Triple Inversion Geometric Transformation

Abstract

The triple inversion geometric transformation can be used as a puzzle, a toy, a teaching aid, a therapeutic device, etc. The transformation comprises a plurality of hinged polyhedra and can be configured between three congruent inversion configurations.

Description

[0001] CROSS-REFERENCE TO RELATED APPLICATIONS

[0002] This application claims the benefit of U.S. patent application No. 17 / 821,178, filed on August 21, 2022, the entire disclosure of which is hereby incorporated by reference. Background Art

[0003] Geometric transformations with components that are coupled together have enjoyed cross-generational appeal as puzzles, toys, teaching aids, therapeutic devices, and the like. Such transformations can be configured between different geometric configurations, as shown, for example, in UK patent application No. GB ​​2,107,200 to Asano. However, the geometry and construction of known transformations inherently limit the number and type of geometric configurations that can be achieved. Therefore, there is a need for geometric transformations that can achieve different configurations and have different properties. Summary of the invention

[0004] In one aspect, the present disclosure provides a geometric transformation puzzle that can be inverted (flipped from the inside out) in three different ways, thereby presenting a common polyhedron in each "inverted configuration", but with a different outermost surface in each of the three instances. For example, a representative embodiment includes a triple inverted geometric transformation that can be manipulated into a common parallelepiped shape (e.g., a box) in three different ways, such that a different outermost surface is presented in each instance. As detailed herein, such transformed embodiments can have many interesting properties that enhance their appeal and usefulness.

[0005] In one aspect, the present disclosure provides a geometric transformation puzzle. The geometric transformation puzzle includes a plurality of hinged polyhedrons, wherein the transformation can be configured between a first inverted configuration, a second inverted configuration, and a third inverted configuration, wherein the first inverted configuration, the second inverted configuration, and the third inverted configuration are congruent. In another aspect, the present disclosure provides a method for manipulating the geometric transformation puzzle into an inverted state.

[0006] In any embodiments, each of the hinged polyhedrons may include one side having a side length of √3 units; two sides having a side length of √2 units; and three sides having a side length of one unit.

[0007] In any embodiment, all outermost surfaces of the first inverted configuration may include a first surface decoration, all outermost surfaces of the second inverted configuration may include a second surface decoration, and all outermost surfaces of the third inverted configuration may include a third surface decoration. The first surface decoration, the second surface decoration, and the third surface decoration may all be different from each other.

[0008] In any embodiments, each of the hinged polyhedrons may include a first face, a second face, a third face, and a fourth face, wherein the plurality of hinged polyhedrons include twelve polyhedrons hingedly connected into a ring, wherein each of the hinged polyhedrons includes a first magnet disposed adjacent to the first face, wherein the first magnets of adjacent polyhedrons in the ring have opposite polarities.

[0009] In any embodiment, each of the hingedly connected polyhedrons may include a second magnet disposed adjacent the second face. The second magnets of adjacent polyhedrons in the ring may have opposite polarities.

[0010] In any embodiment, each of the hingedly connected polyhedrons may include a third magnet disposed adjacent to the third face. The third magnets of adjacent polyhedrons in the ring may have opposite polarities.

[0011] In any embodiment, each of the hingedly connected polyhedrons may include a fourth magnet disposed adjacent to the fourth face. The fourth magnets of adjacent polyhedrons in the ring may have opposite polarities.

[0012] In any embodiments, the outermost surface of the first inverted configuration is a hidden inner surface in the second inverted configuration and the third inverted configuration, the outermost surface of the second inverted configuration is a hidden inner surface in the first inverted configuration and the third inverted configuration, and the outermost surface of the third inverted configuration is a hidden inner surface in the first inverted configuration and the second inverted configuration.

[0013] In any embodiments, each of the hinged polyhedrons may be congruent.

[0014] In any embodiments, each of the polyhedrons may be a tetrahedron.

[0015] In any embodiments, the first inverted configuration may be a first parallelepiped, the second inverted configuration may be a second parallelepiped, and the third inverted configuration may be a third parallelepiped.

[0016] In any embodiment, the outermost surface of the first inverted configuration may consist of the first surface, the outermost surface of the second inverted configuration may consist of the second surface, and the outermost surface of the third inverted configuration may consist of the third surface. The first surface, the second surface, and the third surface may be mutually exclusive.

[0017] In any embodiment, the plurality of hinged polyhedrons may consist of twelve polyhedrons hingedly connected in a ring. Adjacent polyhedrons in the ring may be mirror versions of each other.

[0018] In any embodiment, each of the hinged polyhedrons may include a first side and a second side, and may be hingedly connected to a first adjacent polyhedron of the ring along the first side, and may be hingedly connected to a second adjacent polyhedron of the ring along the second side. The first side may be perpendicular to the second side. BRIEF DESCRIPTION OF THE DRAWINGS

[0019] Representative embodiments are described with reference to the following figures, wherein like reference numerals refer to like parts throughout the various views unless otherwise specified.

[0020] Figure 1 A stereoscopic view of a geometric transformation in three different inverted parallelepiped configurations at three different points in time is shown according to a representative embodiment of the present disclosure.

[0021] Figure 2 shows the geometric transformation in the ring configuration, which is Figure 1 The geometric transformations are the same as shown in .

[0022] Figure 3A Shows that with Figure 1 and Figure 2 Schematic projection of a segment of the same geometric transformation of the structure and features.

[0023] Figure 3B yes Figure 3A Detailed view of the geometric transformation of a polyhedron.

[0024] Figure 4 A schematic diagram of surface decoration of a segment of geometric transformation according to an embodiment of the present disclosure is shown, wherein the geometric transformation is Figure 1 The geometric transformations are the same as shown in .

[0025] Figure 5 A schematic diagram showing magnet placement of a geometrically transformed segment according to one embodiment of the present disclosure.

[0026] Figure 6A-6F A method according to a representative embodiment of the present disclosure is shown. Figure 1 A method for manipulating geometric transformations into inverted configurations. DETAILED DESCRIPTION

[0027] The present disclosure provides a geometric transformation (interchangeably referred to herein as a "transformation") comprising hinged polyhedra, each of which has specific geometric properties. Each of the polyhedra is hingedly connected to other polyhedra of the transformation and optionally has structural features that enable unique functions of the transformation and / or exhibit unique properties of the transformation. As used herein, the term "transformation" means a plurality of hinged polyhedra.

[0028] The transformations described herein have attributes that individually and / or collectively enhance the usefulness and appeal of the transformations such as puzzles, teaching aids, therapeutic devices, and toys. As will be understood from the following description, such attributes may include any one or more of the following:

[0029] · The ability of the transformation to flip the inside out ("inversion") three times into three inverted polyhedral configurations ("inversion configurations"), where in each inversion configuration the overall polyhedron presented is congruent to the overall polyhedron in every other inversion configuration.

[0030] For each inversion configuration, the outermost surface of the polyhedron is different from (e.g., mutually exclusive of) the outermost surface of every other inversion configuration.

[0031] • For each inversion configuration, the outermost surface of the polyhedron has a different appearance and / or texture (surface treatment) than the outermost surface of at least one other congruent inversion configuration.

[0032] • Geometric and magnetic compatibility with other geometric transformations enables the transformation to be assembled with and / or coupled to other transformations.

[0033] As used herein, the term "congruent" means that two geometric figures are identical in shape and size. This includes the case when one of the geometric figures is a mirror image of the other.

[0034] Figure 1 A transformation 100 is shown according to a representative embodiment of the present disclosure. As shown, the transformation 100 has a polyhedral shape, form, or configuration (a parallelepiped in this example). Figure 2-Figure 5 As detailed, transformation 100 includes a plurality of hingedly connected polyhedrons that can be manipulated, repositioned, and optionally stabilized (e.g., magnetically) relative to each other to create different overall forms or configurations. In this disclosure, the term "configuration" refers to the shape, form, or configuration of the overall transformation 100, while the term "polyhedron" refers to the individual polyhedrons that make up transformation 100. Nonetheless, the overall transformation 100 can have one polyhedral configuration.

[0035] In particular, Figure 1The same transformation 100 is shown in three different inversion configurations A, B and C at three different points in time. In each inversion configuration A, B and C, the transformation 100 has a parallelepiped configuration that is congruent with each of the other parallelepiped configurations. It follows that the surface area of ​​the outermost surface of one of the parallelepiped inversion configurations is equal to the surface area of ​​the outermost surface of the other parallelepiped inversion configurations.

[0036] As used herein, an "inverted configuration" means a configuration of transformation 100 in which all outermost surfaces are inner surfaces in another configuration (e.g., another inverted configuration). As used herein, an "inner surface" is a surface that extends through the interior volume of a transformation, other than the outermost surface of a transformation. Depending on the geometry of the transformation and the material from which the transformation is constructed, the inner surface may or may not be visible. Representative inner surfaces include those of PCT Publication No. WO / 2022 / 130285. Figure 2 a, which PCT publication is incorporated herein by reference in its entirety.

[0037] exist Figure 1 In the example of , inverted configuration A is an inverted configuration because all outermost visible surfaces 102a (first surfaces) are hidden as invisible inner surfaces in configurations B and C. Likewise, inverted configuration B is an inverted configuration because all outermost visible surfaces 102b (second surfaces) are hidden as inner surfaces in inverted configurations A and C. In the same way, inverted configuration C is an inverted configuration because all outermost visible surfaces 102c (third surfaces) are hidden as inner surfaces in inverted configurations A and B.

[0038] The ability of transformation 100 to achieve three congruent inversion configurations enables interesting possibilities that enhance the usefulness of transformation 100. For example, a first surface may optionally have an appearance and / or texture (surface decoration) that is different from a second surface and / or a third surface. Similarly, a second surface may optionally have a surface decoration that is different from the first surface and / or the third surface. And in some embodiments, a third surface may optionally have a surface decoration that is different from the first surface and / or the second surface. The surface decoration of any given surface may be produced by the material from which the particular surface is constructed, the application of a graphic to the surface, the treatment of the surface to impart a texture, and / or other reasons.

[0039] exist Figure 1 In the example of FIG. 1 , the first surface, the second surface and the third surface have different surface decorations, which advantageously enables the transformation 100 to present the same parallelepiped inverted configuration with three different surface decorations. Figure 4A representative surface decoration arrangement is detailed that enables transformation 100 to present the same parallelepiped inverted configuration with three different surface decorations.

[0040] In any embodiment, the transform 100 may include a plurality of optional magnets positioned and polarized in a manner that stabilizes the transform 100 in a number of different configurations, including Figure 1 The total number of magnets can vary, for example, 12, 24, 36, 48, 72 or more. Figure 5 Details are configured to Figure 1 The transformation is stabilized on a representative magnet configuration in a parallelepiped inverted configuration.

[0041] Figure 2 Shown with Figure 1 100 is a perspective view of a transformation 200 that is identical to the transformation 100. The transformation 200 includes a plurality of polyhedra 210a-2101 that are hingedly connected to form a continuous ring.

[0042] The representative transformation 200 includes twelve polyhedra, but other embodiments may include a greater number by dividing one or more of the polyhedra 210a-210l into sub-polyhedra. For example, one embodiment may divide each of the polyhedra 210a-210l into two separate complementary polyhedra that, when combined, have the same Figure 1 Each polyhedron 210a-210l has the same polyhedral shape. Therefore, such an embodiment will include 24 polyhedrons. In this way, the present disclosure also includes transformations including 36, 48 or a larger number of polyhedrons.

[0043] In the illustrated embodiment, the polyhedra 210a-210l are congruent and each have Figure 3A The polyhedrons 210a-210l are hingedly connected by a plurality of hinges 212a-212l. In particular, each of the polyhedrons 210a-210l is hingedly connected to two adjacent polyhedrons in the polyhedrons 210a-210l by two of the hinges 212a-212l.

[0044] In the illustrated embodiment, each of the polyhedrons 210a-210l has a solid shell with a cavity formed therein. The cavity may be provided with one or more magnets positioned and polarized to stabilize the transformation 200 in different configurations, such as a parallelepiped configuration corresponding to three inversion configurations. Figure 5One such representative magnet configuration is detailed. By way of example, and not limitation, the solid outer shell of each of the polyhedrons 210a-210l can be formed from a polymer such as high-density and low-density polyethylene (LDPE, HDPE), polypropylene (PP), polystyrene (PS, ABS), polyester (PET), or other suitable durable and safe materials.

[0045] Due to the geometry of the polyhedra 210a-210l and the hinged connections 212a-212l between them, the transformation 100 can be manipulated into many different configurations, including Figure 1 and Fig. 6F The three parallelepiped inversion configurations shown in Figure 2 and Figure 6A-6E The intermediate configuration.

[0046] As from Figure 2 Obviously, each of the polyhedrons 210a-210l may be provided with surface decorations, such as graphics, textures, colors, etc. Figure 1 As can be appreciated, the coordinated placement of the different surface finishes enables the transformation 100 to present each of the different surface finishes in each inverted configuration. Figure 4 One such surface decoration arrangement is detailed.

[0047] Figure 3A Is with Figure 1 and Figure 2 Schematic projection of a transformation segment 300 of the same construction and features as a segment of a geometric transformation. Specifically, the transformation segment 300 includes four hinged polyhedrons 310a-310d, each of which corresponds to one of the polyhedrons of transformations 100, 200. To reiterate, each of the polyhedrons of transformations 100, 200 has a geometry corresponding to polyhedrons 310a-310d.

[0048] Three of the four-polyhedron transformation segments 300 can be hinged to form an end-to-end continuous ring to achieve Figure 1 and Figure 2 The dodeca-polyhedron transformation 100, 200. The polyhedrons 310a-310d are hingedly coupled together by hinges 312b-312d, and the hinge 312a is configured to couple the polyhedron 310a to another adjacent polyhedron or transformation segment (not shown). Figure 3B FIG. 1 is a diagram showing details of the polyhedron 310c and hinges 312c, 312d. Figure 3A Detailed view of the .

[0049] The geometry of the polyhedrons 310a-310d and the hinged connections therebetween enable the geometric transformations of the present disclosure to be manipulated into the configuration shown and described herein. Figure 3A and Figure 3B A representative geometry and hinge configuration is shown. However, Figure 3A and Figure 3B The specific geometries and coupling arrangements shown are representative and not limiting.

[0050] Figure 3A The geometric shape can be realized with a larger number of polyhedrons and with different hinge connection arrangements. For example, each of the polyhedrons 310a-310d can be divided into two or more sub-polyhedrons, as described above. As an example of a different hinge connection arrangement, two polyhedrons can be connected with two hinges instead of two hinges. Figure 3A 310a-310d are hingedly connected with a single hinge. However, it should be understood that of all possible theoretical geometries of polyhedrons 310a-310d, very few such geometries will be able to achieve a geometric transformation that includes three of the transformation segments 300 connected in an end-to-end continuous ring to achieve three congruent inverted configurations. For at least this reason, the geometries described herein are not obvious variations of known geometries.

[0051] As shown, each of the polyhedrons 310a-310d in the illustrated embodiment is a tetrahedron having four faces, six edges, and four vertices, as shown in FIG. Figure 1 and Figure 2 The geometric transformation of the polyhedron is the same as shown in the figure. Figure 3A and Figure 3B The projection onto the two-dimensional plane in doubles the three edges, so in Figure 3A and Figure 3B However, those skilled in the art will understand this characteristic of the projection, which will be further explained below.

[0052] Figure 3B The edge, face and vertex details of a representative polyhedron 310c that is congruent to polyhedrons 310a-310b and 310d are detailed.Polyhedrons 310a and 310c are mirror images or mirrored versions of polyhedrons 310b and 310d.

[0053] As shown, polyhedron 310c includes six sides defining four faces having four vertices. In particular, polyhedron 310c includes a first side 314, a second side 316, a third side 318, a fourth side 320, a fifth side 322, and a sixth side 324. Figure 3A and Figure 3B310c, but the geometry of the polyhedron 310c dictates that in a three-dimensional embodiment of the polyhedron the first side 314 is perpendicular to the second side 316 (eg Figure 2 ).

[0054] The first side 314, the third side 318, and the fourth side 320 define a first face 326. The second side 316, the third side 318, and the fifth side 322 define a second face 328. The second side 316, the fourth side 320, and the sixth side 324 define a third face 330. The first side 314, the fifth side 322, and the sixth side 324 define a fourth face 332. The first face 326 has a first vertex 336, a second vertex 338, and a third vertex 340. The second face 328 has a second vertex 338, a third vertex 340, and a fourth vertex 342. The third face 330 has a first vertex 336, a third vertex 340, and a fourth vertex 342. The fourth face has a first vertex 336, a second vertex 338, and a fourth vertex 342.

[0055] The first face 326 is congruent with the second face 328. The third face 330 is congruent with the fourth face 332. Each of the first face 326, the second face 328, the third face 330, and the fourth face 332 is a right triangle. In addition, the third face 330 and the fourth face 332 are isosceles triangles.

[0056] The relative lengths of the six sides will now be described in detail with reference to the legend 334, which is applicable to Figure 3A and Figure 3B Both. It should be understood that although the following describes edge lengths, such description properly describes the distances between corresponding vertices. Therefore, the following description of "edge lengths" does not limit the present disclosure to geometric transformations having tetrahedral polyhedra with six continuous, linear, uninterrupted edges. In fact, the present disclosure includes geometric transformations formed by polyhedra with discontinuous and / or nonlinear edges, as long as such polyhedra have edges corresponding to Figure 3B The vertices are those shown in , and the relative distances between them are as defined in the legend 334.

[0057] Each of the six sides of each of the polyhedrons 310a-310c has a relative side length (alternatively, vertex distance) indicated by a symbol thereon, which corresponds to the relative side lengths defined in the legend 334. In particular, the first side 314, the second side 316, and the sixth side 324 (carrying a plus symbol) have a relative side length of 1 unit, and in some embodiments (e.g., the embodiment shown) are the only sides with a relative side length of 1 unit. The third side 318 (carrying a triangle symbol) (the longest side of the polyhedron 310c) has a relative side length of √3 units (the square root of three units), and in some embodiments (e.g., the embodiment shown) are the only sides with such a side length. The fourth side 320 and the fifth side 322 (carrying a square symbol) have a relative side length of √2 units (the square root of two units), and in some embodiments (e.g., the embodiment shown) are the only sides with such a side length.

[0058] The side lengths shown are relative and can be scaled up or down as long as the relative lengths between the six sides remain constant. For example, in one representative embodiment, the basic unit is 10 cm. In such an embodiment, the first side 314, the second side 316, and the sixth side 324 will have a side length of 10 cm. According to the relationship defined in the legend 334, the third side 318 (the longest side) will have a side length of 10√3 cm=17.32 cm, and the fourth side 320 and the fifth side 322 will have a side length of 10√2 cm=14.14 cm. In another representative embodiment where the basic unit is 20 cm, each side length will be twice as long as the previously defined embodiment. Therefore, the relative side lengths (alternatively, vertex distances) defined by the legend 334 can be scaled up or down.

[0059] return Figure 3A , two additional features of transformation segment 300 are apparent. First, each polyhedron is a mirror image of two adjacent polyhedrons. For example: polyhedron 310b is a mirror image of polyhedrons 310a and 310c; polyhedron 310c is a mirror image of polyhedrons 310b and 310d, and so on. This property enables similar edges of adjacent polyhedrons to be hinged as described below. Therefore, although all polyhedrons are congruent, there are two types that are mirror images of each other, namely, type one polyhedrons (e.g., polyhedrons 310a, 310c) and type two polyhedrons (e.g., polyhedrons 310b, 310d). In terms of geometry, transformation segment 300 includes a repeating alternating pattern that includes: type one polyhedrons, type two polyhedrons, type one polyhedrons, and so on.

[0060] from Figure 3AAn obvious second property is that adjacent polyhedra are hinged together along similar edges by hinges 312a-312d. Figure 3A and Figure 3B , hinge 312c hinges the first side 314 of polyhedron 310c to the corresponding first side of the mirrored polyhedron 310b. Similarly, hinge 312d hinges the second side 316 of polyhedron 310c to the corresponding side of the mirrored polyhedron 310d.

[0061] Hinged or flexible connections allow the polyhedra to be manipulated relative to each other, allowing geometric transformations to achieve different configurations (such as Figure 1 parallelepiped configuration) and Figure 2 and Figure 6A-6E , while the overall geometric transformation remains a single device rather than an incoherent assortment of parts.

[0062] The geometrically transformed polyhedra described herein are generally assembled so that corresponding edges (directly adjacent edges) of adjacent polyhedra are adjacent or have a spacing of less than 1 mm, such as 0.5 mm. Figure 2 Obviously, Figure 2 A transformation 200 is shown along with its representative hinged connections between adjacent polyhedra.

[0063] The hinges 312a-312d can take many different forms. In some embodiments, each of the hinges 312a-312d is a decal or sticker applied to the faces of at least two adjacent polyhedrons (e.g., mirrored faces of adjacent polyhedrons) such that the hinge extends directly from one of the polyhedrons to the other polyhedron. For example, referring to Figure 3A , if hinge 312c has such a configuration, hinge 312c will be an applique applied to at least first face 326 of polyhedron 310c and extending to the adjacent mirrored face of polyhedron 310b, thereby hingedly connecting the adjacent polyhedron along first side 314 of polyhedron 310c. In some such embodiments, the applique may include more than one hinge. For example, in one embodiment, a single continuous applique is applied to polyhedrons 310a-310d and includes at least hinges 312b-312d, respectively. Representative hinges of this configuration are detailed in U.S. Pat. Nos. 10,569,185 and 10,918,964, which are incorporated herein by reference in their entirety.

[0064] In other embodiments, the hinge is integrally formed with the polyhedron and extends directly from one of the polyhedrons to an adjacent polyhedron. In such embodiments, the hinge may be formed as a flexible polymer strip of the same or similar material as the shell of the polyhedron. For example, referring to Figure 3A If the hinge 312c has such a configuration, the hinge 312c will be integrally formed with the polyhedrons 310b, 310c as at least one polymer strip extending between the polyhedrons 310b, 310c, thereby coupling adjacent polyhedrons along the first edge 314 of the polyhedron 310c. Representative hinges of this configuration are detailed in U.S. Pat. No. 11,358,070, which is incorporated herein by reference in its entirety.

[0065] In other embodiments, the hinge is formed as one or more internal flexible connecting strips (e.g., of a thin flexible polymer or fabric) that extend between adjacent polyhedrons and are configured to be anchored within the interior cavities of adjacent polyhedrons. Figure 3A , if hinge 312c has such a configuration, a portion of hinge 312c will be anchored in the inner cavity of polyhedron 310b, and another portion of hinge 312c will be anchored with the inner cavity of polyhedron 310c, thereby connecting adjacent polyhedrons along first edge 314 of polyhedron 310c. A representative hinge of this configuration is described in detail in PCT Publication No. WO2022 / 030285, which is incorporated herein by reference in its entirety.

[0066] In any embodiment, more than one hinge may extend between adjacent sides of adjacent polyhedrons.The foregoing hinge structures are representative and non-limiting.

[0067] Based on the geometry of polyhedrons 310a-310d and this description, it is apparent that adjacent hinges are perpendicular to each other due to the perpendicular relationship between the first side (e.g., first side 314) and the second side (e.g., second side 316). For example, hinge 312c is perpendicular to hinge 312d. This is apparent from Figure 2 Obviously.

[0068] The geometries and hinges described above enable the geometric transformations of the present disclosure to achieve three inverted configurations, e.g. Figure 1. For example, the geometry described herein enables a first hinge of each polyhedron (e.g., hinge 312c in the example of polyhedron 310c) to have a perpendicular orientation relative to a second hinge of the same polyhedron (e.g., hinge 312d). To reiterate, each polyhedron has a first hinge oriented in the x-direction and a second hinge oriented in the orthogonal y-direction. In addition, the geometry described herein enables the geometric transformations of the present disclosure to form the same parallelepiped inverted configuration in three different ways, wherein each face of the parallelepiped inverted configuration includes a) four isosceles triangle faces of four different polyhedrons (each corresponding to a relatively small third face 330 or fourth face 332), or b) two right triangle faces of two different polyhedrons (each corresponding to a relatively large first face 326 or second face 328).

[0069] The geometric transformations of the present disclosure may include additional optional features that enhance the ability of the transformation to exhibit certain properties that make the transformation more engaging as a teaching tool or puzzle, or otherwise make the transformation more appealing.

[0070] In order to demonstrate the triple inversion capability of the geometric transformation of the present disclosure, different surface decorations may be selectively set on certain surfaces of the polyhedron. Specifically, certain surfaces of the polyhedron may be selectively provided with different surface decorations to demonstrate the property that all outermost surfaces of one inversion configuration are completely hidden as inner surfaces in the other two inversion configurations. Otherwise, the user may not understand the triple inversion capability of the geometric transformation.

[0071] As used herein, one surface decoration is different from another surface decoration, for example, if it has a different color, pattern, surface texture, graphic theme, orientation, or other attributes that impart a different appearance and / or tactile feel to another surface decoration. On the other hand, surface decoration is not limited to a single color or texture, and can include a coordinated theme, however, the theme has different portions that differ in color or texture (e.g., a repeating decorative pattern (motif)). Any given surface decoration can be produced by the material of the construction surface, the color, graphics, decals, stickers, etc., applied to the surface, and / or the texture of the surface.

[0072] Figure 4 An alternative and representative surface decoration arrangement is schematically illustrated, which exhibits a triple reversal capability of geometric transformation. However, the illustrated embodiment is representative and not limiting.

[0073] Figure 4 (Similar to Figure 3A) is a schematic projection of the transformation segment 400. Specifically, the transformation segment 400 includes six hinged polyhedrons 410a-410f, each of which has four faces and can have Figure 3A-3B The geometric shape of the polyhedron of the transformation segment 300. Figure 3A-3B Two of the geometrically shaped transformation segments 400 can be hingedly connected to form an end-to-end continuous ring to achieve Figure 1 and Figure 2 It should be understood that the polyhedra 410a-410f are hingedly connected together (e.g., by Figure 3A-3B ), which has been removed for simplicity. Figure 4 omitted.

[0074] The transformation segment 400 is described with reference to a "first surface", a "second surface" and a "third surface", which are the outermost surfaces in a first, second inverted configuration and a third inverted configuration, respectively, of a geometric transformation having Figure 3A-3B The two segments 400 of the geometric shape are formed, and the segments are hingedly connected to form a continuous ring end to end to achieve Figure 1-Figure 2 Transformation of the dodeca-polyhedron 100, 200.

[0075] In particular, the segment 400 is described with reference to first surfaces 450a-450h, second surfaces 452a-452h, and third surfaces 454a-454h. The first surfaces 450a-450h are the outermost surfaces of the first inverted configuration (e.g., Figure 1 of the parallelepiped inverted configuration A), but in the second and third inverted configurations (e.g., Figure 1 The second surfaces 452a-452h are the outermost surfaces of the second inverted configuration (e.g., Figure 1 The third surfaces 454a-454h are visible surfaces of the parallelepiped inverted configuration B of the first inverted configuration and the third inverted configuration, but are hidden as inner surfaces in the first inverted configuration and the third inverted configuration. Figure 1 The outermost surface of the first inverted configuration C) is the outermost surface of the first inverted configuration, but is hidden as an inner surface in the first inverted configuration and the second inverted configuration. To reiterate, the outermost surface of the first inverted configuration is composed of the first surfaces 450a-450h, the outermost surface of the second inverted configuration is composed of the second surfaces 452a-452h, and the outermost surface of the third inverted configuration is composed of the third surfaces 454a-454h.

[0076] In some embodiments, the first surface decoration is different from the second surface decoration and / or the third surface decoration to exhibit a triple reversal capability of the transformation. Figure 4 In the embodiment of FIG. 4 , the first surfaces 450a - 450h carry concentric circles, the second surfaces 452a - 452h carry parallel lines, and the third surfaces 454a - 454h carry parallel lines and perpendicular lines.

[0077] Although the polyhedron of segment 400 may have Figure 3A-3B The same geometry of the polyhedron of segment 300 is used to describe Figure 4 The term "surface" for the first surface, the second surface, and the third surface does not correspond to the term "surface" used to describe Figure 3A and Figure 3B The term "face" is used to describe the geometry of a polyhedron. For example, the geometry of tetrahedral polyhedrons 410a-410f provides that each polyhedron has a first face, a second face, a third face, and a fourth face; however, polyhedrons 410a-410f do not have all of the first, second, and third surfaces. In fact, Figure 4 Each of the polyhedrons 410a-410l in has only two types of surfaces: a first surface and a second surface; a first surface and a third surface, or a second surface and a third surface. In other words, according to Figure 4 Surface decoration arrangement, each polyhedron 410a-410f has a surface that is the outermost (visible) surface in only two of the three inverted configurations.

[0078] As shown, each of the polyhedrons 410a-410f includes two different types of surfaces. Polyhedrons 410a, 410d include a first surface and a second surface in the relative positions shown; polyhedrons 410b, 410e include a second surface and a third surface; and polyhedrons 410c, 410f include a first surface and a third surface. Two such transformation segments 400 are hingedly connected into an end-to-end continuous ring (assuming that each of the polyhedrons has Figure 3A-3B ) enables the resulting geometric transformation to present only the first surfaces 450a-450h in a first parallelepiped inverted configuration; only the second surfaces 452a-452h in a second parallelepiped inverted configuration; and only the third surfaces 454a-454h in a third parallelepiped inverted configuration. Advantageously, this helps a user and / or observer understand when the transformation is in different inverted configurations.

[0079] The foregoing surface decoration arrangements are representative and non-limiting. For example, in other embodiments, the first and second surfaces may have the same or coordinated surface decoration that is different from the third surface; such a configuration will present the same or coordinated surface decoration in two different inverted configurations, but not in the third configuration. In still other embodiments, the first, second, and third surfaces all have the same or coordinated surface decoration.

[0080] As another optional feature, any geometric transformation of the present disclosure may include magnets positioned and polarized to stabilize the transformation in inverted and intermediate configurations, including Figure 6A-6F Those shown in .

[0081] Figure 5 FIG. 5 shows a representative magnet arrangement in a transformation segment 500 according to an embodiment of the present disclosure. Figure 3A and Figure 4 , Figure 5 is a schematic projection, and the transformation segment 500 has Figure 1 and Figure 2 Specifically, the transformation segment 500 includes four hinged polyhedrons 510a-510d, each of which corresponds to one of the polyhedrons of the transformations 100 and 200, and each of which may have Figure 3A-3B The geometric shapes shown in .

[0082] Three of the transformation segments 500 can be hingedly connected to form an end-to-end continuous ring to achieve Figure 1 and Figure 2 The dodeca-polyhedron transformation 100, 200. The polyhedrons 510a-510d are hingedly coupled together by hinges 512b-512d, and the hinge 512a is configured to couple the polyhedron 510a to another adjacent polyhedron (not shown).

[0083] In some embodiments, at least some of the magnets are positioned and polarized so that when positioned adjacent to each other, the hinged faces of adjacent polyhedrons can be magnetically coupled. For example, polyhedrons 510a, 510b are provided with magnets that are positioned and polarized so that the second face 528a of polyhedron 510a can be magnetically coupled with the second face 528b of polyhedron 510b.

[0084] In some embodiments, at least some of the magnets are positioned and polarized such that when positioned adjacent to one another, the mirror images of the non-hinged polyhedrons are magnetically coupled. Figure 2, the magnets may be arranged on the isosceles faces of the polyhedrons 210a and 210h so that those faces are arranged in certain configurations (such as Figure 6B The configuration shown in FIG. 1 is magnetically coupled together.

[0085] Consistent with these objectives, a representative magnet arrangement will now be described.

[0086] Each of the polyhedrons 510a-510d includes a plurality of magnets, i.e., at least one magnet positioned adjacent to each face, such that the magnetic field from the magnet extends through the face to which the magnet is positioned adjacent. For example, the polyhedron 510a includes a magnet 560a positioned adjacent to the first face 526a, a magnet 562a positioned adjacent to the second face 528a, a magnet 564a positioned adjacent to the third face 530a, and a magnet 566a positioned adjacent to the fourth face 532a. Similarly, the polyhedrons 510b-510d include at least one magnet positioned adjacent to each face.

[0087] As per Figure 5 As is apparent from the symbols in , the magnets positioned adjacent to the hinged faces have opposite polarities to enable magnetic coupling. For example, magnets 562a and 562b (positioned adjacent to the second faces 528a, 528b, respectively) have opposite polarities. Similarly, magnets 560b, 560c (positioned adjacent to the first faces 526b, 526c, respectively) have opposite polarities.

[0088] In addition, magnets positioned adjacent to corresponding (similar) faces of the hinged polyhedron have opposite polarities, even if the faces are not directly hinged. For example, magnets 564a, 564b are positioned adjacent to third faces 530a, 530b, respectively, and have opposite polarities. Similarly, magnets 566a, 566b are positioned adjacent to fourth surfaces 532a, 532b, respectively, and have opposite polarities.

[0089] exist Figure 5 In the embodiment, each of the polyhedrons 510a-510d has a magnet of a single polarity. However, in other embodiments, at least some of the polyhedrons have magnets of two polarities, particularly if the polarity of each magnet is opposite to the polarity of the magnet of the corresponding face of the hinged polyhedron. Thus, Figure 5 The arrangements shown in are representative and not limiting.

[0090] In addition, despite Figure 5A single "+" or "-" symbol is shown for each face of the polyhedrons 510a-510d, but such symbols may represent more than one magnet, i.e., some embodiments include more than one magnet positioned adjacent to each face, e.g., two or three magnets per face. Such a configuration may increase the magnetic force between adjacent polyhedrons. In fact, a single face of a single polyhedron may have magnets of both polarities, e.g., if each magnet has a polarity opposite to that of a corresponding magnet on an adjacent hinged polyhedron.

[0091] although Figure 5 Each polyhedron is shown to include a plurality of magnets, and each face of each polyhedron has at least one magnet disposed adjacent to the face, but the present disclosure contemplates that in some embodiments, some faces of some polyhedrons do not include any magnets positioned adjacent thereto. For example, in some embodiments, polyhedrons 510a-510d may omit magnets 560a-560d (and / or magnets 562a-562d, 564a-564d, or 566a-566d). For example, in some embodiments, one or more of polyhedrons 510a-510d includes only a single magnet. Reducing the number of magnets may advantageously reduce manufacturing costs; however, reducing the number of magnets may impair functionality.

[0092] exist Figure 5 In FIG. 5 , polyhedra 510a and 510c may generally be considered "A-type" polyhedra, and polyhedra 510b and 510d may be considered "B-type" polyhedra, because the magnetic polarities of A-type and B-type polyhedra attract each other. As shown, transformation segment 500 is an ordered segment of an ABAB polyhedron.

[0093] One or more different structures can be utilized to arrange magnets adjacent to the faces of corresponding polyhedrons. In certain embodiments, each magnet is arranged in an inner cavity formed by the outer shell of the polyhedron. In such embodiments, each magnet can be arranged adjacent to the face by adhering the magnet to a face, by fitting the magnet in a support or recess formed integrally with a face, by accommodating the magnet in a groove, track or bracket formed integrally with the inner side of a face, or by other magnet positioning devices. In certain embodiments, the magnet is designed to move relative to its adjacent face, such as by moving in a bracket or track. Representative structures for positioning magnets adjacent to faces include those described in U.S. Patents Nos. 10,569,185 and 10,918,964 and U.S. Patent Publication No. US2022 / 0047960, which are incorporated herein by reference in their entirety.

[0094] Advantageously, the aforementioned magnetic configuration enables the geometric transformation of the present disclosure to be stable at Figure 1 and Fig. 6F The reverse configuration shown in and some intermediate configurations such as Figure 6B-6D As another benefit, the aforementioned magnetic configuration, with Figure 3A and Figure 3B In combination with the geometries detailed in , magnetic and geometric compatibility with other geometric transformations, such as those described in U.S. Pat. Nos. 10,569,185 and 10,918,964, can be achieved.

[0095] Figure 6A-6F A representative method of manipulating the transformation 600 of the present disclosure into a parallelepiped inverted configuration is illustrated. Figure 1 and Figure 2 The geometric transformations are the same, and each of the polyhedrons 610a-610l has Figure 3A and Figure 3B The geometry and hinged connection shown in .

[0096] To aid understanding, transformation 600 has Figure 4 , although this feature is optional. In particular, transformation 600 is provided with three different surface decorations: a first surface (exemplified by first surface 650 of polyhedron 610d carrying concentric circles); a second surface (exemplified by second surface 652 of polyhedron 610b carrying parallel lines); and a third surface (exemplified by third surface 654 of polyhedron 610f carrying parallel and perpendicular lines). Polyhedrons 610a, 610g have corresponding Figure 4 The surface decoration of the polyhedron 410a; the polyhedrons 610b, 610h have corresponding Figure 4 The surface decoration of the polyhedron 410b; the polyhedrons 610c, 610i have corresponding Figure 4 The surface decoration of the polyhedron 410c; the polyhedrons 610d, 610j have corresponding Figure 4 The surface decoration of the polyhedron 410d; the polyhedrons 610e and 610k have the corresponding Figure 4 The surface decoration of the polyhedron 410e; and the polyhedrons 610f, 610l have corresponding Figure 4 The following method is described with respect to different types of surface decoration (eg, first surface, second surface, third surface) to help understand how the method can be applied to achieve all three inversion configurations.

[0097] The following description provides a general method for configuring the transformation 600 into three different parallelepiped inversion configurations, wherein the outermost surface of each inversion configuration consists of the first surface 650, the second surface 652, or the third surface 654. To aid understanding, a specific method for configuring the transformation 600 into a parallelepiped inversion configuration having an outermost surface including the second surface 652 (e.g., consisting of the second surface 652) is also provided. However, the method can be easily applied to configuring the transformation 600 into a parallelepiped inversion configuration having an outermost surface including the first surface 650 or the third surface 654 (e.g., consisting of the first surface 650 or the third surface 654).

[0098] It should be understood that the illustrated method is representative and not limiting. A skilled person using the transformation segment 300 can implement the method using fewer steps than all the steps illustrated and / or by combining certain steps. Fig. 6F The inverted configuration shown in .

[0099] exist Fig. 6A In the optional first step shown in , the transformation 600 is placed in the illustrated open loop configuration, whereby diagonally opposite polyhedra exhibit different surface decorations. For example, polyhedra 610a, 610b, 610g, 610h exhibit a second surface 652, while polyhedra 610e, 610f, 610k, 610l exhibit a third surface 654.

[0100] Next, diagonally opposed polyhedra exhibiting the same surface decoration are translated adjacent to each other, thereby producing four adjacent triangular surfaces exhibiting the same surface decoration. In this example, polyhedron 610a is translated diagonally to abut polyhedron 610h, thereby producing Figure 6B It is worth noting that the outermost surface of the resulting inverted parallelepiped configuration will include a second surface 652 exhibited on diagonally opposite polyhedrons 610a, 610b, 610g, 610h. Therefore, this step can be adapted so that the resulting inverted configuration exhibits a different surface decoration.

[0101] Figure 6B Shown by Fig. 6A The steps of 610 and 610 produce an intermediate configuration that can be described as a triangular rhombus configuration. The end polyhedrons are then rotated inwardly on the corresponding penultimate polyhedrons to which the end polyhedrons are hingedly connected. In this example, polyhedrons 610j, 610k are rotated inwardly on polyhedrons 610l, 610i, respectively, and polyhedrons 610d, 610e are rotated inwardly on polyhedrons 610c, 610f, respectively. This produces a configuration as shown in FIG. Figure 6C The configuration shown in .

[0102] Figure 6C Shown by Figure 6B The steps of . In this intermediate configuration, transformation 600 has a longitudinal axis 656 and a latitudinal axis 658. On each side of the longitudinal axis 656, transformation 600 has three distinct points (a center point and two outer points) that include vertices of one or more polyhedrons. The polyhedrons are then manipulated so that, on a first side of the longitudinal axis 656, the center point intersects with an outer point on a first side of the latitudinal axis 658. For example, the points of polyhedron 610h are brought together with the points of polyhedron 610i. The polyhedrons are further manipulated so that, on a second side of the longitudinal axis 656 (opposite the first side), the center point intersects with an outer point on a second side of the latitudinal axis 658 (opposite the first side). For example, the points of polyhedron 610b are brought together with the points of polyhedron 610c. This produces a configuration such as Fig.6D The configuration shown in .

[0103] Fig.6D Shown by Figure 6C The central vertex 660, which is centrally disposed between the polyhedrons 610a, 610b, 610g, 610h, is then lifted upward while the end points 662a, 662b are rotated downward.

[0104] Fig. 6E Shown by Fig.6D As a final step to achieve the parallelepiped inverted configuration, the endpoints 662a, 662b are brought together to produce Fig. 6F The parallelepiped inverted configuration.

[0105] like Fig. 6F , the resulting parallelepiped inverted configuration has an outermost surface including (e.g., consisting of) a second surface 652 (carrying parallel lines in this example). To reiterate, the first surface 650 and the third surface 654 are internally hidden within the transformation 600 in the illustrated parallelepiped configuration. For the avoidance of doubt, Fig. 6F The view shown in is the same as the view of the opposite side of the transformation 600 (ie only the second surface 652 is shown). The aforementioned method can be adapted so that the outermost surface of the parallelepiped consists of only the second surface or the third surface.

[0106] The foregoing description provides representative examples of geometric transformations configured to achieve a triple inversion configuration, optionally with surface decoration and / or magnetic features that complement the triple inversion functionality.

Claims

1. A geometric transformation puzzle, comprising: transformation, the transformation comprising a plurality of polyhedrons, wherein the plurality of polyhedrons consists of twelve polyhedrons hingedly connected into a ring, and each of the polyhedrons comprises three different side lengths, Wherein the transformation is configurable between a first inverted configuration, a second inverted configuration and a third inverted configuration, wherein the first inverted configuration, the second inverted configuration and the third inverted configuration are congruent parallelepipeds.

2. The geometric transformation puzzle according to claim 1, wherein: Each of the polyhedrons includes one side having a side length of √3 units, two sides having a side length of √2 units, and three sides having a side length of one unit.

3. The geometric transformation puzzle according to claim 2, in, Each of the polyhedrons includes a magnet disposed adjacent a face, wherein the magnets of adjacent polyhedrons in the ring have opposite polarity.

4. The geometric transformation puzzle according to claim 1, wherein: Each of the polyhedrons includes one side having a side length of √3 units, one side having a side length of √2 units, and one side having a side length of one unit.

5. The geometric transformation puzzle according to claim 4, in, Each of the polyhedrons includes a magnet disposed adjacent a face, wherein the magnets of adjacent polyhedrons in the ring have opposite polarity.

6. The geometric transformation puzzle according to claim 1, in, Each of the polyhedrons includes a first face, a second face, a third face, and a fourth face, Each of the polyhedrons includes a first magnet disposed adjacent to the first face, wherein the first magnets of adjacent polyhedrons have opposite polarities.

7. The geometric transformation puzzle according to claim 6, wherein: Each of the polyhedrons includes a second magnet disposed adjacent to the second face, wherein the second magnets of adjacent polyhedrons in the ring have opposite polarities.

8. The geometric transformation puzzle according to claim 7, wherein: Each of the polyhedrons includes a third magnet disposed adjacent to the third face, wherein the third magnets of adjacent polyhedrons in the ring have opposite polarities.

9. The geometric transformation puzzle according to claim 8, wherein: Each of the polyhedrons includes a fourth magnet disposed adjacent to the fourth face, wherein the fourth magnets of adjacent polyhedrons in the ring have opposite polarities.

10. The geometric transformation puzzle according to claim 1, wherein: the outermost surface of the first inverted configuration is a hidden inner surface in the second inverted configuration and the third inverted configuration, The outermost surface of the second inverted configuration is a hidden inner surface in the first inverted configuration and the third inverted configuration, and The outermost surface of the third inverted configuration is a hidden inner surface in the first inverted configuration and the second inverted configuration.

11. The geometric transformation puzzle according to claim 1, wherein: Each of the polyhedrons includes two non-congruent faces.

12. The geometric transformation puzzle according to claim 1, wherein: The outermost surface of the first inverted configuration consists of a first surface, The outermost surface of the second inverted configuration consists of the second surface, The outermost surface of the third inverted configuration consists of the third surface, and The first surface, the second surface and the third surface are mutually exclusive.

13. The geometric transformation puzzle according to claim 1, wherein: Adjacent polyhedrons in the ring are mirror images of each other.

14. The geometric transformation puzzle according to claim 13, wherein: Each of the polyhedrons includes a first side and a second side and is hingedly connected to a first adjacent polyhedron of the ring along the first side and hingedly connected to a second adjacent polyhedron of the ring along the second side, wherein the first side is perpendicular to the second side.

15. A geometric transformation puzzle, comprising: transformation, the transformation comprising twelve polyhedra sequentially and hingedly connected in a ring, wherein the transformation can be configured between a first parallelepiped, a second parallelepiped and a third parallelepiped, wherein the first parallelepiped, the second parallelepiped and the third parallelepiped are congruent, wherein the outermost surface of the first parallelepiped consists of a first surface, the outermost surface of the second parallelepiped consists of a second surface, the outermost surface of the third parallelepiped consists of a third surface, and the first surface, the second surface and the third surface are mutually exclusive, Each of the polyhedrons includes three sides with different lengths and two non-congruent faces.

16. The geometric transformation puzzle according to claim 15, wherein: Each of the polyhedrons includes one side having a side length of √3 units, one side having a side length of √2 units, and one side having a side length of one unit.

17. The geometric transformation puzzle according to claim 16, wherein: Each of the polyhedrons includes two sides having a side length of √2 units and three sides having a side length of one unit.

18. The geometric transformation puzzle according to claim 17, wherein: Each of the polyhedrons includes a magnet disposed adjacent a face, wherein the magnets of adjacent polyhedrons in the ring have opposite polarity.

19. A geometric transformation puzzle, comprising twelve polyhedrons sequentially and hingedly connected into a ring, wherein: Each of the polyhedrons includes a side with a side length of √3 units, a side with a side length of √2 units, and a side with a side length of one unit, wherein each of the hinged polyhedrons includes a magnet arranged adjacent to a face, wherein the magnets of adjacent polyhedrons have opposite polarities, wherein the geometric transformation puzzle is configured to be magnetically stabilized in a first parallelepiped, a second parallelepiped, and a third parallelepiped, wherein the first parallelepiped, the second parallelepiped, and the third parallelepiped are congruent.

20. The geometric transformation puzzle according to claim 19, wherein: Each of the polyhedrons includes two non-congruent faces.

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