Triple inversion geometric transformation

Abstract

The present invention provides a geometric transformation structure. A triple inversion geometric transformation structure can be used as a puzzle, a toy, an educational aid, a therapy device, etc. The transformation structure includes a plurality of hinged polyhedra and is positionable between three congruent inversion configurations.

Description

Detailed Description of the Invention

[0001] (Related Applications) This application claims priority to and the benefit of U.S. Patent Application No. 17 / 821,178, filed August 21, 2022, the entire contents of which are incorporated herein by reference. [Background technology]

[0002] Geometric transformations with interlocking members have appeal across generations as puzzles, toys, educational aids, therapeutic devices, etc. Such transformations can be arranged between different geometric configurations, for example as shown in Asano's UK Patent Application GB2,107,200. However, the geometry and structure of known transformations inherently limit the number and types of geometric configurations that can be implemented. Thus, there is a need for geometric transformations that can realize different configurations and have different attributes. Summary of the Invention

[0003] In one aspect, the present invention provides a geometric transformation structure that is invertible (inverted from the inside) in three different ways, so that it presents a common polyhedron in each "inverted configuration" but has a different outermost surface in each of the three instances. For example, a representative embodiment includes a triple-inverting geometric transformation structure that can be steered into a common parallelepiped shape (e.g., a box) in three different ways, so that it presents a different outermost surface in each instance. As detailed herein, embodiments of such transformations can have many interesting attributes that enhance their appeal and utility.

[0004] In one aspect, the present invention provides a geometric transformation structure, the transformation structure including a plurality of hinged polyhedra, the transformation structure positionable between a first inverted configuration, a second inverted configuration, and a third inverted configuration, the first inverted configuration, the second inverted configuration, and the third inverted configuration being congruent. In another aspect, the present invention provides a method for steering a geometric transformation structure to an inverted state.

[0005] In any embodiment, each of the hinged polyhedrons may include one side with a side length of √(3) units, two sides with a side length of √(2) units, and three sides with a side length of 1 unit.

[0006] In any embodiment, all of the outermost surfaces of the first inverted configuration may include a first surface decoration, all of the outermost surfaces of the second inverted configuration may include a second surface decoration, and all of the 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 be different from each other.

[0007] In any embodiment, each of the hinged polyhedrons may include a first face, a second face, a third face and a fourth face, the plurality of hinged polyhedrons including 12 polyhedrons hinged to form a ring, each of the hinged polyhedrons including a first magnet disposed adjacent the first face, the first magnets of adjacent polyhedrons in the ring having opposite polarities.

[0008] In either embodiment, each of the hinged polyhedrons may include a second magnet disposed adjacent the second face, and the second magnets of adjacent polyhedrons of the annulus may have opposite polarities.

[0009] In any embodiment, each of the hinged polyhedrons may include a third magnet disposed adjacent the third surface, and the third magnets of adjacent polyhedrons of the annulus may have opposite polarities.

[0010] In any embodiment, each of the hinged polyhedrons may include a fourth magnet disposed adjacent the fourth face, and the fourth magnets of adjacent polyhedrons of the annulus may have opposite polarities.

[0011] In any embodiment, the outermost surface of the first inverted configuration is an inner surface hidden between the second inverted configuration and the third inverted configuration, the outermost surface of the second inverted configuration is an inner surface hidden between the first inverted configuration and the third inverted configuration, and the outermost surface of the third inverted configuration is an inner surface hidden between the first inverted configuration and the second inverted configuration.

[0012] In either embodiment, each of the hinged polyhedra may be congruent.

[0013] In either embodiment, each of the polyhedrons may be a tetrahedron.

[0014] In any embodiment, 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.

[0015] In any embodiment, the outermost surface of the first inverted configuration may comprise a first surface, the outermost surface of the second inverted configuration may comprise a second surface, and the outermost surface of the third inverted configuration may comprise a third surface. The first, second and third surfaces may be mutually exclusive.

[0016] In any embodiment, the plurality of hinged polyhedrons may comprise 12 polyhedrons hinged to form a ring, and adjacent polyhedrons in the ring may be mirror versions of each other.

[0017] In any embodiment, each of the hinged polyhedrons may include a first edge and a second edge, may be hinged to a first adjacent polyhedron of the annular body along the first edge, and may be hinged to a second adjacent polyhedron of the annular body along the second edge, and the first edge may be perpendicular to the second edge.

[0018] Exemplary embodiments are described with reference to the following drawings, in which like reference numerals refer to like parts throughout the various drawings unless otherwise noted. [Brief description of the drawings]

[0019] [Figure 1] 1 shows perspective views of a geometric transformation of three different inverted parallelepiped configurations at three different times according to one exemplary embodiment of the present invention. [Diagram 2] 1 shows a geometric transformation that is a ring-shaped construction, which is the same as the geometric transformation shown in FIG. [Figure 3A] 3 shows a schematic projection of a geometric transformation segment having the same structure and characteristics as the geometric transformation of FIGS. 1 and 2. FIG. [Figure 3B] FIG. 3B is a detailed view of one polyhedron of the geometric transformation of FIG. 3A. [Figure 4] FIG. 2 shows a schematic diagram of a surface decoration of a geometric transformation segment according to one embodiment of the present invention, where the geometric transformation is the same as that shown in FIG. [Diagram 5] 1 shows a schematic diagram of a magnet arrangement of a geometric transformation segment according to one embodiment of the present invention. [Figure 6A] 2 illustrates a method for steering the geometric transformation of FIG. 1 into an inverted configuration, according to one exemplary embodiment of the present invention. [Figure 6B] 2 illustrates a method for steering the geometric transformation of FIG. 1 into an inverted configuration, according to one exemplary embodiment of the present invention. [Figure 6C] 2 illustrates a method for steering the geometric transformation of FIG. 1 into an inverted configuration, according to one exemplary embodiment of the present invention. [Figure 6D] 2 illustrates a method for steering the geometric transformation of FIG. 1 into an inverted configuration, according to one exemplary embodiment of the present invention. [Figure 6E] 2 illustrates a method for steering the geometric transformation of FIG. 1 into an inverted configuration, according to one exemplary embodiment of the present invention. [Figure 6F] 2 illustrates a method for steering the geometric transformation of FIG. 1 into an inverted configuration, according to one exemplary embodiment of the present invention. DETAILED DESCRIPTION OF THE PREFERRED EMBODIMENTS

[0020] The present invention provides a geometric transformation structure (also referred to herein as a "transformation") of polyhedra including hinge connections, each of said polyhedra having specific geometric properties. Each of said polyhedra is preferably hinged to other polyhedra of the transformation structure and has unique functions that enable the transformation structure to be realized and / or structural features that represent unique attributes of the transformation structure. As used herein, the term "transformation structure" refers to a plurality of hinged polyhedra.

[0021] The transformations described herein have attributes that individually and / or collectively enhance the usefulness and appeal of the transformations as puzzles, educational aids, therapeutic devices, toys, etc. As will be appreciated from the following description, such attributes can include any one or more of the following: The ability to convert the internal inversion of the transformation ("inversion") three times into three inverted polyhedral configurations ("inversion configurations"). In each inversion configuration, all polyhedra represented are congruent to all polyhedra in each of the other inversion configurations. For each inverted configuration, the outermost surface of the polyhedron is different from (e.g., mutually exclusive with) the outermost surfaces of each of the other inverted configurations. For each inverted 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 inverted configuration. · Geometric and magnetic compatibility with other geometric transformations allows the transformation to be assembled and / or combined with other transformations.

[0022] As used herein, the term "congruent" means that two geometric figures have the same shape and dimensions. This includes cases where one geometric figure is the mirror image of the other.

[0023] FIG. 1 illustrates a transform 100 according to an exemplary embodiment of the present invention. As illustrated, the transform 100 has a polyhedral shape, form or configuration (in this example, a parallelepiped). As detailed in FIGS. 2-5, the transform 100 preferably includes multiple hinged polyhedrons that can be manipulated and repositioned relative to one another to stably (e.g., magnetically) create different overall forms or configurations. In the present invention, the term "configuration" refers to the shape, form or configuration of the entire transform 100, and the term "polyhedron" refers to each polyhedron that comprises the transform 100. Nevertheless, the entire transform 100 can have one polyhedral configuration.

[0024] In particular, Figure 1 shows the same transform 100 in three different inverted configurations A, B and C at three different times. In each inverted configuration A, B and C, the transform 100 has a parallelepiped configuration that is congruent with all of the other parallelepiped configurations, such that the surface area of ​​the outermost surface of one of the parallelepiped inverted configurations is equal to the surface area of ​​the outermost surface of the other parallelepiped inverted configuration.

[0025] As used herein, "inverted configuration" refers to the following configuration of transform 100: in this configuration, all outermost surfaces are inner surfaces in another configuration (e.g., another inverted configuration). As used herein, an "inner surface" is a surface that is not the outermost surface of the transform, but extends through the interior volume of the transform. Depending on the geometry of the transform and the materials from which the transform is constructed, the inner surface may or may not be visible. Exemplary inner surfaces include the inner surface shown in FIG. 2a of PCT Publication WO / 2022 / 130285, which is incorporated herein by reference in its entirety.

[0026] In the example of Figure 1, inverted configuration A is an inverted configuration because all of the outermost visible surfaces 102a (first surfaces) are hidden as non-visible interior surfaces in configurations B and C. Similarly, inverted configuration B is an inverted configuration because all of the outermost visible surfaces 102b (second surfaces) are hidden as interior surfaces in inverted configurations A and C. Similarly, inverted configuration C is an inverted configuration because all of the outermost visible surfaces 102c (third surfaces) are hidden as interior surfaces in inverted configurations A and B.

[0027] The ability to realize three congruent inverse configurations of transform 100 can enhance the utility and interest of transform 100. For example, a first surface can optionally have a different appearance and / or texture (surface decoration) than a second surface and / or a third surface. Similarly, a second surface can preferably have a surface decoration different from a first surface and / or a third surface. Also, in some embodiments, a third surface can preferably have a surface decoration different from a first surface and / or a second surface. The surface decoration of any given surface can be generated by constructing a particular surface material, applying a shape to the surface, treating the surface to impart a texture, and / or other causes.

[0028] In the example of Figure 1, the first, second and third surfaces have different surface decorations, which advantageously allows the transform 100 to represent the same parallelepiped inverted configuration with three different surface decorations. Figure 4 details an exemplary surface decoration arrangement such that the transform 100 can represent the same parallelepiped inverted configuration with three different surface decorations.

[0029] In either embodiment, the transformer 100 can include a number of selectable magnets positioned and polarized in configurations to stabilize the transformer 100 in many different configurations (including the parallelepiped of FIG. 1). The total number of magnets can vary, for example, to 12, 24, 36, 48, 72, or greater. FIG. 5 details a configuration that stabilizes the transformer of FIG. 1 in one representative magnet configuration of a parallelepiped inverted configuration.

[0030] Figure 2 shows a perspective view of a transformation 200 that is the same as transformation 100 of Figure 1. Transformation 200 includes multiple polyhedra 210a-210l that are hinged to form a continuous ring.

[0031] Although the exemplary transform 200 includes 12 polyhedra, other embodiments can include a greater number by dividing one or more of the polyhedra 210a-210l into sub-polyhedra. For example, in one embodiment, each of the polyhedra 210a-210l can be divided into two single, complementary polyhedra. When combined, the two polyhedra have the same polyhedral shape as each of the polyhedra 210a-210l of FIG. 1. Thus, such an embodiment includes 24 polyhedra. Thus, the present invention further includes transforms that include 36, 48, or a greater number of polyhedra.

[0032] In the illustrated embodiment, the polyhedrons 210a-210l are congruent and each have the geometric shape detailed in 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 connected to two adjacent ones of the polyhedrons 210a-210l via two of the hinges 212a-212l.

[0033] In the illustrated embodiment, each of the polyhedrons 210a-210l has a solid case with a cavity formed therein that can accommodate one or more magnets positioned and polarized to stabilize the transducer 200 in different configurations (e.g., parallelepiped configurations corresponding to the three inverted configurations). Such exemplary magnet configurations are described in more detail below with respect to FIG. 5. By way of example and not limitation, the solid case of each of the polyhedrons 210a-210l can be formed from a polymer such as high density polyethylene, low density polyethylene (LDPE, HDPE), polypropylene (PP), polystyrene (PS, ABS), polyester (PET), or other suitable durable and safe material.

[0034] Due to the geometric shapes of the polyhedra 210a-210l and the hinge connections 212a-212l between them, the transform 100 can be manipulated into many different configurations, including the three parallelepiped inverted configurations shown in Figures 1 and 6F, as well as the intermediate configurations of Figures 2 and 6A-6E.

[0035] As can be seen from Fig. 2, each of the polyhedra 210a-210l can be provided with surface decorations such as graphics, textures, colors, etc. As can be seen from Fig. 1, by coordinating different surface decorations, the transform 100 can exhibit each of the different surface decorations in each inverted configuration. Fig. 4 details such a surface decoration arrangement.

[0036] Figure 3A is a schematic projection of a transformation segment 300 having the same structure and characteristics as the segments of the geometric transformations of Figures 1 and 2. Specifically, transformation segment 300 includes four hinged polyhedra 310a-310d, each of which corresponds to one of the polyhedra of transformations 100, 200. That is, each polyhedron of transformations 100, 200 has a geometric shape corresponding to polyhedrons 310a-310d.

[0037] Three of the four-polyhedron transform segments 300 may be hinged together in an end-to-end continuous loop to achieve the twelve-polyhedron transforms 100, 200 of Figures 1 and 2. Polyhedrons 310a-310d are hinged together via hinges 312b-312d, with hinge 312a configured to couple polyhedron 310a to other adjacent polyhedrons or transform segments (not shown). Figure 3B is a detailed view of Figure 3A showing details of polyhedron 310c and hinges 312c, 312d.

[0038] The geometries of the polyhedra 310a-310d and the hinge connections between them allow the geometric transformations of the present invention to be steered into the configurations shown and described herein. Thus, Figures 3A and 3B show representative geometries and hinge configurations. However, the particular geometries and coupling arrangements shown in Figures 3A and 3B are representative and not limiting.

[0039] The geometry of FIG. 3A can be realized using a larger number of polyhedra and different hinge connection arrangements. For example, each of the polyhedra 310a-310d can be divided into two or more sub-polyhedra as described above. As an example of a different hinge connection arrangement, the two polyhedra can be hinged with two hinges rather than a single hinge as shown in FIG. 3A. However, it should be understood that the following geometric transformation can be realized with very few geometries of all possible theoretical geometries of the polyhedra 310a-310d. That is, the geometric transformation realizes three congruent inverted configurations by including three of the transformation segments 300 connected as end-to-end continuous rings. At least for this reason, the geometries described herein are not obvious variations of known geometries.

[0040] As shown, each of the polyhedra 310a-310d in the illustrated embodiment is a tetrahedron having four faces, six edges, and four vertices, similar to the polyhedron of the geometric transformation shown in Figures 1 and 2. Nine edges appear in the schematic diagrams of Figures 3A and 3B because the projection of the three-dimensional tetrahedral shape onto the two-dimensional plane of Figures 3A and 3B doubles three edges, but one skilled in the art should understand this property of the projection, which will be made clearer below.

[0041] 3B details the edges, faces and vertices of representative polyhedron 310c, which is congruent with polyhedrons 310a-310b and 310d. Polyhedrons 310a and 310c are mirror images or versions of polyhedrons 310b and 310d.

[0042] 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. Although shown in a two-dimensional projection in Figures 3A and 3B, the geometry of polyhedron 310c defines that in one three-dimensional embodiment of the polyhedron, first side 314 is perpendicular to second side 316 (shown in Figure 2).

[0043] 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.

[0044] The first surface 326 and the second surface 328 are congruent. The third surface 330 and the fourth surface 332 are congruent. Each of the first surface 326, the second surface 328, the third surface 330, and the fourth surface 332 is a right triangle. Furthermore, the third surface 330 and the fourth surface 332 are an isosceles triangle.

[0045] Next, the relative lengths of the six sides will be described in detail with reference to an example diagram 334 applicable to both Figures 3A and 3B. In the following, side lengths will be described, but it should be understood that such description properly describes the distance between corresponding vertices. Therefore, the following description of "side lengths" does not limit the present invention to geometric transformations of tetrahedral polyhedra having six continuous, linear, discontinuous sides. Indeed, the present invention includes geometric transformations formed from polyhedra having discontinuous and / or non-linear sides, provided that the vertices correspond to those shown in Figure 3B and the relative distances therebetween are as defined in diagram 334.

[0046] Each of the six sides of each of the polyhedrons 310a-310c has a relative side length (alternatively, a vertex distance) indicated by a symbol thereon that corresponds to the relative side length defined in the illustrated example 334. In particular, the first side 314, the second side 316, and the sixth side 324 (marked with a plus symbol) have a relative side length of 1 unit, and in some embodiments (e.g., the illustrated embodiment) are the only sides having a relative side length of 1 unit. The third side 318 (marked with a triangle symbol) (the longest side of the polyhedron 310c) has a relative side length of √(3) units (the square root of 3 units), and in some embodiments (e.g., the illustrated embodiment) are the only sides having such a side length. The fourth side 320 and the fifth side 322 (marked with a square symbol) have a relative side length of √(2) units (the square root of 2 units), and in some embodiments (e.g., the illustrated embodiment) are the only sides having such a side length.

[0047] The side lengths shown are relative and can be scaled proportionally provided that the relative lengths between the six sides remain constant. For example, in one exemplary embodiment, the base unit is 10 cm. In such an embodiment, the first side 314, the second side 316, and the sixth side 324 have side lengths of 10 cm. According to the relationship defined in the example diagram 334, the third side 318 (the longest side) has a side length of 10√(3) cm = 17.32 cm, and the fourth side 320 and the fifth side 322 have side lengths of 10√(2) cm = 14.14 cm. In another exemplary embodiment where the base unit is 20 cm, the length of each side would be twice as long as in the previously defined embodiment. Thus, the relative side lengths (alternatively, the vertex distances) defined by the example diagram 334 can be scaled proportionally.

[0048] Returning to FIG. 3A, two additional features of transform segment 300 are evident. First, each polyhedron is a mirror of two adjacent polyhedrons. For example, polyhedron 310b is a mirror of polyhedrons 310a and 310c, polyhedron 310c is a mirror of polyhedrons 310b and 310d, and so on. This attribute allows similar edges of adjacent polyhedrons to be hinged as described below. Thus, there are two types of polyhedrons that are all congruent but mirror images of each other: type 1 polyhedrons (e.g., polyhedrons 310a, 310c) and type 2 polyhedrons (e.g., polyhedrons 310b, 310d). In terms of geometry, transform section 300 includes a repeating alternating pattern of type 1 polyhedrons, type 2 polyhedrons, type 1 polyhedrons, and so on.

[0049] A second property evident from Figure 3A is that adjacent polyhedrons are hinged along like edges via hinges 312a-312d. For example, referring to both Figures 3A and 3B, hinge 312c hinges a first edge 314 of polyhedron 310c to a corresponding first edge of mirrored polyhedron 310b. Similarly, hinge 312d hinges a second edge 316 of polyhedron 310c to a corresponding edge of mirrored polyhedron 310d.

[0050] Hinged or flexible connections allow the polyhedra to be manipulated relative to one another. In this way, the geometric transformation can realize different configurations (e.g., the parallelepiped configuration of FIG. 1) as well as the configurations shown in FIGS. 2 and 6A-6E, but the overall geometric transformation is still a single device and not an incongruous assortment of parts.

[0051] The geometrically transformed polyhedra described herein are typically assembled such that corresponding edges (directly adjacent edges) of adjacent polyhedra are adjacent or spaced apart by less than 1 mm, e.g., 0.5 mm, as is evident from Figure 2, which shows a representative hinge connection between the transform 200 and adjacent polyhedra.

[0052] Hinges 312a-312d can take many different forms. In some embodiments, each of hinges 312a-312d is a decal or sticker that is applied to the faces of at least two adjacent polyhedrons (e.g., mirror image faces of adjacent polyhedrons) such that the hinge extends directly from one of the polyhedrons to the other. For example, referring to FIG. 3A, if hinge 312c has such a structure, hinge 312c is a decal that is applied to at least first face 326 of polyhedron 310c and extends to the adjacent mirror image face of polyhedron 310b, thus hinge-connecting the adjacent polyhedrons along first side 314 of polyhedron 310c. In some such embodiments, the decal can include multiple hinges. For example, in one embodiment, a single continuous decal is applied to polyhedrons 310a-310d, and accordingly includes at least hinges 312b-312d. Representative hinges of this construction are described in detail in US Patents 10,569,185 and 10,918,964, which are incorporated herein by reference in their entireties.

[0053] In other embodiments, the hinge is integrally formed with the polyhedrons and extends directly from one of the polyhedrons to an adjacent polyhedron. In such embodiments, the hinge can be formed as a flexible polymer tape of the same or similar material as the case of the polyhedrons. For example, referring to FIG. 3A, where hinge 312c has such a structure, hinge 312c is integrally formed with polyhedrons 310b, 310c as at least one polymer tape extending between polyhedrons 310b, 310c, thereby connecting to the adjacent polyhedron along first side 314 of polyhedron 310c. Exemplary hinges of this configuration are described in detail in U.S. Pat. No. 11,358,070, which is incorporated herein by reference in its entirety.

[0054] In other embodiments, the hinge is formed as one or more internal flexible connecting belts (e.g., thin flexible polymers or fabrics) that extend between adjacent polyhedrons and are configured to be anchored within the lumen of the adjacent polyhedron. For example, referring to FIG. 3A, when hinge 312c has such a configuration, a portion of hinge 312c is anchored within the lumen of polyhedron 310b and another portion of hinge 312c is anchored to the lumen of polyhedron 310c, thereby connecting the adjacent polyhedron along first side 314 of polyhedron 310c. Exemplary hinges of this configuration are described in detail in PCT Publication No. WO2022 / 030285, which is incorporated herein by reference in its entirety.

[0055] In any embodiment, multiple hinges can extend between adjacent edges of adjacent polyhedrons. The above hinge structures are exemplary and not limiting.

[0056] Based on the geometrical shape of the polyhedrons 310a-310d and this description, it is clear 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 clear from FIG.

[0057] The above-mentioned geometries and hinges allow the geometric transformation of the present invention to realize three inverted configurations, such as the three parallelepipeds shown in FIG. 1. For example, the geometries described herein allow the first hinge of each polyhedron (e.g., hinge 312c in the example of polyhedron 310c) to have a perpendicular orientation to the second hinge of the same polyhedron (e.g., hinge 312d). Also, each polyhedron has a first hinge oriented in the x-direction and a second hinge oriented in the orthogonal y-direction. Furthermore, the geometries described herein allow the geometric transformation of the present invention to form the same parallelepiped inverted configuration in three different ways, including a) four isosceles triangular faces of four different polyhedrons (corresponding to the relatively small third face 330 or fourth face 332, respectively), or b) two right-angled triangular faces of two different polyhedrons (corresponding to the relatively large first face 326 or second face 328, respectively).

[0058] The geometric transformations of the present invention may include additional optional features that demonstrate the power of some attributes to make the transformation more attractive as an educational tool or puzzle, or to make the transformation more attractive in other ways.

[0059] In order to express the triple inversion capability of the geometric transformation of the present invention, some surfaces of the polyhedron can be selectively set with different surface decorations. Specifically, some surfaces of the polyhedron can be selectively provided with different surface decorations so that all outermost surfaces of one inversion configuration show completely hidden attributes as inner surfaces in the other two inversion configurations. Otherwise, users may not be able to understand the triple inversion capability of the geometric transformation.

[0060] As used herein, one surface decoration may differ from another surface decoration, for example, if it has a different color, pattern, surface texture, graphic theme, orientation, or other attribute that imparts a different appearance and / or tactile sensation than another surface decoration. On the other hand, a surface decoration is not limited to a single color or texture, but may include a coordinating theme, but the theme has different portions of color or texture (e.g., a repeating decorative motif). Any given surface decoration may be produced by the materials that construct the surface, the color, the application of graphics, decals, stickers, etc. to the surface, and / or the texture of the surface.

[0061] 4 illustrates a schematic representation of an optional and representative surface decoration arrangement that shows the triple inversion capability of geometric transformations. However, the illustrated embodiment is representative and not limiting.

[0062] FIG. 4 (similar to FIG. 3A) is a schematic projection of a transform section 400. Specifically, transform section 400 includes six hinged polyhedra 410a-410f, each having four faces and which may have the polyhedral geometry of transform section 300 of FIGS. 3A-3B. Two of transform segments 400 having the geometry of FIGS. 3A-3B may be hinged end-to-end as a continuous ring to achieve the twelve-polyhedron transforms 100, 200 of FIGS. 1 and 2. It should be understood that polyhedra 410a-410f are hinged (e.g., by hinges as shown in FIGS. 3A-3B) and omitted from FIG. 4 for simplicity.

[0063] 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 the first, second, and third inverted configurations of a geometric transformation, respectively. The geometric transformation is formed by two of the segments 400 having the geometric shapes of Figures 3A-3B, which are hinged to form an end-to-end continuous loop, thereby implementing the twelve-polyhedron transformation 100, 200 of Figures 1-2.

[0064] In particular, segment 400 will be described with reference to first surfaces 450a-450h, second surfaces 452a-452h, and third surfaces 454a-454h. First surfaces 450a-450h are the outermost surfaces of the first inverted configuration (e.g., the visible surfaces of parallelepiped inverted configuration A in FIG. 1), but are hidden as inner surfaces in the second and third inverted configurations (e.g., inverted configurations B and C in FIG. 1). Second surfaces 452a-452h are the outermost surfaces of the second inverted configuration (e.g., the visible surfaces of parallelepiped inverted configuration B in FIG. 1), but are hidden as inner surfaces in the first and third inverted configurations. Third surfaces 454a-454h are the outermost surfaces of the third inverted configuration (e.g., inverted configuration C in FIG. 1), but are hidden as inner surfaces in the first and second inverted configurations. Additionally, the outermost surfaces of the first inverted configuration are made up of first surfaces 450a-450h, the outermost surfaces of the second inverted configuration are made up of second surfaces 452a-452h, and the outermost surfaces of the third inverted configuration are made up of third surfaces 454a-454h.

[0065] In some embodiments, the first surface decoration is different from the second surface decoration and / or the third surface decoration to exhibit a triple inversion capability of the transformation. In the embodiment of Figure 4, the first surface 450a-450h has concentric circles, the second surface 452a-452h has parallel lines, and the third surface 454a-454h has parallel and perpendicular lines.

[0066] Although the polyhedron of segment 400 may have the same geometry as the polyhedron of segment 300 of Figures 3A-3B, the term "surface" used to describe the first, second, and third surfaces of Figure 4 does not correspond to the term "face" used to describe the geometry of the polyhedron of Figures 3A and 3B. For example, the geometry of tetrahedral polyhedrons 410a-410f defines each polyhedron as having a first, second, third, and fourth face, however, none of polyhedrons 410a-410f has all of the first, second, and third surfaces. In practice, each of polyhedrons 410a-410l of Figure 4 has a first and second surface, a first and third surface, or a second and third surface. In other words, according to the surface decoration arrangement of FIG. 4, each polyhedron 410a-410f has a surface that is the outermost (visible) surface in only two of the three inverted configurations.

[0067] As shown, each of the polyhedrons 410a-410f includes two different types of surfaces: polyhedrons 410a, 410d include first and second surfaces in the relative positions shown, polyhedrons 410b, 410e include second and third surfaces, and polyhedrons 410c, 410f include first and third surfaces. By hingedly connecting two such transform sections 400 as an end-to-end continuous annulus (assuming each of the polyhedrons has the geometric shape shown in Figures 3A-3B), the resulting geometric transformation can be such that in the first parallelepiped inverted configuration only the first surfaces 450a-450h appear, in the second parallelepiped inverted configuration only the second surfaces 452a-452h appear, and in the third parallelepiped inverted configuration only the third surfaces 454a-454h appear. Advantageously, this helps the user and / or viewer understand when the transform is in a different inverted configuration.

[0068] The surface decoration arrangements described above are representative and not limiting. For example, in other embodiments, the first and second surfaces can have the same or matching surface decoration that is different from the third surface, such that such configurations exhibit the same or matching surface decoration in two different inverted configurations but not in the third configuration. In some other embodiments, the first, second, and third surfaces all have the same or matching surface decoration.

[0069] As another optional feature, any of the geometric transforms of the present invention may include magnets that can be positioned and poled to stabilize them in inverted and intermediate configurations, including those shown in Figures 6A-6F.

[0070] Figure 5 shows a representative magnet arrangement of a transform segment 500 according to one embodiment of the present invention. Similar to Figures 3A and 4, Figure 5 is a schematic projection, and the transform segment 500 has the same structure and characteristics as the segments of the geometric transformations of Figures 1 and 2. Specifically, the transform segment 500 includes four hinged polyhedra 510a-510d, each of which corresponds to one of the polyhedra of the transformations 100, 200, and each of which may have the geometric shape shown in Figures 3A-3B.

[0071] Three of the transformation sections 500 may be hinged as an end-to-end continuous ring to achieve the twelve-polyhedron transformations 100, 200 of Figures 1 and 2. Polyhedrons 510a-510d are hinged via hinges 512b-512d, with hinge 512a configured to couple polyhedron 510a to another adjacent polyhedron (not shown).

[0072] In some embodiments, at least some of the magnets are positioned and polarized such that when positioned adjacent one another, hinged faces of adjacent polyhedrons can magnetically couple, for example, polyhedrons 510a, 510b are provided with magnets positioned and polarized such that second face 528a of polyhedron 510a can magnetically couple with second face 528b of polyhedron 510b.

[0073] In some embodiments, at least some of the magnets are positioned and polarized such that mirror image faces of the non-hinged polyhedrons are magnetically coupled when positioned adjacent to one another. For example, and referring briefly to FIG. 2, magnets may be positioned on the isosceles faces of polyhedrons 210a and 210h. In this manner, the faces may be magnetically coupled in several configurations, such as the configuration shown in FIG. 6B.

[0074] Consistent with these goals, representative magnet configurations are described herein.

[0075] Each of the polyhedrons 510a-510d includes a plurality of magnets, i.e., at least one magnet disposed adjacent to each face, such that a magnetic field from the magnets can extend through the face to which the magnet is adjacently disposed. 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 disposed adjacent to each face.

[0076] As is evident from the labeling in Figure 5, magnets positioned adjacent to hinged faces have opposite polarities to allow for magnetic coupling. For example, magnets 562a and 562b (positioned adjacent to second faces 528a, 528b, respectively) have opposite polarities. Similarly, magnets 560b, 560c (both positioned adjacent to first faces 526b, 526c, respectively) have opposite polarities.

[0077] Also, magnets positioned adjacent corresponding (similar) faces of the hinged polyhedrons have opposite polarities even if the faces are not directly hinged. For example, magnets 564a, 564b are positioned adjacent third faces 530a, 530b, respectively, and have opposite polarities. Similarly, magnets 566a, 566b are positioned adjacent fourth surfaces 532a, 532b, respectively, and have opposite polarities.

[0078] In Figure 5, each of the polyhedrons 510a-510d has magnets of a single polarity. However, in other embodiments, at least some of the polyhedrons have magnets of two polarities, and in particular the polarity of each magnet is opposite to the polarity of the magnet on the corresponding face of the hinged polyhedron. Thus, the configuration shown in Figure 5 is representative and not limiting.

[0079] Additionally, although FIG. 5 illustrates a single "+" or "-" symbol used on each face of polyhedrons 510a-510d, such a symbol may represent multiple magnets, i.e., some embodiments include multiple magnets disposed adjacent each face, e.g., two or three magnets on each face. Such a configuration may increase the magnetic forces between adjacent polyhedrons. Indeed, a single face of a single polyhedron may have magnets of two polarities, e.g., each magnet may have a polarity opposite to that of the corresponding magnet on the adjacent hinged polyhedron.

[0080] 5 shows each polyhedron including multiple magnets, with each face of each polyhedron having at least one magnet disposed adjacent to the face, the present invention contemplates that in some embodiments, some faces of polyhedrons do not include any magnets disposed adjacent thereto. For example, in some embodiments, polyhedrons 510a-510d can 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 include only a single magnet. Reducing the number of magnets is advantageous in reducing manufacturing costs. However, reducing the number of magnets may also impair functionality.

[0081] 5, polyhedra 510a and 510c may be generally considered to be "A-type" polyhedra, and polyhedra 510b and 510d may be considered to be "B-type" polyhedra because the magnetic polarities of the A-type and B-type polyhedra are attracted to each other. As shown, transformation segment 500 is an ordered segment of an ABAB polyhedron.

[0082] The magnets can be positioned adjacent to the corresponding polyhedron faces using one or more different structures. In some embodiments, each magnet is positioned within a bore formed from the housing of the polyhedron. In such embodiments, each magnet can be positioned adjacent to one face by gluing the magnet to the face, fitting the magnet into a support or recess integrally formed with the face, by housing the magnet in a groove, rail or carriage integrally formed with the inside of the face, or by other magnet positioning devices. In some embodiments, the magnet is designed to move relative to its adjacent face by moving within a carriage or rail. Exemplary structures for positioning magnets adjacent to faces include those described in U.S. Pat. No. 10,569,185, U.S. Pat. No. 10,918,964, and U.S. Pat. No. US2022 / 0047960, each of which U.S. patents and U.S. patent publications are incorporated herein by reference in their entirety.

[0083] Advantageously, the above-mentioned magnetic configurations allow the geometric transformation of the present invention to be stable to the inverted configurations shown in Figures 1 and 6F as well as to some intermediate configurations (e.g., intermediate configurations shown in Figures 6B-6D). As another advantage, the above-mentioned magnetic configurations can be combined with the geometries detailed in Figures 3A and 3B to achieve magnetic and geometric compatibility with other geometric transformations (e.g., those described in U.S. Patents 10,569,185 and 10,918,964).

[0084] Figures 6A-6F show an exemplary method of manipulating the transform 600 of the present invention into a parallelepiped inversion configuration. The transform 600 is the same as the geometric transform of Figures 1 and 2, and each of the polyhedra 610a-610l has the geometric shape and hinge connections shown in Figures 3A and 3B.

[0085] Transformation 600 has a surface decoration arrangement as shown in Fig. 4, although for ease of understanding this characteristic can be selected. In particular, transformation 600 includes a first surface (exemplified by first surface 650 having a polyhedron 610d with concentric circles), a second surface (exemplified by second surface 652 passing through polyhedron 610b with parallel lines), and a third surface (exemplified by third surface 654 passing through polyhedron 610f with parallel and perpendicular lines). Polyhedrons 610a, 610g have a surface decoration corresponding to polyhedron 410a of Fig. 4, polyhedrons 610b, 610h have a surface decoration corresponding to polyhedron 410b of Fig. 4, polyhedrons 610c, 610i have a surface decoration corresponding to polyhedron 410c of Fig. 4, polyhedrons 610d, 610j have a surface decoration corresponding to polyhedron 410d of Fig. 4, polyhedrons 610e, 610k have a surface decoration corresponding to polyhedron 410e of Fig. 4, and polyhedrons 610f, 610l have a surface decoration corresponding to polyhedron 410f of Fig. 4. Describing the following method for different types of surface decorations (e.g., first surface, second surface, third surface) is intended to facilitate understanding of how the method can be applied to all implementations of the three inverted configurations.

[0086] The following description provides a general method for configuring the transform 600 into three different parallelepiped inverted configurations, where the outermost surface of each inverted configuration is composed of a first surface 650, a second surface 652, or a third surface 654. For ease of understanding, a specific method for configuring the transform 600 to have a parallelepiped inverted configuration having an outermost surface that includes the second surface 652 (e.g., composed of the second surface 652) is also provided. However, this method can be easily applied to a parallelepiped inverted configuration having an outermost surface that includes the first surface 650 or the third surface 654 (e.g., composed of the first surface 650 or the third surface 654).

[0087] It should be understood that the exemplary method is representative rather than limiting, and that the engineer conversion section 300 may implement the inverted configuration shown in FIG. 6F using fewer than all of the illustrated steps and / or by combining some steps.

[0088] In the first optional step shown in Fig. 6A, the transform 600 is arranged in an exemplary open ring configuration. Diagonally opposing polyhedra thus represent different surface decorations. For example, polyhedra 610a, 610b, 610g, 610h represent the second surface 652, and polyhedra 610e, 610f, 610k, 610l represent the third surface 654.

[0089] Next, the diagonally opposed polyhedrons that exhibit the same surface decoration are translated in parallel to be adjacent to each other, thereby generating four adjacent triangular surfaces that exhibit the same surface decoration. In this example, polyhedron 610a is translated diagonally to be adjacent to polyhedron 610h to generate the configuration shown in FIG. 6B. It should be noted that the outermost surface of the resulting parallelepiped inverted configuration includes a second surface 652 that appears on the diagonally opposed polyhedrons 610a, 610b, 610g, 610h. This step can therefore be adapted so that the resulting inverted configurations exhibit different surface decorations.

[0090] Figure 6B shows an intermediate configuration generated by the steps of Figure 6A, which can be described as a Mitsubishi-type configuration. The end polyhedrons are then rotated inwards on their corresponding penultimate polyhedrons hinged to them. In this example, polyhedrons 610j and 610k are rotated inwards on polyhedrons 610l and 610i, respectively, and polyhedrons 610d and 610e are rotated inwards on polyhedrons 610c and 610f, respectively. This produces the configuration shown in Figure 6C.

[0091] FIG. 6C shows an intermediate configuration produced by the steps of FIG. 6B. In this intermediate configuration, the transform 600 has a longitudinal axis 656 and a latitudinal axis 658. On each side of the longitudinal axis 656, the transform 600 has three distinct points (one center point and two outer points) that comprise the vertices of one or more polyhedra. The polyhedra are then manipulated such that on a first side of the longitudinal axis 656, the center point intersects with an outer point on the first side of the latitudinal axis 658. For example, the point of polyhedron 610h is placed with the point of polyhedron 610i. The polyhedra are further manipulated such that on a second side of the longitudinal axis 656 (opposite the first side) the center point intersects with an outer point on the second side of the latitudinal axis 658 (opposite the first side). For example, the point of polyhedron 610b is placed with the point of polyhedron 610c. This produces the configuration shown in FIG. 6D.

[0092] Figure 6D shows an intermediate configuration produced by the steps of Figure 6C, in which a central vertex 660, centrally located between the polyhedra 610a, 610b, 610g, 610h, is raised and end points 662a, 662b are rotated downward.

[0093] Figure 6E shows an intermediate configuration produced by the steps of Figure 6D. As a final step to achieve the inverted parallelepiped configuration, the end points 662a, 662b are placed together to produce the inverted parallelepiped configuration of Figure 6F.

[0094] As shown in Figure 6F, the resulting parallelepiped inverted configuration has an outermost surface that includes a second surface 652 (which in this example has parallel lines) (e.g., consists of the second surface 652). Also, the first surface 650 and the third surface 654 are internally hidden within the illustrated transformation 600 of the parallelepiped configuration. For the avoidance of doubt, the view shown in Figure 6F is the same as the view of the opposite side of the transformation 600 (i.e., showing only the second surface 652). The above-described method can be adapted such that the outermost surface of the parallelepiped consists of only the second surface or the third surface.

[0095] The foregoing description provides representative examples of geometric transformations configured to achieve three inversion configurations, preferably with surface decorations and / or magnetic features that complement the triple inversion functionality.

Claims

1. A geometric transformation structure, the geometric transformation structure includes a transformation structure; the transformation structure includes twelve polyhedrons hinged to form a ring; the transformation structure is positionable between a first parallelepiped, a second parallelepiped and a third parallelepiped, the first parallelepiped, the second parallelepiped and the third parallelepiped being congruent, an outermost surface of the first parallelepiped comprises a first surface, an outermost surface of the second parallelepiped comprises a second surface, an outermost surface of the third parallelepiped comprises a third surface, the first surface, the second surface and the third surface being mutually exclusive; A geometric transformation structure, wherein each of the polyhedra has three different side lengths.

2. 2. The geometric transformation structure of claim 1, wherein each of the polyhedra includes one side whose side length is the square root of three units (√(3) units), one side whose side length is the square root of two units (√(2) units), and one side whose side length is 1 unit.

3. The geometric transformation structure of claim 2 , wherein each of the polyhedra includes two sides whose side lengths are square root of 2 units (√(2) units) and three sides whose side lengths are 1 unit.

4. The geometry transformation structure of claim 3 , wherein each of the polyhedrons includes a magnet disposed adjacent a face, the magnets of adjacent polyhedrons of the annulus having opposite polarities.

5. each of the polyhedrons includes a first surface, a second surface, a third surface, and a fourth surface; The geometric transformation structure according to claim 1 , wherein each of the polyhedrons includes a first magnet disposed adjacent to the first face, the first magnets of adjacent polyhedrons having opposite polarities.

6. The geometric transformation structure of claim 5 , wherein each of the polyhedrons includes a second magnet disposed adjacent the second face, the second magnets of adjacent polyhedrons of the annular body having opposite polarities.

7. The geometric transformation structure of claim 6 , wherein each of the polyhedrons includes a third magnet disposed adjacent the third surface, and the third magnets of adjacent polyhedrons of the annular body have opposite polarities.

8. The geometric transformation structure of claim 7 , wherein each of the polyhedrons includes a fourth magnet disposed adjacent the fourth face, the fourth magnets of adjacent polyhedrons of the annular body having opposite polarities.

9. 2. The geometric transformation structure of claim 1, wherein each of the polyhedrons includes a first side and a second side, and is hinged to a first adjacent polyhedron of the annular body along the first side and hinged to a second adjacent polyhedron of the annular body along the second side, the first side being perpendicular to the second side.

10. A geometric transformation structure, the geometric transformation structure includes a transformation structure; the transformation structure includes a plurality of polyhedrons, the plurality of polyhedrons being comprised of 12 polyhedrons hinged to form a ring, each of the polyhedrons having three different edge lengths; A geometric transformation structure, the transformation structure being positionable between a first inverted configuration, a second inverted configuration, and a third inverted configuration, the first inverted configuration, the second inverted configuration, and the third inverted configuration being congruent parallelepipeds.

11. 11. The geometric transformation structure of claim 10, wherein each of the polyhedra includes one side whose side length is the square root of three units (√(3) units), two sides whose side length is the square root of two units (√(2) units), and three sides whose side length is 1 unit.

12. 11. The geometric transformation structure of claim 10, wherein each of the polyhedra includes one side whose side length is the square root of three units (√(3) units), one side whose side length is the square root of two units (√(2) units), and one side whose side length is one unit.

13. each of the polyhedrons includes a first surface, a second surface, a third surface, and a fourth surface; The geometric transformation structure according to claim 10 , wherein each of the polyhedrons includes a first magnet disposed adjacent the first face, the first magnets of adjacent polyhedrons having opposite polarities.

14. The geometric transformation structure of claim 13 , wherein each of the polyhedrons includes a second magnet disposed adjacent the second face, the second magnets of adjacent polyhedrons of the annulus having opposite polarities.

15. The geometric transformation structure of claim 14 , wherein each of the polyhedrons includes a third magnet disposed adjacent the third surface, the third magnets of adjacent polyhedrons of the annular body having opposite polarities.

16. The geometric transformation structure of claim 15 , wherein each of the polyhedrons includes a fourth magnet disposed adjacent the fourth face, the fourth magnets of adjacent polyhedrons of the annular body having opposite polarities.

17. an outermost surface of the first inverted configuration is an inner surface hidden by the second inverted configuration and the third inverted configuration; an outermost surface of the second inverted configuration is an inner surface hidden by the first inverted configuration and the third inverted configuration; The geometric transformation structure of claim 10 , wherein an outermost surface of the third inverted configuration is an inner surface hidden by the first inverted configuration and the second inverted configuration.

18. The geometric transformation structure of claim 10 , wherein each of the polyhedra includes two non-congruent faces.

19. an outermost surface of the first inverted configuration comprises a first surface; an outermost surface of the second inverted configuration comprises a second surface; an outermost surface of the third inverted configuration comprises a third surface; The geometric transformation structure of claim 10 , wherein the first surface, the second surface, and the third surface are mutually exclusive.

20. The geometric transformation structure of claim 10 , wherein adjacent polyhedra of the annulus are mirror versions of each other.

21. 21. The geometric transformation structure of claim 20, wherein each of the polyhedra includes a first side and a second side, and is hinged to a first adjacent polyhedron of the annular body along the first side and hinged to a second adjacent polyhedron of the annular body along the second side, the first side being perpendicular to the second side.

22. 13. The geometric transformation structure of claim 11 or 12, wherein each of the polyhedrons includes a magnet disposed adjacent a face, the magnets of adjacent polyhedrons of the annulus having opposite polarities.

23. A geometric transformation structure, A geometric transformation structure comprising twelve polyhedrons hinged to form a ring, each of the polyhedrons having one side whose side length is the square root of three units (√(3) units), one side whose side length is the square root of two units (√(2) units), and one side whose side length is 1 unit, each of the hinged polyhedrons including magnets adjacent to a face, the magnets of adjacent polyhedrons having opposite polarities, the transformation structure being configured to be magnetically stabilized by a first parallelepiped, a second parallelepiped, and a third parallelepiped, the first parallelepiped, the second parallelepiped, and the third parallelepiped being congruent.

24. 24. The geometric transformation structure of claim 23, wherein each of the polyhedra includes two non-congruent faces.

25. 24. The geometric transformation structure of claim 23, wherein an outermost surface of the first parallelepiped comprises a first surface, an outermost surface of the second parallelepiped comprises a second surface, and an outermost surface of the third parallelepiped comprises a third surface, the first surface, the second surface, and the third surface being mutually exclusive.

26. 24. The geometric transformation structure of claim 23, wherein adjacent polyhedra of the annulus are mirror versions of each other.

27. A geometric transformation structure, the geometric transformation structure includes a transformation structure; the transforming structure includes a plurality of hinged polyhedra; A geometric transformation structure, the transformation structure being positionable between a first inversion configuration, a second inversion configuration, and a third inversion configuration, the first inversion configuration, the second inversion configuration, and the third inversion configuration being congruent.

28. 28. The geometric transformation structure of claim 27, wherein each of the hinged polyhedra includes one side with a side length of √(3) units, two sides with a side length of √(2) units, and three sides with a side length of 1 unit.

29. all of the outermost surfaces of the first inverted configuration include a first surface decoration; all of the outermost surfaces of the second inverted configuration include a second surface decoration; all of the outermost surfaces of the third inverted configuration include a third surface decoration; The geometric transformation structure of claim 27 , wherein the first surface decoration, the second surface decoration, and the third surface decoration are different from each other.

30. each of the hinged polyhedrons includes a first face, a second face, a third face, and a fourth face, and the plurality of hinged polyhedrons includes twelve polyhedrons hinged to form a ring; 28. The geometric transformation structure of claim 27, wherein each of the hinged polyhedrons includes a first magnet disposed adjacent the first face, the first magnets of adjacent polyhedrons of the annulus having opposite polarities.

31. 31. The geometric transformation structure of claim 30, wherein each of the hinged polyhedrons includes a second magnet disposed adjacent the second face, the second magnets of adjacent polyhedrons of the annulus having opposite polarities.

32. 32. The geometric transformation structure of claim 31 , wherein each of the hinged polyhedrons includes a third magnet disposed adjacent the third face, the third magnets of adjacent polyhedrons of the annulus having opposite polarities.

33. 33. The geometric transformation structure of claim 32, wherein each of the hinged polyhedrons includes a fourth magnet disposed adjacent the fourth face, the fourth magnets of adjacent polyhedrons of the annulus having opposite polarities.

34. an outermost surface of the first inverted configuration is an inner surface hidden by the second inverted configuration and the third inverted configuration; an outermost surface of the second inverted configuration is an inner surface hidden by the first inverted configuration and the third inverted configuration; 28. The geometric transformation structure of claim 27, wherein an outermost surface of the third inverted configuration is an inner surface hidden by the first inverted configuration and the second inverted configuration.

35. 28. The geometric transformation structure of claim 27, wherein each of the hinged polyhedra are congruent.

36. 36. The geometric transformation structure of claim 35, wherein each of the polyhedra is a tetrahedron.

37. 28. The geometric transformation structure of claim 27, wherein the first inverted feature is a first parallelepiped, the second inverted feature is a second parallelepiped, and the third inverted feature is a third parallelepiped.

38. an outermost surface of the first inverted configuration comprises a first surface; an outermost surface of the second inverted configuration comprises a second surface; an outermost surface of the third inverted configuration comprises a third surface; 28. The geometric transformation structure of claim 27, wherein the first surface, the second surface, and the third surface are mutually exclusive.

39. 28. The geometric transformation structure of claim 27, wherein the plurality of hinged polyhedra consists of 12 polyhedra hinged to form a ring, adjacent polyhedra in the ring being mirror versions of each other.

40. 40. The geometric transformation structure of claim 39, wherein each of the hinged polyhedrons includes a first side and a second side, and is hinged to a first adjacent polyhedron of the annular body along the first side and is hinged to a second adjacent polyhedron of the annular body along the second side, the first side being perpendicular to the second side.

41. 29. The geometric transformation structure of claim 28, wherein all outermost surfaces of the first inverted configuration include a first surface decoration, all outermost surfaces of the second inverted configuration include a second surface decoration, and all outermost surfaces of the third inverted configuration include a third surface decoration, the first surface decoration, the second surface decoration, and the third surface decoration being different from one another.

42. each of the hinged polyhedrons includes a first face, a second face, a third face, and a fourth face, and the plurality of hinged polyhedrons includes twelve polyhedrons hinged to form a ring; 42. The geometric transformation structure of claim 41, wherein each of the hinged polyhedrons includes a first magnet disposed adjacent the first face, the first magnets of adjacent polyhedrons of the annulus having opposite polarities.

43. 43. The geometric transformation structure of claim 42, wherein the first inverted feature is a first parallelepiped, the second inverted feature is a second parallelepiped, and the third inverted feature is a third parallelepiped.

44. each of the hinged polyhedrons includes a first face, a second face, a third face, and a fourth face, and the plurality of hinged polyhedrons includes twelve polyhedrons hinged to form a ring; 30. The geometric transformation structure of claim 28, wherein each of the hinged polyhedrons includes a first magnet disposed adjacent the first face, the first magnets of adjacent polyhedrons of the annulus having opposite polarities.

45. each of the hinged polyhedrons includes a first face, a second face, a third face, and a fourth face, and the plurality of hinged polyhedrons includes twelve polyhedrons hinged to form a ring; 30. The geometric transformation structure of claim 29, wherein each of the hinged polyhedrons includes a first magnet disposed adjacent the first face, the first magnets of adjacent polyhedrons of the annulus having opposite polarities.

46. 46. ​​The geometric transformation structure of claim 45, wherein the first inverted feature is a first parallelepiped, the second inverted feature is a second parallelepiped, and the third inverted feature is a third parallelepiped.

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