrication of the ribbons is based on the designs in FIG. 3G. For example, the design can be used to make ribbons from a sheet of material with a laser cutter. Each fabricated ribbon is then bent at the bending axis by the indicated dihedral angle for the associated matching pair of wings. FIG. 3H depicts a view of a ribbon 380 fabricated from the design 370 shown in FIG. 3E, bent and placed according to the polygon mesh. FIG. 31 depicts a view of two ribbons bent andplaced. The ends of the ribbons overlap each other at the end connecting tab surfaces to complete the weaver. The ribbons are placed by matching up a wing with its wing pair wing by using the wing pair indices, placing the b-surface facing the b-surface of the other wing in the wing pair with the bending axes aligned, and imposing the dihedral angle at each bending axis. The ends of the ribbons are placed such that each first end of each ribbon overlaps a second end of the same or another ribbon such that the a-side of the connecting tab at the first end shares a surface with the b-side of the connecting tab at the second end. One or more ribbons are placed in a sequence in a weaver. For a closed loop weaver, the first end of the first ribbon overlaps the second end of the last ribbon to complete a weaver. For the current example, each of 6 closed loop weavers include one ribbon each. FIG. 3J depicts a view of three ribbons bent and placed. FIG. 3K depicts a view of the completed 3D ribbon woven structure with all six ribbons.

[0252] The ribbon topology for the ribbon 380 in FIG. 3H is depicted in FIG. 3L. All six weavers have the same topology and each topology consists of a figure 8, whereby each figure 8 crossing is associated with an edge of the inner tetrahedron. There is a twist due to writhe in the weaver in each loop of each figure 8 such that when the association at the crossing is removed, the twist in each loops cancels the other out and the weaver is a loop with no twist. FIG. 3M depicts a ribbon topology of the two ribbons in FIG. 3H. FIG. 3N depicts a ribbon topology for three ribbons for the structure in FIG. 3K. The loops of the figure 8’s of the ribbons do not intersect each other. Each figure 8 loop forms a Borromean link with two other loops of two adjacent figure 8 weavers. FIG. 30 depicts the ribbon topology 390 of six ribbons for the completed structure in FIG. 3K. A Borromean link 398 is shown for one of the augmenting tetrahedrons involving three weavers,392, 394 and 396. There are four such Borromean links, one for each outer tetrahedron.

[0253] As another example, a target shape is for an open box with a divider. A polygon mesh 400 for the box shape is shown in FIG. 4A and includes 12 vertices, 9 polygons and 18 faces. The polygon for the divider is given by the vertex list (2, 3, 8, 9). The edges this polygon have vertex lists (3,8), (9,8) and (2,9) and are shared by three polygons each.

[0254] Referring to the Figures, FIG. 4B shows a view of the faces of the polygon mesh in FIG. 4A subdivided into wings, one wing for each edge of a face. Contiguous sequences of connecting pairs of wings and matching pairs of wings form weavers. In this example, wings are designed to be wide such that there is a very small hole in the center of each polygon. For a polygon mesh with square polygons, each weaver is approximately a straight strip when flattened. For the polygon mesh of the box, there are two weavers. FIG. 4C depicts designs for the two weavers using ribbons with 2 or 8 matching pairs each. The first weaver has 10 matching pairs and the design includes ribbon 470 of 8 matching pairs and ribbon 472 of two matching pairs. Each design is based on the contiguous surface of a weaver. The contiguous surfaces are loops that are opened to make one or more ribbons. The weaver is flattened at bending axes where there are dihedral angles other than 180 degrees between planes of wings in a matching pair. The ends of each ribbon design include connecting tab surfaces from the wings in the first and last wings in the contiguous sequence for each ribbon. In this example, the ends of each ribbon include both wings in the connecting pair with the shared connecting tab giving each end a connecting tab and two strut tabs. In an embodiment the ends of each ribbon design include a connecting pair of wings. A dihedral angle for each matching pair of wings at each edge is indicated on the design for each ribbon. Wing pair indices indicate where strut tabs of each ribbon are positioned so that the ribbons can be assembled into the structure.

[0255] The fabrication of the ribbons is based on the designs in FIG. 4C. Each ribbon is placed and bent at the bending axis by the indicated dihedral angle for the associated matching pair of wings. FIG. 4D depicts a view of a ribbon 480 based on design 470 that has been bent and placed according to the placement of its wings in the polygon mesh. FIG. 4E depicts a view of ribbon 482 based on design 472 for the next contiguous sequence in the first weaver. The two ribbons 480 and 482 are placed in a sequence such that wings in a wing pair share a surface and the first end 474 of the first ribbon overlaps the second end 476 of the second ribbon and the first end of the second ribbon overlaps the second end of the first ribbon to complete the first weaver. The a-side of the strut tab at ribbon end 474 shares a surface with the b-side of the strut tab at ribbon end 476. FIG. 4F depicts a view of the four ribbons from the four bottom designs in FIG. 4C bent and placed to complete the second weaver. FIG. 4G depicts a view of the completed 3D ribbon woven structure with all six ribbons placed using the polygon mesh in FIG. 4A.

[0256] The ribbon topology for the 3D woven structure in FIG. 4G is depicted in FIG. 4H. There are two twisted weavers in the topology and three associations with three matching pairs each. Each of these associations corresponds to each edge shared by three polygons in the polygon mesh in FIG. 4A

[0257] As another example, a target shape is a frame for a quadcopter drone. A polygon mesh for the quadcopter shape is shown in FIG. 5A and includes 9 vertices and 17 polygons. The mesh includes a central upside-down pyramid with base (2, 3, 5, 4). Each triangle of the pyramid is augmented by an elongated tetrahedron. The edges of the pyramid base (5,3), (2,3), (2,4) and (4,5) are shared by three polygons. The triangular polygons of the pyramid (1,2,3), (1,3,4), (1,4,5), (1,5,2) result in internal structure that strengthens the frame. The edges of the triangular polygons of the pyramid (1,2), (1,3), (1,4) and (1,5) are shared by four polygons.

[0258] Referring to the Figures, FIG. 5B shows two views of faces of the polygon mesh in FIG. 5A subdivided into wings, one wing for each edge of a face. Contiguous surfaces of connecting pairs of wings and matching pairs of wings form weavers. For the quadcopter frame, there are five weavers. FIG. 5C depicts designs for ribbons with 8 orlO matching pairs each. Four weavers have designs 570, 571, 572 and 573 of one ribbon each and a fifth weaver has designs 574 and 575 for two ribbons of 10 matching pairs each. Each design is based on a closed loop weaver opened at a connecting pair and flattened. In this example, the fifth weaver is again opened to make the resulting ribbons have 10 matching pairs each. A dihedral angle for each matching pair of wings is indicated at each bending axis. Wing pair indices indicate where strut tabs of each ribbon are positioned. Each end wing of the ribbons is a wing in a connecting pair. Each end wing is used twice in the design of the ribbons. Twin wings are two or more wings of the same or different ribbons that are copies of the same wing design in a weaver. Twin wings are in the same or different ribbons of the same weaver. Each end wing has a twin which is a connecting pair wing of another end wing of a ribbon designed from the same weaver. This can be seen from the wing pair indices and shapes of the end wings. For example, each ribbon end in 570 has connecting pair of wings 586

[0259] The fabrication of the ribbons is based on the designs in FIG. 5C. Each ribbon is placed and bent at the bending axis by the indicated dihedral angle for the associated matching pair of wings. FIG. 5D depicts a view of a ribbon 580 based on design 570 that has been bent and placed to complete the first weaver. The polygon mesh is overlayed on the ribbons for clarity. The ends of the weaver overlap at connecting tab 586. FIG. 5E depicts a view of ribbon 581 based on design 571 for the second weaver. The a-side of one ribbon end 587 shares a surface with the b-side of the other ribbon end. The second ribbon overlaps an end wing of the first ribbon with connectingtab 586. The overlapped end wing is sandwiched in a pocket between its twin wing and a wing of the second ribbon that is its wing pair wing. This creates a layered arrangement of three wings that holds the ribbons in place by mechanical interference. FIG. 5F depicts a view of all five ribbons from the designs in FIG. 5C bent and placed to form the 3D woven structure in the shape of a quadcopter frame. FIG. 5G depicts a view from above of the structure in FIG. 5F. FIG. 5H depicts a perspective view of the structure in FIG. 5F.

[0260] The ribbon topology for the 3D woven quadcopter frame structure in FIG. 5F is depicted in FIG. 51. The four legs are interwoven as an integral structure making it stiff and resilient.

[0261] As another example, a target shape is a unit cell for a gyroid lattice. A polygon mesh for a gyroid lattice unit cell is shown in FIG. 6A and includes 14 vertices, 3 non-planar polygons and 6 faces with 10 edges each. For example, the face f(l) and outward direction is given by directed vertex list {8,10,12,13,14,7,6,3,2,1 } and the arrow, respectively in the FIG. 6A. Each edge in the mesh is shared by two polygons. Each polygon shares five edges with a second polygon and the other five edges with a third polygon.

[0262] The six faces are subdivided into wings. A contiguous surface of matching pairs and connecting pairs of wings forms a single weaver of thirty matching pairs. Referring to the Figures, FIG. 6B depicts designs 670, 671 and 672 for three ribbons for the weaverforthree ribbons with 10 matching pairs each. Each design is based on the weaver that has been opened at a connecting pair and flattened. The weaver is again opened at the connecting pair after each tenth matching pair. A dihedral angle is indicated for each matching pair of wings at each bending axis. Wing pair indices indicate where strut tabs of each ribbon are positioned so that the ribbons can be assembled into the structure. The dashed lines such as 674 indicate the location of the bending axes.

[0263] The fabrication of the ribbons is based on the designs in FIG. 6B. Each ribbon is placed and bent at the bending axis by the indicated dihedral angle for the associated matching pair of wings. FIG. 6C depicts a view of first ribbon 680 based on design 670 that has been bent and placed. FIG. 6D depicts a view of the first ribbon and the second ribbon 681 based on design 671 placed and bent. End 683 of ribbon 681 shares a surface with the end 682 of ribbon 680. FIG. 6E depicts a view of the entire weaver from the designs in FIG. 6B bent and placed to form the completed 3D ribbon woven structure in the shape of a gyroid unit cell. The ends of the ribbons are obscured by other parts of ribbons.

[0264] As another example, a target shape is a table. Views of a polygon mesh for the table shape are shown in FIG. 7A and includes 13 vertices and 20 triangular polygons.

[0265] The faces are subdivided into wings and the contiguous surfaces are determined to include four weavers. Referring to the Figures, FIG. 7B depicts designs for two weavers of three ribbons, 770, 771 and 772, and 773, 774 and 775. There are also two weavers of two ribbons, 776 and 777, and 778 and 779. Each design is based on an opened and flattened weaver that has again been opened at a connecting pair after each sixth matching pair. The designs include a dihedral angle for each matching pair of wings at each bending axis. Wing pair indices indicate where strut tabs of each ribbon are positioned so that the ribbons have wing pair wings sharing or approximately sharing their strut tab surfaces.

[0266] The fabrication of the ribbons is based on the designs in FIG. 7B. Each ribbon is placed and bent at the bending axis by the indicated dihedral angle for the associated matching pair of wings. FIG. 7C depicts a view of a ribbon 780 based on design 770 that has been bent and placed for the first weaver according to the polygon mesh. FIG. 7D depicts a view of all ribbons bent and placed in the polygon mesh for the designs in FIG. 7B to form a 3D woven structure in theshape of a table. FIG. 7E depicts a view from below of the structure in FIG. 7D. The edges shared by more than two triangles strengthen the table structure.

[0267] As another example, a target shape is a frame for a rocket. A polygon mesh 800 including 41 vertices and 58 polygons for the rocket frame is shown in FIG. 8A. The rocket frame includes an outside shell, three legs 802, and five bulkheads 804. In the polygon mesh, each bulkhead is a horizontal hexagon at each of five levels in the rocket frame. Each edge of each bulkhead hexagon is shared by three polygons, except the bottom bulkhead 806 which is shared by four polygons.

[0268] Referring to the Figures, FIG. 8B shows a view of faces of the polygon mesh in FIG. 8A subdivided into wings, one wing for each edge of a face. Contiguous surfaces of matching pairs and connecting pairs include three weavers of 70 matching pairs each. FIG. 8C depicts designs for 9 ribbons making up one of the weavers. The ribbons have a maximum of 8 matching pairs each and the last ribbon has 6 matching pairs. A dihedral angle is indicated for each matching pair of wings at each bending axis. Wing pair indices indicate where strut tabs of each ribbon are positioned so that the ribbons can be assembled into the structure according to the polygon mesh.

[0269] The fabrication of the ribbons is based on the designs in FIG. 8B. Each ribbon is placed and bent at the bending axis by the indicated dihedral angle for the associated matching pair of wings. FIG. 8D depicts a view of the ribbons from the design for the first weaver in FIG. 8C placed according to the dihedral angles and wing pair indices. The two other weavers are also similarly placed. FIG. 8E depicts a view of all ribbons for all three weavers bent and placed to complete the 3D woven structure in the shape of a rocket frame. The ribbons need not be designed so that the final structure has holes. For example, if the subdivision points are at a center of each polygon, there would be no or a small hole. The structure can serve as a frame for a cladding material or the structure can be the cladding itself. The bulkheads are integrated into the shell ofthe structure thereby stiffening it and providing opportunities to reduce the weight of the structure. The highly integrated nature of the structure improves resilience.

[0270] The ribbon topology for the rocket frame structure in FIG. 8E is depicted in FIG. 8F.

[0271] As another example, a target shape is a space frame including a layer of pyramids and tetrahedrons. A polygon mesh for the space frame shape is shown in FIG. 9A and includes 25 vertices: 16 vertices form a top plane of 9 square polygons and 9 vertices form a bottom plane of 4 square polygons. The two planes are joined by a layer of edges that define tetrahedrons and pyramids with square bases. One such tetrahedron has vertex list (10,11,21,24).

[0272] Referring to the Figures, FIG. 9B shows a view of the faces of the polygon mesh in FIG. 9A subdivided into wings, one wing for each edge of each face. The polygons of the polygon mesh in this example are square and triangular cycles in the mesh. The surfaces of the polygons are planar. Contiguous surfaces of connecting pairs of wings and matching pairs of wings form weavers. Opened, flattened weavers are used as designs for ribbons. FIG. 9C depicts designs for ribbons of 4 matching pairs each. Wings in connecting pairs of the four ribbons make up part of the tetrahedron faces with vertex list (10,11,21,24). The four ribbons share strut tab surfaces associated with edges (10,11) and (21,24). Other wings in the ribbon designs in FIG. 9C form faces of adjacent tetrahedrons and square bases of pyramids.

[0273] The fabrication of the ribbons is based on the ribbon designs in FIG. 9C. Each ribbon is placed and bent at the bending axis by the indicated dihedral angle for the associated matching pair of wings. Each end of each ribbon is a connecting pair of wings. FIG. 9D depicts separate views of each of the four ribbons 980, 981, 982 and 983 based on designs 970, 971, 972 and 973, respectively, in FIG. 9C, with each ribbon bent according to the indicated dihedral angles in the designs in FIG. 9C. FIG. 9E depicts a view of two ribbons 980 and 983 placed in relation to eachother according to the polygon mesh in FIG. 9A. FIG. 9F depicts a view of all four ribbons placed so as to share strut tab surfaces associated with edges (10, 11) and (21, 24). The arrangement of ribbons forms a unit cell or module based on a tetrahedron of the space frame. FIG. 9G depicts another view of the four ribbons in FIG. 9F for the unit cell. Each end of each ribbon in the unit cell can share a surface or connect with an end of a ribbon of another unit cell or the end of a ribbon that forms the boundary of the space frame. Each end of each ribbon is a free end such as 920, 922, 924 and 926 that slides into a pocket formed by a wing pair of wings of other unit cells, thereby connecting unit cells and forming a larger structure. In the present example, there are four such unit cells. The rest of the ribbons form the boundaries of the space frame. FIG. 9H, FIG. 91 and FIG. 9J depict views of the completed 3D woven structure with all ribbons assembled using designs from the space frame polygon mesh in FIG. 9A.

[0274] The ribbon topology for the 3D woven structure in FIG. 9J is depicted in FIG. 9K.

[0275] A great variety of woven structures that function as modules can be assembled and connected to make larger structures. Modules can be based on part of a polygon mesh with other modules consisting of adjacent parts so that when the modules are connected, they form a larger structure based on the whole polygon mesh. Modules can have one or more free ribbon ends extending from the module. A ribbon end or an end wing is termed free if the end wing of a ribbon of a first module does not have a wing pair wing in the module. Instead, the wing pair wing is part of another module. In addition, the end wing of a module has a twin wing in another module. The twin wing and wing pair wing of the other module form a pocket into which the end wing of the first wing slides thus connecting modules. An example is a tetrahedron with four free ribbon ends, two at each of two adjacent edges of the tetrahedron, based on the space frame such as that in FIG. 9A, but without the square pyramid bases. The module consists of three ribbons, two ribbons 10and 30 with six connecting pairs and one ribbon 20 with five connecting pairs as shown in FIG. 9L. The ends of ribbons 10 and 30 have end wings 12 and next wings 16 in connecting pairs. The ends of ribbon 20 have end wings 22 and 26 and next wings 24 and 28. The wing indices and dihedral angles are as indicated, the dashed lines are bending axes and the areas between bending axes are connecting pairs of wings, with each wing having a lightening hole. The connecting pairs are hook-shaped, with an indent such as 18. FIG. 9M depicts the first ribbon 10 bent according to the dihedral angles. End wings 12 of the first ribbon extend from the tetrahedron being formed. FIG. 9N depicts second ribbon 20 bent and placed according to the dihedral angles. Second ribbon 20 forms a loop, with each end wing overlapping with the connecting pair wing of the other wing. End wing 26 overlaps its twin wing 24 and end wing 22 overlaps wing 28. The module is formed by placing ribbon 20 inside ribbon 10 and interweaving ribbon 30 so that wings are placed sharing a surface with their wing pair wings according to the wing pair indices. The assembled module 40 is shown FIG. 90. Each end wing of the second ribbon is placed between the wings of a wing pair. Each such wing pair consists of the twin wing next to the other end wing of the second ribbon and a wing from one of the other two ribbons. Therefore, there are two adjacent arrangements of three wings in a layer, one based on each end wing of the second ribbon. This arrangement locks the ribbon in place due to mechanical interference. The module has four free ribbon ends consisting of connecting pairs of wings, 12 and 16 extending from two adjacent edges of a tetrahedron. Modules such as 40 can be connected by means of the hook-shaped connecting pairs of wings at free ribbon ends. Two modules are aligned and shown in FIG. 9P. The two modules are connected by placing two end wings 12 of one module 40 into two pockets formed by wing pairs 52 of a second module 50 to obtain a larger structure of two connected modules as shown in FIG. 9Q.The structure is held together by mechanical interference created by the free ribbon ends locking into the pockets, without the need for bonding, adhesive, welding, etc.

[0276] This method can be repeated with many tetrahedron modules to create larger structures such as the space frame in the shape of a plane shown in FIG. 9R. This is useful for building components such as a ceiling or a wall or the shell of a structure or vessel. As can be imagined, by changing the relative vertex positions of the tetrahedrons for the modules and by changing the number and positions of the free ribbon ends extending out, a great variety of different shapes of modules can be constructed for target structures with a great variety of shapes. Further embodiments include modules based on shapes other than a tetrahedron. Each edge of a tetrahedron can have zero, one or two free ribbon ends extending from it, making for a great variety of modules and possible structures. For example, a linear of connected tetrahedrons, a tetrahelix, can be assembled from a tetrahedron with three free ribbon ends at each edge of one of the triangular faces of a tetrahedron module. Each tetrahelix can branch off another tetrahelix, making it easy to build up larger frameworks consisting of interconnected tetrahelices.

[0277] As another example, a target shape is a frame for a sports car. A side view and bottom view of a polygon mesh for the car is shown in FIG. 10A. The car frame includes an outside shell, inside compartments for the driver and engine and features such as wheel hubs. The inclusion of triangles and internal structure subdividing compartments, stiffens the frame.

[0278] Referring to the Figures, FIG. 10B shows side and bottom views of the 3D woven structure in the shape of the frame for a sports car. Cycles in the polygon mesh define polygons. The frame includes ribbons for weavers of wings derived from the subdivision of the faces for the polygon mesh in FIG. 10A. The ribbons need not be designed so that the final structure has holes. For example, if the subdivision points are at a center of each polygon, there would be no or a smallhole. The structure can serve as a frame for a cladding material or the structure can be the cladding itself.

[0279] The ribbon topology for the 3D woven car frame structure in FIG. 10B is depicted in side and bottom views in FIG. 10C. As can be seen the structure is highly integrated thereby improving structural resilience.

[0280] As an example, the desired structure has a target shape described by a cube with an internal structure that is a tetrahedron sharing its four vertices with diagonally opposite vertices of the cube, as shown in FIG. 11A. This is the same shape as the shape of the pentet block in FIG. 3A. In this example, a representation of the target shape is generated using a tetradihedron polygon mesh. A tetradihedron is a tetrahedron wherein each triangular polygon is degenerate to order 2, i.e. for each first polygon there is a second polygon that shares all the same edges, and for each face there is a second face that shares all the same directed edges. The tetradihedron polygon mesh has vertices, v(l), v(2), v(3) and v(4) and edges (1,2), (1,3), (1,4), (2,3), (2,4), (3,4) as shown in FIG. 11B. Each edge of the tetradihedron polygon mesh is shared by four polygons because the polygons are degenerate . In this example the first of the degenerate polygons are four planar triangles that make up the tetrahedron surface and have vertex list (1,2,3), (1,2,4), (1,3,4) and (2,3,4). The second of the degenerate polygons are given by the same edges and the same vertex lists as the first. The second polygons are four non-planar triangles with surfaces that extend outwards to subdivision vertices v(9), v(10), v(l l) and v(12) as shown in FIG. 11C. These are vertices of the cube not shared with the tetrahedron and the non-planar polygons make up the surface of the cube. For example, non-planar polygon (1,2,3) has subdivision vertex v(10) in its surface. The non-planar polygons have three right triangular surfaces such as (1,2,10).

[0281] Each polygon in the polygon mesh has a first face and an opposite face, wherein each face has an outward direction, wherein the outward directions of the faces point in opposite directions. The method includes determining a shape and dimensions of a plurality of wings by subdividing the faces, wherein each particular wing is subdivided from a particular face of a particular polygon of the polygon mesh, and has a particular edge of the particular polygon and a particular directed edge of the particular face. Each wing has part of the surface and has the same outward direction as that of the face from which the wing is subdivided. For example, non-planar face { 1,3,2} has subdivision point v(10) and is subdivided into wings { 1,3,10}, {3,2,10} and {2,1,10}, and the outward direction of each wing is normal to the surface and outward from the cube. In general, for non-planar polygons, the outward direction changes with location as the orientation of the surface of the face changes. For the planar triangular faces that are part of the tetrahedron surface, the wings are obtained by subdividing each planar face into wings by using subdivision vertices, v(5), v(6), v(7) and v(8) in a center each face, as shown in FIG. 11D. For example, planar face { 1,3,2} has subdivision point v(6) and is subdivided into wings { 1,3,6}, {3,2,6} and {2,1,6}. These wings have outward directions pointing outward from the tetrahedron because of the right-hand thumb rule used.

[0282] Depicted in FIG. 1 IE are four wings of a weaver based on the tetradihedron polygon mesh. Particular wing 2 is subdivided from non-planar face { 1,2,3}, has vertex list {3,1,10}, edge (1,3), and directed edge {3,1 } that points to vertex v(l). Using the right-hand thumb rule, it has an outward direction that points inwards into the volume between the cube and the tetrahedron.

[0283] There is a second wing 4 that is subdivided from the adjacent face to the particular face and shares the particular edge with the particular wing. The term “adjacent faces” refers to two faces that share at least one edge and have outward directions that point into a same volume.Second wing 4 is subdivided from adjacent, planar face { 1,3,2}, has vertex list { 1,3,6}, a shared edge (1,3) with the particular wing, directed edge { 1,3 } that is opposite to the directed edge of the particular wing, and an outward direction that points into the same volume between the cube and the tetrahedron as that of the particular wing. Particular wing 2 and the second wing 4 are a matching pair of wings and have a bending axis at the shared edge (1,3) and a dihedral angle as determined by the angle between the faces.

[0284] There is a third wing 6 that is subdivided from the opposite face to the particular face, has an edge of the polygon adjacent to the edge of the particular wing, and has a directed edge that points to a same vertex as the directed edge of the particular wing. Third wing 6 is subdivided from opposite non-planar face {1,3,2} of the particular face {1,2,3}, has vertex list { 1,3,10}, an adjacent edge, (1,2), and a directed edge, {2,1 } that points to the same vertex, v(l), as that of particular wing 2. Particular wing 2 and the third wing 6 are a connecting pair of wings and share line (1,10) in the surface of the non-planar polygon.

[0285] There is a fourth wing 8 that is subdivided from the opposite face of the particular face and has a shared edge with the particular wing. Fourth wing 8 is subdivided from opposite non- planar face { 1,3,2}, has vertex list { 1,3,10}, shared edge, (1,3) with the particular wing, and directed edge { 1,3} that points in the opposite direction to that of the particular wing. The particular wing and the fourth wing are a wing pair of wings.

[0286] Each wing has a unique matching pair, a unique connecting pair and a unique connecting pair.

[0287] Wing 10 is subdivided from planar face { 1,2,3}, which is the opposite face of the face of wing 4, has vertex list {2,3,6}, an edge, (2,3), and a directed edge, {2,3}, that points to the same vertex, v(3), as that of wing 4. Wing 10 and wing 4 are a connecting pair of wings and shareboundary (3,6). Boundary (3,6) is in the surface of the planar polygon. Wing 10 is a connecting pair with wing 4.

[0288] Alternating connecting pairs and matching pairs of wings form a sequence of wings. The sequence of wings in FIG. HE is (10, 4, 2, 6). Wing 8 is not in the same sequence because it is not a matching pair or connecting pair of either of wings 2, 4, 6, 10. FIG. 11F shows eight wings in a sequence of four connecting pairs and three matching pairs.

[0289] Weavers are surfaces that comprise sequences of alternating connecting pairs of wings and matching pairs of wings. For this example, there are 3 closed-loop weavers, each including 8 matching pairs and 8 connecting pairs of wings. FIG. 11G depicts a weaver 60 for the polygon mesh in FIG. 11B, that is an alternating sequence of eight matching pairs and eight connecting pairs of wings in a closed loop. There are three identical such weavers, one for each direction in 3D space. For the design of ribbons, each weaver is opened at a connecting pair of wings and flattened out. FIG. 11H depicts a view of opened and flattened, planar weaver 60. Weaver 60 has wings in an alternating sequence of matching pairs and connecting pairs. Wings 20 and 22 are in a connecting pair, wings 22 and 31 are in a matching pair, and wings 31 and 32 are in a connecting pair of wings. The ends of the weaver are matching pairs of wings because the weaver was opened at a connecting pair. It could also be opened at a matching pair and then the ends would be a connecting pair. But it does not matter how it is opened as the entire weaver at this point is a mathematical object that is used as the template for one or more ribbons that can include any sequence of adjacent wings in the weaver. The dashed lines indicate the bending axes of matching pairs. The interior solid lines indicate shared boundaries that serve as connecting tabs of connecting pairs of wings. As in all the examples, the design is generated by using a computer with analgorithm to generate wings and to match up wings in matching pairs, connecting pairs and wing pairs.

[0290] Ribbons are sequences of one or more matching pairs and connecting pairs of wings. This means a ribbon comprises a whole or part of a weaver. FIG. Ill depicts a design for a ribbon 70 comprising four connecting pairs and three matching pairs from the middle part of weaver 60 in FIG. 11H. Dashed lines 50 indicate bending axes at the shared edges of the matching pairs. Dashed lines 52 and 54 indicate the shared boundaries of wings in connecting pairs from the non-planar faces, where a 90° bend occurs because of the non-planar surface. Dihedral angles 54 and wing pair indices such as 56 are indicated at each bending axis 50. For each weaver, there are two ribbons such as the one shown in FIG. 111. For the entire structure there are six ribbons with the same design and dihedral angles. The dihedral angles 54 indicate how much the ribbon is bent when fabricated and assembled. Wings with the same pair indices 56 identify wings in a wing pair and are placed with their bending axes at their shared edge according to the polygon mesh. End wing 20 is in a connecting pair with wing 22. The other end wing 21 is also in a connecting pair with wing 23. Lightening holes in each wing have been included in the design.

[0291] In an embodiment the ribbons are cut from a sheet. Each ribbon is placed and bent at the bending axis by the indicated dihedral angle for the associated matching pair of wings and placed according to the polygon mesh. FIG. 11J depicts a view of ribbon 70 placed according to the polygon mesh. The ribbon is bent according to the dihedral angles so that wings of the ribbon can be placed according to their location in the polygon mesh to form part of the target structure in the shape of the tetrahedron in a cube. For example, ribbon 70 has end wing 20 in a connecting pair with wing 22, which are placed according to the placement of the wings in the original mesh as shown in FIG. 11F. FIG. 11K depicts another ribbon 72 with the same shape as the ribbon inFIG. Ill bent and placed according to the tetradihedron polygon mesh. This ribbon has end wing 24 in a connecting pair with wing 26. FIG. 11L depicts the ribbons in FIG. 11 J and FIG. 11K placed together. The two ribbons are placed according to the tetradihedron mesh and share a surface where they have wings 32 and 34 in a wing pair.

[0292] During assembly, ribbons are fit into place by means of a multi-flap interlock closure. FIG. 11M depicts three ribbons overlapping each other at one of the planar triangular faces (outlined by the dashed-lined equilateral triangle in the middle), prior to bending and imposing a three-flap interlock closure to form a corner of the cube. The dashed lines depict where the ribbons are bent. The three end wings 20, 24 and 28 form the flaps for the closure and are in connecting pairs with wings 22, 26 and 30, respectively. These three ribbon ends are bent towards the viewer along the dashed lines and placed with end wing 20 under wing pair wing 30, end wing 24 under wing pair wing 22 and end wing 28 under wing pair wing 26. Each end wing is placed under its wing pair wing which is a connecting pair wing of another end wing. This forms a three-flap interlock closure with corner 44 extending towards the viewer, as shown in FIG. UN. A pocket is formed between each end wing and its wing pair wing.

[0293] The assembled woven structure is depicted in FIG. 11O with all six ribbons bent according to the dihedral angles and placed according to the wing pair indices of the polygon mesh. Four of the corners of the cube correspond to the subdivision points of the non-planar polygons and are formed by three-flap interlock closures as described above. These include those the depicted corners 40, 42 and 44.

[0294] The topology for the structure in FIG. 11K is depicted in FIG. IIP. The topology is represented by weaver sequences of alternating connecting pairs and matching pairs of wings wherein the two wings of a wing pair are represented by the same wing pair index. This identifiesthe interrelation of the weavers at their wing pairs. Weaver sequences are given for the three weavers of the tetradihedron in the TABLE 1 below.TABLE 1: Weaver Sequences for the Tetradihedron

[0295] In an embodiment, structures such as that in FIG. 11O are modules that can be connected to each other by means of side wings. One or more ribbons have one or two side wings 36 attached to connecting pair wing 35 of an end wing, as shown for ribbon 72 in FIG. 11Q. The dashed line separating the side wing from the rest of the ribbon indicates a bend of 90° in ribbon of the module. FIG. HR depicts a view of module structure 82 with two ribbons such as 72 that have two side wings each for a total of four side wings 36 oriented in the same plane. These ribbons with side wings are bent and placed as shown such that there are two side wings on the same side and two side wings on the opposite side. Two modules 82 and 84 rotated and aligned are shown in FIG. IIS. Two module structures can be connected by means of the side wings of one module sliding inside a pocket in the other module, forming a new structure as shown in FIG. 11T. Side wings 36 of module 82 slide into pockets 38 formed by wing pairs of wings forming part of a three-flap closure. In an embodiment there are a plurality of modules connected by means of sliding side wings into pockets of adjacent modules. FIG. HU depicts a view of a module with four ribbons such as 72 that has two side wings each for a total of eight side wings. Four modules such as thatin FIG. 11U are connected by means of their side wings, forming a larger structure as shown in FIG. 11V. This can be repeated indefinitely to make planar or shell-like structures. In an embodiment, the structure has all six ribbons with two side wings each as shown in FIG. 11 W making a total of twelve side wings. A great many of these modules or building blocks can be interconnected to form larger 3D structures of a great variety of shapes.

[0296] The shape of the module structure can be changed by changing the relative positions of the vertices of the cube in the polygon mesh template thereby making some other shape. In general, this shape is a dodecahedron with twelve triangular faces, where the triangular sides are the wings subdivided from the non-planar polygons of a tetradihedron. Any volume of any shape can be filled by this shape. If the positions of the vertices are changed, then the shape of the wings and ribbons changes. A free-form or curved shell can be made by first describing the shape with a curved mesh consisting of a layer of different shaped tetradihedron meshes. In embodiments, tetradihedron is used as a polygon mesh to form a woven module consisting of a tetrahedron in a dodecahedron with triangular sides.

[0297] In an example to demonstrate curved structures with curved surfaces, the target structure is a spherical shell based on an icosahedron. Depicted in FIG. 12A is a tetradihedron placed at an edge of an icosahedron and is used to form a module for the shell. Because the vertices for the tetradihedron in this example are different from the previous cube example in FIG. UN, the design shapes of the six ribbons are also different from that for the cube shape, as shown in FIG. 12B. Each ribbon ends in a wing in a connecting pair of wings. The ribbons are woven together and placed according to the wing pair indices on the ribbon designs to form the module structure in FIG. 12C. The module structures formed from the tetradihedron polygon mesh at each edge of the icosahedron fit together to make the sphere-like shell structure, as shown in FIG. 12D, FIG. 12E,FIG. 12F and FIG. 12G. In FIG. 12G, a module is placed at each of the 30 edges of the icosahedron to complete the spherical shell.

[0298] In embodiments, 3D woven structures have a great variety of shapes and uses. In embodiments, 3D woven structures have a great variety of sizes, including from the nanometer scale to kilometers.Fabrication and Assembly

[0299] There are a variety of methods for fabrication and assembly of structures including manufacture, forming, placement, and connection of ribbons. Using CNC methods such as laser cutting or by other means, ribbons are made by cutting and bending sheets or strips of material. In embodiments, ribbons are formed in some way such as by stamping, milling, casting, curing epoxy, curing carbon fiber composite in a mold, injection molding, thermoforming, additive manufacturing, etc. in their predetermined shapes.

[0300] In an embodiment, ribbon material can have a wide range of properties selected from a group including but not limited to rigid, flexible, elastic, hyper-elastic, electrically conducting, semiconducting, insulating, translucent, opaque, transparent, and reflecting. The ribbon material can be made of many types of materials chosen from the group including but not limited to metals, fiber composite, carbon fiber composite, ceramic, polymer, copolymer, rubber, paper, cardboard, textile, wood, leather, stone, concrete, sandwich core materials, honeycomb core plates, natural, foam, elastomers, alloys, graphene, carbon nanotubes, beta-protein sheets, and glasses. An advantage is that alloys such as Inconel that are difficult to weld can be assembled into a structure.

[0301] In an embodiment, the bend or fold around each bending axis in the ribbon can be made by a variety of means, chosen from a group including but not limited to bending a sheet of material using a brake or some other physical means, applying heat and bending, folding, pinching,molding, casting, joining separate pieces representing each wing at the dihedral angle, using prepreg carbon fiber composite material curing, heat and / or vacuum or some other type of forming using methods known to a person skilled in the art. The bend can be a gradual bend.

[0302] Embodiments provide for ribbons wherein the ribbon material along the bending axis is rigid. Embodiments provide for flexible material or a hinge along the bending axis that allows bending of the ribbon. The hinge is selected from a group consisting of but not limited to a mechanical hinge, a living hinge, a strip of flexible material such as poly-para-phenylene terephthalamide, thin metal, fabric, polymer, etc. A flexible or weakened line of material can be made by scoring or perforating the material along the bending axis and then bending.

[0303] Embodiments provide for ribbons including material that is flexible such as tape, film, fabric, flexible polymers, rubber, organic materials, leather, poly-para-phenylene terephthalamide, bendable metal, etc.

[0304] In embodiments, a ribbon end is locked in an interference fit by sliding a ribbon end under another ribbon. The ribbon end may be deformed to slide it under the other ribbon. Embodiments include placing an end wing of a ribbon between the wings of a wing pair, wherein one of the wings of the wing pair is a twin wing of the end wing. In embodiments, there is a layered arrangement of three or more wings of ribbons at the same strut tab surface. Embodiments include placing end wings in a flap interlock closure.

[0305] The order of assembly can include placing ribbons by matching up wing pair wings on one side of the structure and working across to the other side or some other order.

[0306] Assembly can include assembly of ribbons into modules. The modules can then be connected to form a larger structure.

[0307] Embodiments provide for means of joining wings together to physically connect a wing pair using means including but not limited to adhesive bonding, welding, brazing, melting, gluing, sintering, using fasteners, rivets, bolts, nails, screws, latches, buckles, catches, clasps, stitching, sewing, buttons, tying, self-fasteners, jigsaw puzzle tabs, hook and loop, hole and pin, closures, clamps, couplings, links, magnets, molecular bonds, Van Der Waals forces, etc.

[0308] Embodiments provide for means of joining ribbon ends together using means including but not limited to friction coupling with or without a retaining agent, interference coupling, adhesive bonding, welding, brazing, melting, gluing, sintering, using fasteners such as, rivets, bolts, nails, screws, latches, buckles, catches, clasps, stitching, sewing, buttons, tying, self-fasteners, jigsaw puzzle tabs, hook and loop, hole and pin, closures, clamps, couplings, links, magnets, molecular bonds, etc.

[0309] Embodiments provide for means of holding ribbon ends together, including by making the end of each first ribbon fit into place between the end of a second ribbon and a part of a strut tab of a third ribbon. The a-side of the first ribbon faces the b-side of the second ribbon. The second ribbon is in a sequence of connecting pairs with the first ribbon to form part of a weaver. The second ribbon can also be the first ribbon. The b-side of the first ribbon faces the b-side of a third ribbon with the wing pair wing. The third ribbon can also be the first ribbon or the second ribbon.

[0310] Embodiments provide for means of alignment of ribbon surfaces chosen from the group including but not limited to alignment holes, guide holes, recesses, and protrusions on the ‘b’ sides of the ribbon wings that fit together.

[0311] In embodiments, each wing on each ribbon is identified with a code or indices to indicate wing pair and connecting pair. In embodiments, the wings and ribbons are tagged and tracked bya code and / or by indices that indicate the wing pair, connecting pair and face. In embodiments, the tagging is done by numbers, letters or symbols, colors, QR tags, RF tags, bar codes, etc.

[0312] In accordance with an embodiment, the dihedral angle at the bending axis associated with each matching pair in each ribbon can be imparted by a great variety of means. These means include but are not limited to by using a brake, a bending or roll machine, heating and forming, casting, curing epoxy, molding, 3D printing, and various additive manufacturing means such as 3D printing or laser sintering, etc. In an embodiment, carbon fiber composite ribbons or parts of ribbons are fabricated by curing in a mold and / or in a vacuum bag with the dihedral angle set in place. In an embodiment, prepreg carbon fiber composite fabric is cured in an autoclave or oven with vacuum capabilities to form the bent ribbons. In an embodiment, a hinge material connects two matching pair wings.

[0313] In accordance with an embodiment, the straight wing method is used to make wings in a wing pair that have the same constant width and ribbons include matching pairs of wings that are straight strips.

[0314] In accordance with an embodiment, the ribbons include contiguous surfaces including sequences of matching pairs and connecting pairs to form parts of or whole weavers. The designs for the fabrication of ribbons are based on the shape of the contiguous surfaces of sequences of matching pairs and connecting pairs. Each contiguous surface is flattened into a plane by making each dihedral angle 180 degrees. Each flattened contiguous surface shape is the design for a ribbon and the design includes the location of each bending axis, the wing pair indices, and the dihedral angles between faces from the polygon mesh as well as optional guide holes. The design is used to fabricate the ribbons by casting, molding, forming, mechanical cutting from sheets, laser cutting, water jet cutting, plasma cutting, etc. the ribbon material. Ribbons are bent and placed andoptionally connected to assemble the structure. Each fabricated ribbon is bent, if not fabricated already bent, at each bending axis by the indicated dihedral angle. Each ribbon is placed according to the placement of its associated wings in the polygon mesh. Ribbons are placed by placing wings according to their wing pair indices so that they each share a strut tab surface with their wing pair wing in the same or other ribbons. This is done by placing the b-side surface of each wing facing the b-side of the other wing in each wing pair with the bending axes aligned. Guide holes can be used to align wing pairs or the shape of wing edges can be used as a guide for placement. If needed, ribbons are threaded through a partial structure. The ribbon ends overlap each other at their connecting tab surfaces to form parts of or whole weavers. In embodiments, the ends of the ribbons are placed such that each first end of each ribbon overlaps a second end of the same or another ribbon. The ends are placed so that the a-side of the connecting tab at the first end shares a surface with the b-side of the connecting tab at the second end. One or more ribbons are placed in a sequence to complete a weaver. When a polygon mesh has one or more open polygons, then there are one or more open weavers. If a weaver is not open, then the first end of the first ribbon overlaps the second end of the last ribbon. All the ribbons are bent, placed, and optionally connected to complete the structure.

[0315] Each end of each ribbon overlaps an end of the same or another ribbon at a shared connecting tab surface. In embodiments, the ends of the ribbons are connecting tab surfaces and one wing on each end overlaps. In embodiments, two wings in a connecting pair make up the ends of ribbons and the ends overlap at their connecting tab surface as well as their two strut tab surfaces. In an embodiment, each first wing at an end of a ribbon is held in place at its strut tab surface in a sandwich between a strut tab from an overlapping second wing at an end of a ribbonand a strut tab from the wing in the wing pair with the first wing. In embodiments, the ribbon ends are held in place by friction or by an interference coupling.Features and Advantages

[0316] In an embodiment, the present invention is a method to design, fabricate and assemble ribbons to form a 3D woven structure in a shape represented by a polygon mesh. In an embodiment, the present invention is a 3D woven structure of ribbons in a shape represented by a polygon mesh. In an embodiment, the present invention is a structure with the topology of a 3D woven structure of ribbons in a shape represented by a polygon mesh.

[0317] A feature of embodiments is that a 3D woven structure can include internal structure such as in a lattice or space frame, in a free form that is internally subdivided and are represented by polygons in the interior, and in articles with internal compartments. The structure can include external structures such as fins. A feature of embodiments is that the structure is based on a polygon mesh which has an edge shared by three or more polygons. This arrangement allows there to be internal or external structure. An advantage is that a wide range of shapes is available to embodiments including freeform shapes. An advantage of internal structure is that it can significantly increase the strength and stiffness of the structure.

[0318] A feature of embodiments is that at least one end wing of at least one ribbon in the structure is in a connecting pair of wings. This feature is distinct from the previous application, WO 2023 / 069785, wherein weavers comprise connected matching pairs of wings, which means that each end wing of each ribbon is in a matching pair of wings, and none are in a connecting pair of wings. An unexpected result of a ribbon with end wing that is in a connecting pair of wings is that the end wing can be placed with another wing overlapping it in such a way that it is held in place without the need for bonding, adhesive, welding, etc. There are several ways this can be done.

[0319] One way is by fitting an end wing of a ribbon in a pocket between two other wings that are a wing pair. A first wing is an end wing of a ribbon that is in a connecting pair with a second wing. The first wing is placed in a pocket between a third wing and its wing pair wing. The first wing and the third wing are twin wings. The third wing is in a connecting pair with a fourth wing and is part of the same or another ribbon as the ribbon of the first and second wings. The fourth wing can also be an end wing of a ribbon but it does not have to be. The fourth wing is placed in a pocket between the second wing and its wing pair wing. The second wing and the fourth wing are twin wings. Twin wings use the same wing design but can be part of the same or different ribbons. A feature of the resulting structure is two adjacent layered arrangements of three wings. A middle wing is in a pocket between two outer wings that are in a wing pair and the middle wing is a twin wing of one of the outer wings of the layered arrangement. The adjacent layered arrangements of three wings together with the bends in the ribbons at their bending axes lock the ribbons in place by means of mechanical interference. An advantage is that the structure is held in place without the need for bonding, adhesives, welding, etc. In WO 2023 / 069785, each ribbon end is a wing in a matching pair of wings. This means the ribbon end does not extend to a pocket between wings in a wing pair and so cannot form a mechanical interference coupling. In WO 2023 / 069785, wing pair wings are connected by means of bonding, etc. , thereby holding the ribbons in place.

[0320] Another way is by overlapping ribbon ends with each other in a cyclic pattern, thereby forming an interlocking flap closure. An example of a type of interlocking flap closure is that formed by four overlapping flaps of a cardboard box, wherein each flap is under the next flap. For an interlocking three flap closure, three ribbon ends come together in a triangle of the polygon mesh. A triangle has two faces and six wings. A first wing is an end wing of a ribbon and is in a connecting pair with a second wing. The first wing is placed under a fourth wing that is aconnecting pair with a third wing that is an end wing. The third wing is placed under a sixth wing that is a connecting pair with a fifth wing that is an end wing. The fifth wing is placed under the second wing. The first and fourth wings are a wing pair. The third and sixth wings are a wing pair. The fifth and second wings are a wing pair. The triangular face can be non-planar. Some bending may be required to fit a ribbon end under another ribbon. Any number of flaps can be part of the interlocking flap closure. An advantage of an interlocking flap closure is that the structure is held in place without the need for bonding, adhesives, welding, etc. Mechanical interference of the ribbons holds them together to form the structure. In WO 2023 / 069785, each ribbon end is a wing in a matching pair of wings and it is not possible to form an interlocking flap closure with these ribbon ends.

[0321] Another advantage of ribbons ending in connecting pairs of wings is that it allows for assembly of modules into a larger structure, wherein each module is a 3D woven structure with free ribbon ends. Modules are assembled and connected by means of their free ribbon ends. A free ribbon end of a first module is a connecting pair of wings that has wing pair wings in a second module. A ribbon end of one module slides into a pocket or sleave formed by ribbons of another module thereby connecting the two modules and forming a larger structure. A pocket is formed between wings in a wing pair. The connecting pair of wings at a free ribbon end can be in the shape of a hook. In an embodiment, a free end of a ribbon is a connecting pair that is in the shape of a hook formed by the concave boundary of the connecting pair of wings. A free ribbon end of one module can mechanically couple with ribbons of another module to form a larger structure. An advantage is that hooks lock modules in place without the need for bonding, adhesives, welding, etc. In WO 2023 / 069785, there are no free ribbon ends.

[0322] A feature of embodiments is that one or more ribbon ends can have a side-wing. Each side wing is connected to the connecting pair wing of an end wing. This means it is connected to the second to last wing of a ribbon end. An advantage of side-wings is that modules can be formed wherein the side-wings of one module slide into pockets formed by wing pair wings on other modules to assemble a larger structure from several modules. In WO 2023 / 069785, there are no side wings.

[0323] A feature of embodiments is that the polygon mesh includes one or more triangles. The resulting structure includes struts of ribbons in triangular arrangements. An advantage of struts in triangular arrangements is improved mechanical stability, stiffness and strength, less weight or both.

[0324] A feature of embodiments is that the polygon mesh includes one or more tetrahedrons that can be degenerate. The resulting structure then includes struts in tetrahedral arrangements or tetrahedral plate cells, depending on the shape of the wings. An advantage of struts in tetrahedral arrangements or tetrahedral plate cells is improved mechanical stability, stiffness and strength.

[0325] A feature of embodiments is that polygons in the polygon mesh have no holes or a small hole in the surface. An advantage of no holes or small holes is that ribbons can interlock more tightly without bonding or fasteners to stay in place.

[0326] A feature of embodiments is that polygons in the polygon mesh have one or more holes in the surface. An advantage of holes is that the resulting structure has lightening holes to reduce weight.

[0327] A feature of embodiments is that the structure is based on a polygon mesh with degenerate polygons. An advantage of this is that structures can be designed with less wings, meaning fewercomponents to fit in place. An advantage of this is that modules based on degenerate polyhedrons can be used to efficiently assemble larger structures.

[0328] A feature of embodiments is that the ribbons are of sufficiently thin material that they have some pliability. An advantage of this is that they can be deformed to assist with interweaving and placing ribbons during assembly. Another advantage is that the material can snap back and be held in place mechanically.

[0329] A feature of embodiments is that the thickness and material of the ribbons can be individually tailored. This can customize the local properties of the structure or result in structures with different properties in different places of the structure. Another feature is that the width along the length of each ribbon can be kept approximately constant. This can simplify fabrication of the ribbons which can be cut from a strip or roll of tape of material. In embodiments, a feature is that the ribbon edges are curved for improved performance or aesthetic purposes. An advantage is that the strength or some other mechanical property such as elasticity of the structure can be predetermined locally.

[0330] Embodiments of the present invention include struts which are arrangements of one or two or more ribbons at each of the edges of the polygon mesh. The ribbons at each edge share strut tab surfaces, and couplings or bonding optionally joins them together. An advantage is that structural features are coupled by means of interwoven ribbons and so can improve the resiliency of the structure.

[0331] A feature of embodiments is that the cross-sectional profile of stmts at shared edges includes members that emanate from axes. This type of profile is efficient in preventing buckling because the radius of gyration can be higher than that of a tube. An advantage of this is that the overall structure can be lighter or stiffer and stronger or both.

[0332] A feature of embodiments is that the structure has surfaces connected to other surfaces in other planes. An advantage of this is that the structure resists twisting and torsion.

[0333] A feature of embodiments is that articles can be made to be continuous or integrated, meaning that different parts of the article or structure are integrated into a continuous structure. In accordance with an embodiment, a structure is defined from a polygon mesh. For example, the structure could be the airframe for an airliner. The wings and the fuselage structures can be integrated as one due to the interwoven ribbons. This can have structural advantages such as resilience and stiffness. Another example is for a building with improved earthquake resistance. The ribbons interweave the structure of the building framework thereby making it more resilient.

[0334] A dihedral angel is defined as the angle on the ‘a’ side between the tangent planes of the ‘b’ sides at the shared edge of each matching pair of wings in a ribbon. The dihedral angle is greater than or equal to 0 degrees and less than or equal to 360 degrees. For a flat matching pair of wings with no bend, the dihedral angle is 180 degrees which means the surfaces of the wings are in the same plane. In embodiments, struts associated with an edge at the boundary of a surface mesh include one ribbon bent onto itself such that the ‘b’ sides of the wings in a matching pair at the boundary edge are facing towards each other and so have a dihedral angle of 360 degrees.

[0335] A strut is associated with each edge in the polygon mesh and includes one or more matching pairs and one or more wing pairs of one or more ribbons that share the edge. The dihedral angles of the matching pair of wings in the ribbons making up the struts are determined from the polygon mesh and are such that the sum of the dihedral angles of the matching pairs making up the strut is equal to 360 degrees. Parts of the strut tab surfaces of the wings making up each strut are attached ‘b’ side to ‘b’ side such that their bending axes are parallel with each other. The axis of each strutis the shared edge in the polygon mesh and is substantially parallel to the bending axes of the matching pairs making up that strut.

[0336] In embodiments, the ribbons have notches removed at comers corresponding to the ends of the bending axes, where the vertices in the polygon mesh are located. An advantage of this is to allow for bending and fitting ribbons with a finite thickness at adjacent edges without interfering with each other.

[0337] In embodiments, the wings of each face extend to the center of the face to leave no substantial openings. In embodiments, the wings of each face do not extend to the center of the face, thereby leaving a hole. In embodiments, wings have holes in them. In embodiments, the structure has panels covering openings near the center of polygons in the mesh. An advantage of this is that this arrangement of fitting panels into or over the openings over a surface can be used as a cladding system for the structure, such as for a building or an aircraft skin over an airframe.

[0338] In embodiments, one or more ribbons are interwoven to approximate a structure, such as the form of a surface or the volume of a structure. The surface can have any shape in three- dimensional space and can enclose a volume or be open. The shape of the structure of interconnected polygons can be made to approximate any shape by determining the design of the polygon mesh and each individual polygon, including the number of wings, length, shape and dihedral angle of each wing or design equations for each wing. In an embodiment, the surface or volume can be defined mathematically, computationally, graphically, parametrically, manually, by scanning, etc. In an embodiment, ribbon design parameters such as wing shape, width, thickness, boundary shape and dihedral angle are predetermined using geometric calculations which are performed by means of a computer. The structure can be extended in any three- dimensional direction by adding additional polygons to surfaces at the boundaries of the structure.An advantage of using a polygon mesh is that it enables structures of a great variety of shapes including lattice materials and freeform structures.

[0339] In an embodiment, a target structure has a desired three-dimensional shape. The volume defined by the shape is subdivided into polyhedrons so that the polyhedrons fit together to approximate the volume and describe the shape. In an embodiment, the volume can be subdivided into the cells of a 3-D mesh. In embodiments, the cells are polyhedrons. The cells can enclose a volume or have openings or edges. The polyhedrons include two or more polygons. The surface of the shape is approximated by the outside polygonal faces of the set of polyhedrons, known as surface faces. In embodiments, the structure approximates volumetric shapes, layers or lattices in three-dimensional space by subdividing it into tetrahedrons and / or other polyhedrons. There can be multiple polyhedrons that fit together to approximate the shape. The faces of the polyhedrons are either surface faces or are within the volume of the shape and are shared by or are coincident with the face of an adjacent polyhedron. The faces of the polygons within the body of the shape are referred to as inside faces. Once the size, shape and surface of each polygon in each polyhedron is determined, the wing dimensions, and dihedral angles can be calculated. The dihedral angles are determined by the angles between the faces of the polyhedrons or polygons. The plane of a wing is substantially parallel to the plane of the surface of the polygon of which the wing is an edge, even when the polygon is non-planar or skew.

[0340] In embodiments, a collection of connected polygons or polyhedrons is repeated to form a larger sheet of lattice material or a space frame. The lattice material can have any geometry including simple cubic, centered cubic, octahedral, octet, cube-octet, gyroid, etc. An advantage is that the structure can be a lattice material, a cellular material, a metamaterial, etc.

[0341] In accordance with an embodiment, polygons are chosen from a group including triangles, quadrangles, pentagons, hexagons, heptagons, octagons, nonagons, decagons, hendecagons, dodecagons and other polygons.

[0342] In embodiments, the polyhedrons are chosen from a group including, tetrahedrons, pyramids, pentahedrons, hexahedrons, cuboids, heptahedrons, octahedrons, nonahedrons, decahedrons, hendecahedrons, dodecahedrons, and polyhedrons with more than twelve faces.

[0343] In embodiments, the polyhedrons are chosen from a group including, polyhedrons with curved surfaces such as dihedrons, curved-surface polyhedrons and hosohedrons.

[0344] In embodiments, the ribbon material can have a wide range of properties selected from a group including but not limited to rigid, flexible, elastic, electrically conducting, semiconducting, insulating, translucent, opaque, transparent, and reflecting. The ribbon material can be made of many types of materials chosen from the group including but not limited to metals, alloys, fiber composite, carbon fiber composite, graphene, carbon nanotubes, ceramics, polymers, copolymers, textile, paper, cardboard, wood, leather, stone, concrete, sandwich core materials, honeycomb core plates, natural, foam, elastomeric, alloys of metals, and glasses. An advantage of this is that mechanical, electrical and optical properties of the structure can be changed by changing the ribbon material.

[0345] Different ribbons in the same structure can be of different materials. For example, a structure can comprise ribbons of polymer and ribbons of metal interwoven to form a polymer- metal structure. For example, a structure can comprise ribbons of ceramic and ribbons of metal interwoven to form a ceramic-metal structure with high strength and toughness.

[0346] In embodiments, the structure is assembled by robots.

[0347] In embodiments, the bends at each bending axis in the ribbon can be made by means of a number of methods, chosen from the group including but not limited to by bending a sheet of material using a brake or some other physical means, by applying heat and bending, by casting, by attaching two separate plates at an angle, by casting, by using prepreg carbon fiber composite material curing with a mold, heat and / or vacuum or some other form of compression using methods known to a person skilled in the art. When a bend is imparted on the material a bend allowance is used to adjust the ribbon design at each bending axis to ensure a good fit.

[0348] In embodiments, the shape of the structure is spherical or close to spherical. In an embodiment, the shape of the structure is cylindrical. In an embodiment the structure has a freeform shape. In an embodiment, the structure has branches or legs extending from a volume or between volumes. In an embodiment, the shape of the structure is chosen from the group including but not limited to flat planes, curved planes, and free-form surfaces and combinations thereof.

[0349] Embodiments provide for a method of determining the number and design of ribbons including, dihedral angles, wing shapes, and geometric associations between ribbons for a predetermined shape.

[0350] In embodiments, the wings have a great variety of shapes. Embodiments provide a method for determining the size and shape of the wings. In embodiments, wings are shaped such that their connecting tab and the strut tab make up their entire surface, i.e. with no gap between them. This means that when the ribbons are interwoven, no ‘b’ side surfaces are visible. The wing shapes can take on a myriad of shapes. They can be substantially triangular, quadrangular, pentagonal, etc. or can have curved edges. Shapes can be chosen for their decorative effect or so that they interlock in some way, for example like jigsaw puzzle pieces and the like. The wing shapes can be such thatwhen joined, they resemble a weave pattern. This can be beneficial when the structure is used for architecture or clothing or sports equipment.

[0351] Various embodiments of the structure and methods of making same can provide some or all the following advantages:• The structure is light yet strong and resilient.• The structure has, no fasteners, couplings or bonding.• Design, fabrication, manufacture and assembly are simplified by the ribbon design because ribbons can be cut or stamped and / or manufactured by computer numerical control (CNC) methods such as laser cutting of sheets of material.• The repetitive nature of the assembly with limited steps means that robots can be used.• There are no nodes to complicate fabrication and assembly and to increase cost.• The structure can be made from a great variety of materials.• The structure can have a great variety of shapes such as free-form, lattice and branched volumes and are easier or faster to construct than with traditional methods such as 3D printing.• The structure can have different parts continuously integrated by ribbons.• The ribbons can be cut from flat sheets, which is relatively easier and cheaper than alternatives such as frames of tubes connected by nodes or forming a monocoque.• The shape of the ribbons and struts can be individually designed for lighter weight, higher stiffness, easier manufacture, aesthetics, etc.• The structure can be on a great range of length scales.The structure is repairable.• The structure presents a flat surface for attachment of cladding or other components or equipment.• The structure can be built to a predetermined shape or on-the-fly without a predetermined shape.

[0352] Embodiments of the present invention can be used for a range of purposes. Some example applications using the structure may include but are not limited to the following:• As a space-frame, truss, and the like.• As a lattice material.• As part of an aircraft fuselage, wing, body, airframe, etc.• As part of an architectural structure (floors, walls, pillars, building structure, outer cladding).• As part of a fixed structure such as a tower, a dome, an airport building, a parking structure, a stadium, public buildings, housing, a theatre, tunnel cladding, spans / decking of a bridge, and the like.• As the frame for a crane, a stage and the like.• As part of a hyperloop tube.• As a pressure vessel, fuel tank, etc.• As the frame structure for a machine, a robot, a lift, etc.• As a frame, chassis, body panel, of a vehicle, automobile, train, bus, tram, etc.• As a marine structure, a body, hull, frame, deck, partition, etc. of a ship, submarine, yacht, oil / gas platform, etc.• As a rocket frame, body, fuel tank, etc.As part of personal protective gear, such as a helmet.As part of sports equipment.• As part of a space craft structure.• As a blast, ballistic, shock, or impact resistant structure.• As a prosthetic or graft implant.• As a heat exchanger.• As a battery.• As the structure for a vacuum balloon.• As furniture.• As clothing or shoes.• As a toy, game or puzzle.• As a model for another structure.• As part of artwork, sculpture or an installation.

[0353] While embodiments of the invention have been shown and described, modifications thereof can be made by one skilled in the art without departing from the spirit and teachings of the invention. The embodiments described and the examples provided herein are exemplary only, and are not intended to be limiting. Many variations and modifications of the invention disclosed herein are possible and are within the scope of the invention. Accordingly, other embodiments are within the scope of the following claims. The scope of protection is not limited by the description set out above, but is only limited by the claims which follow, that scope including all equivalents of the subject matter of the claims.

[0354] The disclosures of all patents, patent applications, and publications cited herein are hereby incorporated herein by reference in their entirety, to the extent that they provide exemplary, procedural, or other details supplementary to those set forth herein.Definitions

[0355] In addition to the terms as utilized and defined above, the following terms used herein shall be defined as follows.

[0356] As used herein, the term “vertex” shall mean a point in 3D space whose position is identified by coordinates such as Cartesian coordinates in 3D space. A set of vertices can approximate or define a shape in 3D space. The vertices can define the positions of comers or key features of a target structure. Each vertex, v(i), in a set v(l), v(2), . .. , v(N) is identified by its vertex index i, which is an integer between 1 and the total number of vertices, N. FIG. 1A depicts a set of 10 vertices, v(l), v(2), . . ., v(10). FIG. 2A depicts a set of 12 vertices, v(l), v(2), . . ., v(12).

[0357] As used herein, the term “edge” shall mean the edge structure that connects two vertices, also called its endpoints. An edge is represented by a pair of vertex indices associated with the connected vertices. Edges are referred to by the indices of the two vertices that the edge connects. For example, (2,6) refers to edge 204 between v(2) and v(6) as shown by in FIG. 2A. There can be more than one edge connecting the same two vertices. There is associated with each edge two opposite “directed edges,” which point in opposite directions from each of the vertices to the other.

[0358] As used herein, the term “polygon” shall mean a sequence of two or more edges connected at vertices. Each edge connects two vertices at each end. In an embodiment, a polygon is a triangle with three vertices and three edges. Polygons in the polygon mesh can be selected from the group consisting of triangles, quadrangles, pentagons, hexagons, heptagons, octagons, nonagons, and decagons, hendecagons, dodecagons, polygons with more than twelve edges, and combinations thereof.

[0359] A polygon can be planar or non-planar, open or closed. A polygon can be a polygon with holes which is a polygon with one external boundary and one or more interior boundaries. Eachpolygon has a surface whose boundary coincides with the edges of the polygon. The surface can be planar or non-planar, curved or flat or faceted, open or closed. Polygons can have surfaces that bulge or are otherwise curved or faceted where the facets are not all coplanar. Planar polygons can have non-planar surfaces that bulge or have other surface curvature features.

[0360] As used herein, “directed edge” shall be defined as follows: each edge has associated with it two directed edges, one for each direction from one vertex to the other. Each pair of adjacent vertex indices in the ordered vertex list of each face of each polygon defines a directed edge, pointing from the first vertex in the pair to the second. For the faces of closed polygons, the last and first vertices are also a pair because the list cycles around to define a directed edge pointing from the last to the first vertex. The two directed edges associated with an edge connecting vertices with indices il and i2 are given by ordered lists {il, i2} and {i2, il } and are termed opposite directed edges because they point in opposite directions. The direction of the directed edge {il, i2} is from v(i 1) to v(i2). If a face of a closed polygon has vertex index list {il, i2, i3 }, then it has three directed edges, {il, i2}, {i2, i3} and {i3, il }.

[0361] As used herein, the term “polygon mesh” shall mean an assembly of polygons. It can be used to define or approximate the shape of a structure. A polygon mesh is also referred to as a polyhedron because it has many polygons. A polygon mesh is an arrangement of adjacent polygons that are defined by vertices and edges. The polygons can be connected by means of shared edges, which renders these as “adjacent polygons.” FIG. 2A depicts polygon mesh, 200, with a set of twelve vertices connected by edges in the form of three connected polygons by means of three shared edges.

[0362] Two or more polygons can also share one or more edges. That means, for example, that three polygons can share one edge and two polygons can share two edges. FIG. 1A depicts polygonmesh 100 including two edges shared by three polygons each. FIG. 6A depicts a polygon mesh including 3 sets of 5 edges shared by two polygons each. A polygon mesh can include parts that are concave, convex, open, closed and / or internally subdivided by polygons. Polygon mesh surfaces can be non-planar, curved or flat or faceted. Polygon meshes can have degenerate polygons, that is two or more polygons can share the same set of edges. The order of a degenerate polygon is defined as the number of polygons that share the same set of edges. An ordinary polygon is a degenerate polygon of order one. An example of a polygon mesh with degenerate polygons is a dihedron with two polygons sharing all their edges. One or more of the degenerate polygons can be non-planar so that the surfaces of the degenerate polygons have some volume between them. The surfaces of degenerate polygons can bulge away from each other. The non-planar polygons can be faceted by subdividing the surface of the non-planar polygon into planar facets. This is done by introducing a new vertex called a subdivision vertex on the surface of the polygon, for example by placing a new vertex at a center of the polygon surface. A triangular facet can be formed from each edge of the non-planar polygon and the lines from the two vertices of each edge to the subdivision vertex. A polygon mesh can include smaller polyhedrons that are connected at shared faces, shared edges and / or shared vertices or not at all. Polyhedrons that are connected at shared polygons are adjacent polyhedrons. A polygon mesh can be decomposed into smaller adjacent, non-overlapping polyhedrons or polygons. The vertices of the smaller polyhedrons are vertices of the original polyhedron and additional vertices that are added in the volume of the polygon mesh. Tetrahedralization, where the volume of the shape is composed of adjacent tetrahedrons, is an example of this method. In an embodiment, polygons of the mesh are triangles. Like other polygons, triangular polygons can have non-planar surfaces and can have holes. The polygon mesh can have fins, which are polygons that are attached to the rest of the polygon meshby means of an edge shared with two or more polygons of the polygon mesh and have one or more edges not shared with any other polygon. A polygon mesh can be derived from a graph in mathematics, where cycles in the graph are closed polygons in the polygon mesh. A polygon mesh can be obtained by triangulation of a surface or tetrahedralization of a volume.

[0363] As used herein, the term “degenerate polyhedron” shall mean a polyhedron wherein one or more of the polygons making up the polyhedron is a degenerate polygon to order one or more. A tetrahedron is a degenerate tetrahedron to order one. As used herein, the term “tetradihedron” shall mean a tetrahedron wherein each of its four triangular sides is a dihedron.

[0364] As used herein, the term “face” shall mean an oriented surface (surface with an outward direction) associated with the opposite sides of a polygon, z.e., each polygon has two faces. Each face of a polygon has a sequence of directed edges, each pointing to the tail of the next directed edge in the sequence, a surface that is the surface of the polygon, and an outward direction pointing. The two faces of a polygon have the same surface as the polygon surface but are associated with opposite sides. For a planar, closed, polygon, the surface can be the plane bounded by the edges and vertices of the polygon, but the surface can also be curved or faceted. For a non-planar polygon, the surface can be curved or faceted. The surface of the face can be faceted into triangles by selecting a point in the surface and then constructing triangles from adjacent vertices of the face and the point. Each face is represented by an ordered list of vertex indices identifying adjacent vertices going around the polygon. Each face has a direction associated with it that is indicated by the order of the list of indices.

[0365] Unless otherwise stated, the following rule has been utilized for the order that the indices are listed. They are listed in order going around the polygon in the counterclockwise direction with the viewer facing the face on the side of the polygon with which the face is associated.Another way of stating this rule is as a right hand-thumb rule. Curl the fingers of the right hand and hold the thumb up in a “thumbs up” gesture pointing in the direction of the outward direction of the face. Then the curled fingers point in the direction of vertex indices going around the face counterclockwise. For a closed polygon, any vertex can be the first in the list, with the next index in the list indicating the next vertex in the counterclockwise direction, etc. Each polygon has two faces, one on either side. Opposite faces of the same polygon share the same surface but have opposite outward directions. Therefore, opposite faces of the same polygon have vertex lists containing the same indices but in reverse order. For closed polygons, it does not matter with which vertex the list starts. For open polygons, the ordered list starts at one end of the polygon for one face and at the other end for the opposite face. In embodiments, the opposite rule is used where a clockwise ordered vertex list is used and a left-hand thumb rule is used to indicate the outward normal direction.

[0366] As used herein, “index notation” shall be defined as follows: let f(j) denote a face having index j and given by the ordered list of vertex indices using the right-hand thumb rule. Here j is an integer referred to as the face index, which identifies each face and can be from 1 to the total number of faces. f(j) = {i 1 , i2, ..., iN(j)} , where il, etc. are the vertex indices of the vertices going around face f(j) in the counterclockwise direction and N(j) is the number of vertices in the face, so iN(j) is the index of the last vertex in the face. For a triangular face, N(j) is 3 and f(j) = {il, i2, i3}. A after the face index denotes the index of the opposite face of the same polygon. Then j* denotes the index of the face opposite to face f(j). The ordered vertex index list of f(j*) is the reverse of that of the list of f(j), and f(j*) = {iN(j), i(N(j)-l), ..., il }. For example, the polygon mesh depicted in a view from above in FIG. 2A has a pentagon with face 206 denoted by f(2) with vertex list { 10, 7, 11, 1, 9}. It does not matter at what vertex the ordered vertex index list of a facestarts, so f(2) could have been denoted as {7, 11, 1, 9, 10}. Polygon mesh 200 is depicted from below in FIG. 2B with opposite face 208 denoted by f(2*) with vertex list {9, 1, 11, 7, 10}. FIG. 2C depicts polygon mesh 200 in a perspective view slightly from above, showing counterclockwise direction, 210, of the face f(2) and outward direction, 212, pointing upwards in accordance with the right hand-thumb rule. FIG. 2E depicts polygon mesh 200 in a perspective view slightly from below. The opposite face f(2*) to f(2) of the pentagon has outward direction, 214, pointing downwards as depicted in FIG. 2E.

[0367] As used herein, an edge is “adjacent” to another edge if the edge and the adjacent edge are of the same polygon and share a vertex. For example, as shown in FIG. 2A, in the pentagon { 10, 7, 11, 1, 9}, (11,1) is adjacent to (7, 11). While (10,7) is also an adjacent edge to (7,11), (10, 7) is not an adjacent edge of (11,1) in that (10, 7) and (11,1) do not share a vertex. Two vertices are adjacent if they are connected by an edge.

[0368] Two edges are adjacent if they share a same vertex. Two polygons are adjacent if they share a same edge. Two directed edges are adjacent if they point to a same vertex. Directed edge {z, y} is adjacent to directed edge {x, y} because they both point to vertex y. Two polygons are adjacent if they share a same edge. Two faces are adjacent if they have one or more shared edges and have outward directions that point into the same volume, where pointing into the same volume means that there is no other face that shares the same edge and is between the two faces. The face {x,y,z} is adjacent to the face {y,x,w} if there are no other faces dividing the volume between {x,y,z} and {y,x,w}.

[0369] Two polyhedrons are adjacent if they share a same polygon. Two polyhedrons (v,w,x,y) and (v,w,x,z) share polygon (v,w,x) and are adjacent if (v,w,x) is not degenerate. If polygon (v,w,x) is degenerate then the two polyhedrons would have different polygons both indicated by the samevertex list (v,w,x). In that case since the two polyhedrons would have different polygons, they would not be adjacent. However, there would be a third polyhedron, which is a dihedron with vertex list (v,w,x), that is adjacent to the other two polyhedrons.

[0370] Two directed edges are opposite if they have opposite directions and are associated with the same edge. Each directed edge has exactly one opposite directed edge. The directed edge {y, x} is opposite to directed edge {x, yj.Two faces are opposite if they have opposite outward directions and are from the same polygon. The face {z,y,x} is opposite to the face {x,y,z}. For a polygon (x,y,z) with no other polygon sharing edge (x,y), the face{z,y,x) is both opposite and adjacent to face {x,y,z} and the outward directions of the face and the opposite face point into the same volume, i.e. the volume around the polygon.

[0371] As used herein, the term “wing” shall mean a surface in 3D space obtained by segmenting or subdividing the surface of a face. A shape and dimension of each wing is determined by subdividing each face. The surface of each face is subdivided such that each wing has part of the subdivided surface, the outward direction of the face and one directed edge of the face. There is one wing associated with each directed edge of each face. The directed edge of a wing is called the wing edge. The total number of wings (“Nw”) is equal to the sum of the directed edges of all the faces. Each wing is identified by the index of the face and the index of the first vertex in the directed edge with which it is associated. Consider a face with index j 1 and with vertex list {il, i2, i3, ..., iN}. The wing associated with the first directed edge, {il, i2} in face f(j 1) is denoted w(j 1, il). The wings for each face are obtained by subdividing the surface of the face in such a way that each wing satisfies the following requirements: (a) its surface boundary includes the directed edge with which it is associated, (b) its surface boundary does not cross the boundary of another wing associated with the same face, and (c) its surface or boundary is partly shared with the surface orboundary of the wing associated with the adjacent directed edge and subdivided from the opposite face. Sharing part of a surface includes coinciding boundaries of surfaces. This means that part of the surface or boundary of a wing coincides with part of the surface or boundary of the next wing from the opposite face j 1 *. Each wing has an outward direction that is the outward direction of the face. If the face is non-planar then the outward direction is the local outward direction of the face. A method of subdividing each face is by introducing one or more new points on the surface of the face, known as subdivision or intersection points. Each wing is a triangular or other polygonal surface comprising a wing edge and one or more of the one or more subdivision points. The subdivision can include adding a subdivision vertex on the surface of the face, wherein the wings are triangular with vertices including the subdivision vertex and two adjacent vertices of the face. In an embodiment the same subdivision points are used on a face and the opposite face. Parts of the boundaries of the wings of the face can coincide with parts of the boundaries of wings of the opposite face.

[0372] Each wing has a “wing direction” which is the direction of the directed edge with which the wing is associated and is the direction in which the wing points. Each wing has an outward direction which is the outward direction of the face from which the wing is subdivided. Requirement (c) above means that there is a point in the boundary of each wing surface that is in the surface or boundary of the wing with the adjacent edge and associated with the opposite face. A new vertex with index k(j 1, il), is defined on the boundary of wing w(j 1, il) such that it is on the surface or on the boundary of the wing identified by w(j 1 *, i 3 ), where i3 is two vertices ahead of il in the counterclockwise direction of face j l. The new vertex is identified by index k(j 1, il) and is identified with vertex il in face j l. If il is the second to last or last index in the ordered vertex list of the face, then cycling around to the beginning, i3 becomes the first or second index,respectively. Each wing w(jl, il) is associated with a surface defined by the ordered vertex index list {il, i2, k(j l, il)} going around the wing boundary in the counterclockwise direction, wherein12 is the index of the next vertex in the ordered vertex list of face j 1. The outward direction of the wing can be identified by using the right-hand thumb rule with the ordered index list and is the same as the outward direction of the face j 1 from which it was subdivided. Requirement (c) also means that there is a point identified by index k(j l*, i3) on the boundary of wing w(j l*, i3) associated with the opposite face, j l*, and directed edge {i3,i2}. Wing w(j l*, i3) is associated with the ordered vertex index list {i3, i2, k(j 1*, i3)} going around the wing boundary in the counterclockwise direction when facing the opposite face, j l*. This is repeated for each wing of each face. Each wing has an outward direction which is the same as the outward direction of the face from which it was subdivided. For example, FIG. 2D depicts polygon mesh 200 in FIG. 2A in a view from above. The intersection points such as k(j 1, i 1), associated with vertex index i 1 in face f(j 1), are shown for the faces j l, j2 and j3 that have outward directions pointing up. Wing, w(j l, il), is shown by the triangle {il, i2, k(j 1, il)}. FIG. 2F depicts polygon mesh 200 in FIG. 2A in a view from below. The intersection points such as k(j 1 *, i3), associated with vertex index13 in face f(j 1 *), are shown for the faces j l*, j2* and j3* that have outward directions pointing down. Wing, w(j l*, i3), is shown by the triangle {i3, i2, k(j l*, i3)}. FIG. 2G depicts polygon mesh 200 from above showing the wings and intersection points from the faces on both sides of each polygon. As can be seen, the intersection point k(j 1, il) is within the surface of the wing associated with the next edge but opposite face, w(j 1*, i3). Also, the intersection point k(j 1*, i3) is within the surface of the wing, w(j 1, il).

[0373] In an embodiment, wings are triangles with ordered index list, {il, i2, k(j 1, il)}, {i2, i3, k(j 1, i2)}, etc. In embodiments, part of the boundary of a wing is shared with that of a wing withan adjacent directed edge. In embodiments, there is one subdivision point k(j 1) that is shared by all the wings in a face j 1, and the wings are identified as {il, i2, k(j 1)}. In embodiments, wings are quadrilaterals with ordered index list, {il, i2, k(j 1, il), k(j 1, iN(j 1))}, where k(j 1, iN(j 1)) is the index of a new vertex in the wing associated with the directed edge in face j that points to vertex il. That is, the wing includes the vertex with index k(j l, iN(j 1)) from the wing of the previous directed edge. Requirement (b) for wings of the same face not overlapping means that the vertex with index k(j 1, iN(j 1 )) does not lie within the wing w(j, il). The wings can also be pentagons, hexagons, etc. or can have parts of the boundary that are curved.

[0374] As used herein, the term “connecting pair” shall mean a pair of wings that have adjacent directed edges from opposite faces of the same polygon, wherein the two directed edges point towards the same vertex. The two wings of a connecting pair have adjacent directed edges and are subdivided from opposite faces. This means that the wings in a connecting pair are associated with directed edges from opposite faces of the same polygon and share the same second vertex in their directed edge list. Because of requirement (c) for wings, a connecting pair is made of two wings that share parts of their surfaces or boundaries. These coinciding surface or boundary parts are referred to as connecting tabs. Therefore, wings w(j 1, il) and w(j l*, i3) are a connecting pair for the polygon with face j 1 that contains sequence il, i2, i3 in its ordered vertex index list.

[0375] A connecting pair of wings with shared vertex i2 is shown in FIG. 2H. The subdivision point, k(j 1, il), for wing w(j 1, il), is in the surface or the boundary of the wing w(j 1*, i3). Likewise, subdivision point, k(j 1 *, i3), for wing w(j 1*, i3), is in the surface or the boundary of the wing w(j 1, il). The outward directions of the wings are given by the arrows at the corresponding wing subdivision points and are in opposite directions. The part of the wing surface 220 shared by a connecting pair is referred to as the connecting tab.

[0376] As used herein, the term “connecting tab” shall mean the part of the wing surface shared by a connecting pair. Sharing a surface includes sharing a boundary. In FIG. 2H connecting tab 220 is shared by the wings in connecting pair w(j l, il) and w(j l*, i3). FIG. 21 depicts the same connecting pair (with connecting tab 220) in a view from above. The connecting tab is the region in the quadrilateral with vertices i2, k(j 1, il), c(j 1, i2), k(j 1*, i3). As shown here, a new vertex with index c(j 1, i2) is the intersection point of the edges of the wings for the connecting pair at vertex i2. s. In embodiments, the subdivision and intersection points of wings in a connecting pair are in the same location and the connecting tab is the shared wing boundary extending from the shared vertex i2 to the shared point.

[0377] As used herein, the term “wing pair” shall mean a pair of wings associated with the same edge from opposite faces of the same polygon. The two wings of a wing pair have opposite directed edges and opposite faces. Both wings are subdivided from the same surface, both have boundaries that include the same edge, and both have a part of the surface adjacent to the edge that is shared. The directed edges associated with the wings of a wing pair are in opposite directions. Therefore, wings w(j l, il) and w(j l*, i2) shown in FIG. 2 J are a wing pair for the polygon with face j l containing edge (il, i2). The number of wing pairs sharing each edge of the polygon mesh is the same as there are polygons sharing that edge. The outward directions of the wings are given by the arrows at the corresponding wing subdivision points and are in opposite directions. The part of the wing surface shared by a wing pair is referred to as the strut tab. Strut tab, 230, in FIG. 2H is the surface shared by wing pair w(j 1, i 1) and w(j 1 *, i2). FIG. 2K depicts the same wing pair in a view from above. The strut tab is the region in the triangular surface with vertices il, i2, s(j 1, il). Here a new vertex with index s(j 1, il) is the intersection point of the edges of the wings of the wing pair sharing edge (il, i2).

[0378] As used herein, the term “strut tab” shall mean the part of the wing surface shared by a wing pair. The term “shares a surface” can also mean that the surfaces are in contact, partly in contact or almost in contact with each other.

[0379] As used herein, the term “matching pair” shall mean a collection of two wings which have (a) a shared edge, and (b) outward directions that point into the same volume. The two wings of a matching pair have opposite directed edges and adjacent faces. Pointing into the same volume means that there is no other polygon that shares the same edge and is between the wings in the volume into which the outward directions of the wings point. The number of polygons sharing an edge is the valence of the edge. If the valence is one, the two wings in the matching pair are associated with opposite faces of the same polygon and the edge is a boundary of the polygon mesh. If the valence is two, the wings in the matching pair are associated with two faces on the same side of two polygons sharing the edge. If the valence is one or more, one wing in the matching pair is the first wing encountered when rotating around the shared edge as axis in the direction of the outward direction of the other wing. The angle of rotation between the two wings of a matching pair is the dihedral angle. Wings w(j 1, il) and w(j2, i2) subdivided from faces with adjacent vertices il and i2 are a matching pair if the outward directions of faces j 1 and j2 point into the same volume. If the shared edge is at the boundary of the polygon mesh, then the matching pair is also a wing pair, j2 is j 1* and the dihedral angle is 360 degrees. Each matching pair has a bending axis at the shared edge and a dihedral angle determined by the angle between the wings at the shared edge. Degenerate polygons with surfaces that bulge away from each other can have dihedral angles greater than zero at their shared edges because of the bulging.

[0380] Shown in FIG. 2L, for polygon mesh 200, is a matching pair consisting of w(jl, il) and w(j2, i2). The outward directions of the wings are given by the arrows at the corresponding wingsubdivision points. Both outward directions point into the volume above the polygon mesh. The matching pair on the opposite faces consisting of w(j l*, i2) and w(j2*, il) is shown in FIG. 2M. Both outward directions point into the volume below the polygon mesh. Each edge in the polygon mesh has the same number of matching pairs as there are polygons sharing the edge. The edges (3,4) and (4,5) are shared by three polygons each in polygon mesh 100 as depicted in FIG. 1A and have three matching pairs.

[0381] As used herein, the term “end wing” shall mean the wing at the end of a ribbon, which can be in a matching pair or in a connecting pair. In embodiments of the present invention, at least one of the ribbons has at least one end wing that is in a connecting pair.

[0382] As used herein, the term “twin wings” shall mean two or more wings of the same or different ribbons that are designed from the same wing in a weaver. Twin wings are in different places of the same or different ribbons of the same weaver. When the ribbons are placed, twin wings are in the same place in the polygon mesh. An end wing of a ribbon optionally has a twin wing which is a connecting pair wing of another end wing of the same or another ribbon designed from the same weaver. The advantage of this is that when assembled, the twin wings are placed on top of each other and the ribbon ends overlap. The overlapping of the ribbon ends holds the ribbons together by means of mechanical interference, without the need for bonding, etc.

[0383] As used herein, the term “side wing” shall mean a wing that is connected to the wing in the connecting pair wing of the end wing. This means that the second wing from the end of a ribbon is connected to the end wing in a connecting pair, can be connected to matching pair wing in the weaver sequence, and can also be connected to a side wing. A side wing of a ribbon of one structure or module can be placed into a pocket formed by two wings in a wing pair in another module thereby connecting the two modules. The shape of the side wing is similar to the shape ofthe wings in the wing pair so that it fits tightly in the pocket. Many modules with side wings can be connected by means of their side wings to form larger structures.

[0384] A compact wing notation refers to the wing associated with the i-th directed edge and the j-th face as W(i,j). W(i,j) is an element of the incidence matrix of the directed edges and faces of the polygon mesh. The following shorthand can be used: the connecting pair wing of W(i,j) is W(iA,j*), the matching pair wing of W(i,j) is W(i*,jA), and the wing pair wing of W(i,j) is W(i*j*), where means opposite and “A” means adjacent directed edge or face. The connecting, matching and wing pairs can be determined using matrix multiplication of W(i,j) with adjacency and opposition matrices determined from the polygon mesh.

[0385] As used herein, the term “weaver” is a contiguous surface that is the union of the surfaces of wings in sequences of matching pairs and connecting pairs in the polygon mesh. A weaver can be determined as follows: (a) the first wing in the sequence is chosen arbitrarily; (b) the second wing in the sequence is decided to be either the connecting pair wing or the matching pair wing of the first wing; (c) the third wing in the sequence is the matching pair wing of the previous wing if the previous wing was the connecting pair wing of the first wing, or the third wing in the sequence is the connecting pair wing of the previous wing if the previous wing was the matching pair wing of the first wing; (d) step (c) is repeated for each next wing until either the next wing is the first wing or there is no next wing available. If the next wing is the first wing, then the weaver is a closed loop. If no next wing is available then the weaver is open and the last wing is one end of the open weaver. The rest of the weaver if there is a rest is added to the sequence by going backwards from the first wing until the other end. Each weaver is a sequence of wings inalternating matching pairs and connecting pairs. Each weaver is a surface with a local outward direction corresponding to the outward directions of the wings making up the weaver. There are one or more weavers that result from any polygon mesh. A weaver can be a closed loop or open. If each matching pair in a weaver is connected to other matching pairs at each wing by connecting tabs, then there is no end to the weaver and it is a closed loop. The weaver is a closed loop when it is a sequence of connected matching pairs where the last matching pair is connected to the first matching pair with their wings forming a connecting pair. This occurs for polygon meshes wherein all the polygons are closed. Weavers alternatingly cross over and under another or the same weaver at each strut tab to form a 3D woven structure. The crossing over or under direction is determined by the outward direction of the wing in the weaver at that strut tab. At a connecting tab, the outward direction of the weaver is in both directions because each strut tab includes both wings in a connecting pair. A weaver sequence is an ordered list of indices, with a unique index for each wing in the weaver.

[0386] The weavers and their interrelation define a topology of the structure. The topology can include a knot topology, a ribbon topology, a link topology, associations, and combinations thereof. Knot and link topology includes the interrelation of weavers. Ribbon topology includes the twist and writhe of the ribbons and weavers. An association is a bundle of one or more weavers sharing or almost sharing strut tab surfaces at an edge of the polygon mesh. An association locks ribbon topology in place at each edge shared by two or more polygons of the polygon mesh. No matter the shape or dimension of the wings, a structure has the same topology if its weavers have the same interrelation, meaning its weavers have the same connecting pairs, same matching pairs and same wing pairs, no matter the shape of the wings. This means that topology can be identified by replacing the index of each wing in each weaver sequence by its wing pair index. The topologyis then described by a list of a sequences of numbers that appear in pairs. Instead of numbers, any symbols could be used. Each number occurs twice and represents a wing pair. For each polygon mesh geometry there is a ribbon topology even if the polygon mesh is a mesh with an edge shared by more than two polygons, or a has degenerate polygons. The topology is set by the polygon mesh and the convention used to define the outward direction of polygon faces. In embodiments, a method is directed to producing a target structure consisting of one or more interwoven ribbons. In embodiments, the invention is a ribbon woven structure. In embodiments, a structure has the topology of a ribbon-woven structure. The interrelation of the weavers is represented by the shared wing pair indices of each wing. The topology is represented by weaver sequences of alternating connecting pairs and matching pairs of wings wherein each wing is represented by its wing pair index. For a closed loop weaver, the weaver sequence can start at any of the wings in the sequence. For an open weaver the sequence starts at one of the end wings. The direction of the sequence does not change the topology. This wing index weaver sequence representation of the topology is like a Dowker notation in knot theory, suitable for nonplanar graphs. The ribbon topology for the woven structure based on a tetrahedron is given by the three weaver sequences in TABLE 2 below. This topology is the same as a Borromean link of three ribbons.TABLE 2: Weaver Sequences for a Tetrahedron

[0387] As used herein, the term “ribbon” is defined as follows: A ribbon is a three-dimensional object that is fabricated using a weaver for the design. A ribbon comprises a sequence of wings in alternating matching pairs and connecting pairs. If it is closed loop then a weaver is opened at one of the connecting pairs of wings. Weavers are flattened at the bending axis of each matching pair so that the dihedral angle is 180 degrees. Weavers can be divided into smaller parts by opening them at selected connecting pairs or matching pairs of wings. The result is one or more open flattened weavers or smaller parts of weavers. Each ribbon is fabricated using a weaver or a part of a weaver as a design. The length of ribbons is chosen. It can be chosen to be short enough so that the ribbons can be easily interwoven. Ribbons can be fabricated for example by cutting out a sheet of material in the shape of a flattened weaver or part of a weaver and then bending the material at each bending axis by the dihedral angle. When placed in the structure being assembled, each ribbon has a volume and is placed such that it extends from the wing surfaces of a weaver in the outward direction of the wings. The surface of the wings of the ribbons is referred to as the b- side of the ribbon. The b-side is the side that has a surface that is shared with or is in the neighborhood of another b-side surface from a wing in the same wing pair and included in the same or another ribbon. At the strut tabs, the ribbons have an a-side opposite to the b-side. At a connecting tab the a-side can be on both sides of the ribbon and all or parts of the b-sides are in the interior of the ribbon. The a-side is on the opposite side from the b-side and is generally visible. Ribbons have two ends. The ends of ribbons can include connecting tab surfaces or strut tab surfaces. The ends include wings that can be in a matching pair with the next wing in the ribbon or in a connecting pair with the next wing in the ribbon.

[0388] One or both ends of a ribbon can comprise wings that are twin wings with the wings in the same or another ribbon. The first or last wing of a ribbon is called an end wing. The same wingof a weaver is used in the design of overlapping ribbons. This is so that a ribbon end overlaps another ribbon in the same place as the same or another ribbon. A feature of embodiments is that the ribbon ends overlap the same or other ribbons and are placed in a pocket or sleave between wing pair wings.

[0389] If wings are triangular then flattened matching pair surfaces are quadrilaterals. Each matching pair has a bending axis that coincides with the shared edge of the wings in the matching pair. Each matching pair of wings has a dihedral angle, wherein the dihedral angle is the angle about the bending axis from one wing surface through the other wing surface of the matching pair.

[0390] Optionally, in embodiments, a notch can be removed from the corners at the vertices of the shared edge of the matching pair. For example, for matching pair 240 in FIG. 2N given by {il, k(j2, i2), i2, k(j l,il)}, notches can be made at v(il) and v(i2). Ribbons share strut tab surfaces with other ribbons and are placed ‘b’ side to ‘b’ side and can connect at these shared surfaces. Optionally, in embodiments, guide holes from the surfaces of the wings define guide holes in the ribbons. Guide-holes can be used to align the ribbons during assembly by means of guide pins and to ensure that the wings are placed accurately according to the polygon mesh.

[0391] As used herein, the term “strut” is defined as follows: a strut is a collection of matching pairs of wings whose bending axes coincide with part of or all of an edge in the original polygon mesh and is also referred to as a “winged strut.” The shared edge also defines the strut axis. The matching pairs of wings in a strut share strut tab surfaces with each other in wing pairs. For example, a strut of two matching pairs is depicted in FIG. 2N. Wings 240 and 242 connect at strut tabs 230 and 232 to form strut 250. Two more matching pairs, 244 and 246, connect at shared strut tabs, 234 and 236, to form an adjacent strut 252 as depicted in FIG. 20. A contiguous surface is formed from the sequence of matching pairs 242, 246 and 248. The contiguous surface forms partof the design for a ribbon. This is repeated for all matching pairs of wings until contiguous surfaces defining ribbons include all the wings

[0392] The ribbons are bent at their bending axes, interwoven and placed in the structure. Repeating for ribbons for each contiguous surface completes the structure 280 depicted in FIG. 2Q which has a 3D woven pattern with the topology of a trefoil knot. The ribbon topology has 11 twists and three associations, one at each edge shared by two polygons.

[0393] As used herein, the term “placed according to the polygon mesh” is defined as follows. Each ribbon is placed by placing each wing of the ribbon according to its wing pair indices so that each wing shares a strut tab surface with its wing pair wing in the same or other ribbons. Each ribbon is placed such that at each edge of the polygon mesh, the ribbon is bent according to a dihedral angle at a bending axis that is aligned with the edge.

[0394] As used herein, the term “3D weaving” is defined as follows: (1) a three-dimensional structure whose components are assembled in a woven pattern. The structure includes 3D structures that can be approximated or are defined by a polygon mesh. (2) A method, algorithm and / or process for fabricating a 3D weaving. As used herein, a “3D woven structure” is any structure of the present invention. Therefore, 3D woven structures include structures based on 2D polygon meshes as well as 3D polygon meshes.

[0395] As used herein, the term “guide holes” means holes (which are optionally present) in the polygon surface which are then holes in the wings. The purpose of the holes is to help in lining up ribbon surfaces when they are connected ‘b’ side to ‘b’ side in a connecting pair or wing pair. The holes are placed in the part of the polygon surface that is shared by the wing pairs and connecting pairs.

[0396] Amounts and other numerical data may be presented herein in a range format. It is to be understood that such range format is used merely for convenience and brevity and should be interpreted flexibly to include not only the numerical values explicitly recited as the limits of the range, but also to include all the individual numerical values or sub-ranges encompassed within that range as if each numerical value and sub-range is explicitly recited. For example, a numerical range of approximately 1 to approximately 4.5 should be interpreted to include not only the explicitly recited limits of 1 to approximately 4.5, but also to include individual numerals such as 2, 3, 4, and sub-ranges such as 1 to 3, 2 to 4, etc. The same principle applies to ranges reciting only one numerical value, such as “less than approximately 4.5,” which should be interpreted to include all of the above-recited values and ranges. Further, such an interpretation should apply regardless of the breadth of the range or the characteristic being described.

[0397] Unless defined otherwise, all technical and scientific terms used herein have the same meaning as commonly understood to one of ordinary skill in the art to which the presently disclosed subject matter belongs. Although any methods, devices, and materials similar or equivalent to those described herein can be used in the practice or testing of the presently disclosed subject matter, representative methods, devices, and materials are now described.

[0398] Following long-standing patent law convention, the terms “a” and “an” mean “one or more” when used in this application, including the claims.

[0399] Unless otherwise indicated, all numbers expressing quantities of ingredients, reaction conditions, and so forth used in the specification and claims are to be understood as being modified in all instances by the term “about.” Accordingly, unless indicated to the contrary, the numerical parameters set forth in this specification and attached claims are approximations that can vary depending on the desired properties sought to be obtained by the presently disclosed subject matter.

[0400] As used herein, the term “about” and “substantially” when referring to a value or to an amount of mass, weight, time, volume, concentration or percentage is meant to encompass variations of in some embodiments ±20%, in some embodiments ±10%, in some embodiments±5%, in some embodiments ±1%, in some embodiments ±0.5%, and in some embodiments ±0.1% from the specified amount, as such variations are appropriate to perform the disclosed method.

[0401] As used herein, the term “substantially perpendicular” and “substantially parallel” is meant to encompass variations of in some embodiments within ±10° of the perpendicular and parallel directions, respectively, in some embodiments within ±5° of the perpendicular and parallel directions, respectively, in some embodiments within ±1° of the perpendicular and parallel directions, respectively, and in some embodiments within ±0.5° of the perpendicular and parallel directions, respectively.

[0402] As used herein, the term “and / or” when used in the context of a listing of entities, refers to the entities being present singly or in combination. Thus, for example, the phrase “A, B, C, and / or D” includes A, B, C, and D individually, but also includes any and all combinations and subcombinations of A, B, C, and D.REFERENCES

[0403] “ The Knot Book,” Colin C. Adams, American Mathematical Society, 2004.

Claims

WHAT IS CLAIMED IS:

1. A method to form a structure comprising one or more ribbons, wherein the method comprises:(a) identifying a shape of the structure;(b) generating a representation of the shape with a polygon mesh, wherein(i) the polygon mesh comprises an assembly of polygons,(ii) each polygon comprises vertices, edges and a surface bounded by the edges,(iii) for each edge there is a first directed edge and an opposite directed edge, and(iv) each polygon has a first face and an opposite face;(c) subdividing the faces into a plurality of wings , wherein each particular wing(i) is from a particular face and has a particular directed edge,(ii) is a matching pair with a second wing, wherein the particular wing and the second wing(A) have a shared edge and opposite directed edges,(B) are from faces with outward directions pointing into a same volume, and(C) have a bending axis at the particular edge and a dihedral angle, and(iii) is a connecting pair with a third wing, wherein the particular wing and the third wing(A) have directed edges that point to a same vertex, and(B) are from opposite faces of the particular polygon;(d) determining a shape and dimensions of the one or more ribbons, wherein(i) each ribbon of the one or more ribbons comprises an alternating sequence of a matching pair and a connecting pair of wings, and(ii) at least one of the ribbons has an end wing that is in a connecting pair of wings,(e) fabricating the one or more ribbons based on the shape and dimensions of the ribbons and the dihedral angles at the bending axes; and(f) forming the structure by bending and placing the one or more ribbons according to the polygon mesh.

2. The method of Claim 1, wherein the one or more ribbons is a plurality of ribbons.

3. The method of any of Claims 1-2, wherein, for at least some of the ribbons in the one or more ribbons, the alternating sequence of a matching pair and a connecting pair of wings comprises an alternating sequence that comprises a first matching pair of wings, a first connecting pair of wings, a second matching pair of wings, and a second connecting pair of wings.

4. The method of Claim 3, wherein, for at least some of the ribbons in the one or more ribbons, the alternating sequence of a matching pair and a connecting pair of wings comprises an alternating sequence that further comprises a third matching pair of wings and a third connecting pair of wings, such that, for at least some of the ribbons in the one or more ribbons the alternating sequence comprises the first matching pair of wings, the first connecting pair of wings, the second matching pair of wings, the second connecting pair of wings, the third matching pair of wings, and the third connecting pair of wings.

5. The method of any of Claims 1-4, wherein at least one of the ribbons has a side wing connected to the connecting pair wing of an end wing of the ribbon.

6. The method of any of Claims 1-4, wherein each of the ribbons of at least 50% of the ribbons in the one or more ribbons has at least one end wing that is in a connecting pair of wings.

7. The method of Claim 6, wherein each of the ribbons of the at least 50% of the ribbons in the one or more ribbons has a side wing connected to the connecting pair wing of an end wing of the ribbon.

8. The method of any of Claims 1-4, wherein at least one of the ribbons in the one or more ribbons has (a) a first end wing that is in a first connecting pair of wings and (b) a second end wing that is in a second connecting pair of wings.

9. The method of Claim 8, wherein the at least one of the ribbons has (a) a first side wing connected to the connecting pair wing of the first end wing and (b) a second side wing connected to the connecting pair wing of the second end wing.

10. The method of Claim 8, wherein each of the ribbons of at least 50% of the ribbons in the one or more ribbons has a first end wing that is in a connecting pair of wings and a second end wing that is in a connecting pair of wings.

11. The method of Claim 8, wherein each of the ribbons of the at least 50% of the ribbons in the one or more ribbons has (a) a first side wing connected to the connecting pair wing of the first end wing of the ribbon and (b) a second side wing connected to the connecting pair wing of the second end wing of the ribbon.

12. The method of any of Claims 1-11, wherein the polygon mesh comprises an edge shared by three or more polygons.

13. The method of any of Claims 1-11, wherein an end wing that is in a connecting pair of wings is placed in a pocket formed by a wing pair of wings.

14. The method of any of Claims 1-11, wherein an end wing that is in a connecting pair of wings is placed in flap-interlock closure.

15. The method of any of Claims 1-14, wherein the structure is a module based on a degenerate polyhedron polygon mesh.

16. The method of any of Claims 1-15, wherein the ribbons are connected at shared strut tab and / or connecting tab surfaces.

17. The method of any of Claims 1-16, wherein the step of forming the structure comprises elastically deforming the one or more ribbons while bending and placing the one or more ribbons according to the polymer mesh.

18. The method of Claim 17, wherein after the ribbons are placed according to the polymer mesh, the elastic deformation provides forces upon the structure that maintains placement of the one or more ribbons.

19. A structure comprising one or more ribbons, wherein(a) the shape of the structure is represented by a polygon mesh, wherein,(i) the polygon mesh comprises an assembly of polygons,(ii) each polygon comprises vertices, edges and a surface bounded by the edges,(iii) for each edge, there is a first directed edge and an opposite directed edge, and(iv) each polygon has a first face and an opposite face;(b) the faces are subdivided into a plurality of wings, wherein each particular wing(i) is from a particular face and has a particular directed edge,(ii) is a matching pair with a second wing, wherein the particular wing and the second wing(A) have a shared edge and opposite directed edges,(B) are from faces with outward directions pointing into a same volume, and(C) have a bending axis at the particular edge and a dihedral angle, and(iii) is a connecting pair with a third wing, wherein the particular wing and the third wing(A) have directed edges that point to a same vertex, and(B) are from opposite faces of the particular polygon;(c) each ribbon of the one or more ribbons comprises an alternating sequence of a matching pair and a connecting pair of wings, wherein(i) at least one of the ribbons has an end wing that is in a connecting pair of wings; and(d) each ribbon is bent and placed according to the polygon mesh.

20. The structure of Claim 19, wherein the one or more ribbons is a plurality of ribbons.

21. The structure of any of Claims 19-20, wherein, for at least some of the ribbons in the one or more ribbons, the alternating sequence of a matching pair and a connecting pair of wings comprises an alternating sequence that comprises a first matching pair of wings, a first connecting pair of wings, a second matching pair of wings, and a second connecting pair of wings.

22. The structure of Claim 21, wherein, for at least some of the ribbons in the one or more ribbons, the alternating sequence of a matching pair and a connecting pair of wings comprises an alternating sequence that further comprises a third matching pair of wings and a third connecting pair of wings, such that, for at least some of the ribbons in the one or more ribbons the alternating sequence comprises the first matching pair of wings, the first connecting pair of wings, the second matching pair of wings, the second connecting pair of wings, the third matching pair of wings, and the third connecting pair of wings.

23. The structure of any of Claims 19-22, wherein at least one of the ribbons has a side wing connected to the connecting pair wing of an end wing of the ribbon.

24. The structure of any of Claims 19-22, wherein each of the ribbons of at least 50% of the ribbons in the one or more ribbons has at least one end wing that is in a connecting pair of wings.

25. The structure of Claim 24, wherein each of the ribbons of the at least 50% of the ribbons in the one or more ribbons has a side wing connected to the connecting pair wing of an end wing of the ribbon.

26. The structure of any of Claims 19-22, wherein at least one of the ribbons in the one or more ribbons has (a) a first end wing that is in a first connecting pair of wings and (b) a second end wing that is a in second connecting pair of wings.

27. The structure of Claim 24, wherein the at least one of the ribbons has (a) a first side wing connected to the connecting pair wing of the first end wing and (b) a second side wing connected to the connecting pair wing of the second end wing.

28. The structure of Claim 27, wherein each of the ribbons of at least 50% of the ribbons in the one or more ribbons has a first end wing that is in a connecting pair of wings and a second end wing that is in a connecting pair of wings.

29. The structure of Claim 28, wherein each of the ribbons of the at least 50% of the ribbons in the one or more ribbons has (a) a first side wing connected to the connecting pair wing of thefirst end wing of the ribbon and (b) a second side wing connected to the connecting pair wing of the second end wing of the ribbon.

30. The structure of any of Claims 19-29, wherein the polygon mesh is a 3D mesh comprising an edge shared by three or more polygons.

31. The structure of any of Claims 19-29, wherein an end wing that is in a connecting pair of wings is placed in a pocket formed by a wing pair of wings.

32. The structure of any of Claims 19-29, wherein an end wing that is in a connecting pair of wings is placed in flap-interlock closure.

33. The structure of any of Claims 19-29, wherein the structure is a module based on a degenerate polyhedron polygon mesh.

34. The structure of any of Claims 19-29, wherein the ribbons are connected to each other at shared strut tab and / or connecting tab surfaces.

35. The structure of Claim 19 wherein a topology of the structure is represented by one or more weaver sequences of wings in which each wing is indicated by its wing pair index.

36. The structure of Claim 19, wherein the structure is made by the method of Claim 1.

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