Triple inversion geometric transformations are useful as puzzles, toys, teaching aids, therapy devices, and the like. The transformations include a plurality of hingedly connected polyhedrons, each of the polyhedrons having at least one of a first surface, a second surface, or a third surface. The transformations are configurable between three congruent inverted configurations.

A polyhedron (100) configured to be removably coupled by shape with another polyhedron (200) to form a Yoshimoto cube (1), such polyhedron (100) comprising: eight half-cubes (101, 102) equal to each other, each of which being delimited by three outer faces (1011, 1012, 1013), having a substantially square configuration, arranged orthogonally to each other whereby each outer face (1011, 1012, 1013) is delimited by two edges (10111, 10112) in common with the other outer faces (1012, 1013) and two free edges (10113, 10114), and by 6 inner faces (1014, 1015, 1016, 1017, 1018, 1019), having a substantially triangular configuration, wherein each inner face (1014, 1015, 1016, 1017, 1018, 1019) extends between a vertex (V), in common with the other inner faces of the half-cube (101), substantially coinciding with the geometric centre of a cube delimited by said three outer faces (1011, 1012, 1013), and a respective base, corresponding to a free edge (10113) between two free edges of one of said outer faces (1011, 1012, 1013); and eight hinge members (300), each configured to connect a free edge (10113, 10113′) of each half-cube (101, 101′) with a free edge (10113′, 10113) of another half-cube (101′, 101) adjacent thereto, wherein each hinge member (300) comprises at least one axis of rotation (301) around which the two half-cubes (101,101′) connected by such hinge member (300) may mutually move; characterized in that each hinge member (300) of such polyhedron (100): extends along each of the two free edges (10113, 10113′) of the two respective adjacent half-cubes (101, 101′) connected by the hinge member (300), starting from a same end of each free edge (10113, 10113′) of the two free edges and for an overall length lower than the length of each free edge (10113, 10113′), optionally substantially other than or equal to half the length of each free edge (10113, 10113′), defines an axis of rotation (301) parallel to each free edge (10113, 10113′), and allows a mutual angular movement between the two respective adjacent half-cubes (101, 101′) connected by it, comprised between a first configuration, wherein the outer faces delimited by the respective free edges (10113, 10113′), at which the connection with the hinge member (300) is made, match, and the corresponding portions of the respective free edges (10113, 10113′) which are not connected by the hinge member (<

Provided is a polyhedral toy which has a wider variety of playing patterns and cannot be predicted in terms of change in forms to thereby enable enhancement of elements of puzzles A first polyhedral piece for forming an upper face of a cube is coupled to a second polyhedral piece for forming a first side face of the cube and a third polyhedral piece for forming a second side face adjacent to the first side face at respective sides of bottom faces of the first polyhedral piece, the second polyhedral piece, and the third polyhedral piece. The second polyhedral piece is coupled to a fourth polyhedral piece for forming a third side face opposed to the second side face of the cube at respective sides of the bottom face of the second polyhedral piece and a bottom face of the fourth polyhedral piece. The fourth polyhedral piece is coupled to a fifth polyhedral piece for forming a lower face opposed to the upper face of the cube at respective sides of the bottom face of the fourth polyhedral piece and a bottom face of the fifth polyhedral piece. The fifth polyhedral piece is coupled to a sixth polyhedral piece for forming a fourth side face opposed to the first side face of the cube at respective sides of the bottom face of the fifth polyhedral piece and a bottom face of the sixth polyhedral piece. The third polyhedral piece of a first cube is coupled to the sixth polyhedral piece of a second cube at respective sides of the bottom faces of the third polyhedral piece of the first cube and the sixth polyhedral piece of the second cube, and the third polyhedral piece of the second cube is coupled to the sixth polyhedral piece of the first cube at respective sides of the bottom faces of the third polyhedral piece of the second cube and the sixth polyhedral piece of the first cube.

Provided is a polyhedral toy which has a wider variety of playing patterns and cannot be predicted in terms of change in forms to thereby enable enhancement of elements of puzzles. A first polyhedral piece for forming an upper face of a cube is coupled to a second polyhedral piece for forming a first side face of the cube and a third polyhedral piece for forming a second side face adjacent to the first side face at respective sides of bottom faces of the first polyhedral piece, the second polyhedral piece, and the third polyhedral piece. The second polyhedral piece is coupled to a fourth polyhedral piece for forming a third side face opposed to the second side face of the cube at respective sides of the bottom face of the second polyhedral piece and a bottom face of the fourth polyhedral piece. The fourth polyhedral piece is coupled to a fifth polyhedral piece for forming a lower face opposed to the upper face of the cube at respective sides of the bottom face of the fourth polyhedral piece and a bottom face of the fifth polyhedral piece. The fifth polyhedral piece is coupled to a sixth polyhedral piece for forming a fourth side face opposed to the first side face of the cube at respective sides of the bottom face of the fifth polyhedral piece and a bottom face of the sixth polyhedral piece. The third polyhedral piece of a first cube is coupled to the sixth polyhedral piece of a second cube at respective sides of the bottom faces of the third polyhedral piece of the first cube and the sixth polyhedral piece of the second cube, and the third polyhedral piece of the second cube is coupled to the sixth polyhedral piece of the first cube at respective sides of the bottom faces of the third polyhedral piece of the second cube and the sixth polyhedral piece of the first cube.

Pentahedral module puzzles include a plurality of pentahedral modules connected by hinges in a continuous loop. Each pentahedral module comprises at least one magnet. The pentahedral modules include mirror image pentahedral modules connected by the hinges in an alternating sequence.

Dual geometry puzzles are formed of a continuous loop of polyhedrons connected by hinges. The polyhedrons include first type polyhedrons having a first geometry and second type polyhedrons having a different second geometry. Each of the polyhedrons includes at least one magnet disposed proximal to at least one face thereof. Eight of the twelve polyhedrons are the first type polyhedron, and four of the twelve polyhedrons are the second type polyhedron. The puzzles may be configurable between a first inverted configuration and a second inverted configuration. A first face of each of the first type polyhedrons may be congruent with a first face and a second face of each of the second type polyhedron.

An object is to provide, in place of a cube toy such as Yoshimoto Cube, a wide variety of playing patterns, playing patterns which cannot be predicted in terms of change in forms, a polyhedral toy which can be enjoyed in terms of change in form even when only one polyhedral toy is used, and a polyhedral toy having extensibility for achieving a wide variety of contents of play through combination of two or more polyhedral toys. Therefore, when a polyhedral piece (U1) forming the toy is formed with three quadrangular pyramids (G1 to G3) having, among six triangular faces (T1 to T6) of a cube (R), three faces (R1 to R3) as bottom faces, the polyhedral piece (U1) includes a triangle surface (10) formed of six triangular faces (T1 to T6) other than respective side faces of the quadrangular pyramids (G1 to G3) placed on one another. When polyhedral pieces (U1 to U8) each having the configuration described above form a toy (1) unfolded into a cuboid shape of two rows and four columns as illustrated in FIG. 1, in plan view on a top face of the toy (1), in each of the first row and the second row, the triangle surface (10), the third face (R3), the third face (R3), and the triangle surface (10) are arranged in the stated order.

Systems and methods which provide for interactive haptic proxies for hand-graspable virtual reality (VR) objects using reconfigurable artifacts and facilitated through use of a haptic reconfigurable artifact toolkit are described. Embodiments provide a haptic reconfigurable artifact toolkit including a set of reconfigurable artifacts for use in forming reconfigurable artifact-based haptic proxies, a set of haptic hardware components for use with respect to reconfigurable artifact-based haptic proxies, and a haptic reconfigurable artifact software platform configured to facilitate generation of reconfigurable artifact-based haptic proxies using items of the set of reconfigurable artifacts and the set of haptic hardware components. Reconfigurable artifacts may be shaped to form reconfigurable artifact-based haptic proxies covering a wide range of hand-grasp postures.

A polyhedron (100) configured to be removably coupled by shape with another polyhedron (200) to form a Yoshimoto cube (1), such polyhedron (100) comprising: eight half-cubes (101, 102) equal to each other, each of which being delimited by three outer faces (1011, 1012, 1013), having a substantially square configuration, arranged orthogonally to each other whereby each outer face (1011, 1012, 1013) is delimited by two edges (10111, 10112) in common with the other outer faces (1012, 1013) and two free edges (10113, 10114), and by 6 inner faces (1014, 1015, 1016, 1017, 1018, 1019), having a substantially triangular configuration, wherein each inner face (1014, 1015, 1016, 1017, 1018, 1019) extends between a vertex (V), in common with the other inner faces of the half-cube (101), substantially coinciding with the geometric centre of a cube delimited by said three outer faces (1011, 1012, 1013), and a respective base, corresponding to a free edge (10113) between two free edges of one of said outer faces (1011, 1012, 1013); and eight hinge members (300), each configured to connect a free edge (10113, 10113′) of each half-cube (101, 101′) with a free edge (10113′, 10113) of another half-cube (101′, 101) adjacent thereto, wherein each hinge member (300) comprises at least one axis of rotation (301) around which the two half-cubes (101,101′) connected by such hinge member (300) may mutually move; characterized in that each hinge member (300) of such polyhedron (100): extends along each of the two free edges (10113, 10113′) of the two respective adjacent half-cubes (101, 101′) connected by the hinge member (300), starting from a same end of each free edge (10113, 10113′) of the two free edges and for an overall length lower than the length of each free edge (10113, 10113′), optionally substantially other than or equal to half the length of each free edge (10113, 10113′), defines an axis of rotation (301) parallel to each free edge (10113, 10113′), and allows a mutual angular movement between the two respective adjacent half-cubes (101, 101′) connected by it, comprised between a first configuration, wherein the outer faces delimited by the respective free edges (10113, 10113′), at which the connection with the hinge member (300) is made, match, and the corresponding portions of the respective free edges (10113, 10113′) which are not connected by the hinge member (<

A transformational toy comprises at least six polyhedron bodies, at least one connection strip for connecting the polyhedron bodies in a chain, wherein the connection strip provides hinges between every pair of adjacent polyhedron bodies of the chain, wherein the hinges facilitate movement of the polyhedron bodies between at least two different geometric transformations of a combined body of all polyhedron bodies, wherein at least one of the connection strips is connecting at least three adjacent polyhedron bodies.