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truncated 6-cube (or truncated hexeract) is a convex uniform 6-polytope, being a truncation of the regular 6-cube. There are 5 truncations for the 6-cube
Truncated_6-cubes
Archimedean solid with 14 faces
In geometry, the truncated cube, or truncated hexahedron, is an Archimedean solid. It has 14 regular faces (6 octagonal and 8 triangular), 36 edges, and
Truncated_cube
Uniform 7- polytope
geometry, a truncated 7-cube is a convex uniform 7-polytope, being a truncation of the regular 7-cube. There are 6 truncations for the 7-cube. Vertices
Truncated_7-cubes
Convex uniform 8-polytope in 8-dimensional geometry
a truncated 8-cube is a convex uniform 8-polytope, being a truncation of the regular 8-cube. There are unique 7 degrees of truncation for the 8-cube. Vertices
Truncated_8-cubes
6-cube is a convex uniform 6-polytope with 3rd order truncations (runcination) of the regular 6-cube. There are 12 unique runcinations of the 6-cube with
Runcinated_6-cubes
Type of tesseract
envelope is a cube. Two of the truncated cube cells project onto a truncated cube inscribed in the cubical envelope. The other 6 truncated cubes project onto
Truncated_tesseract
geometry, a stericated 6-cube is a convex uniform 6-polytope, constructed as a sterication (4th order truncation) of the regular 6-cube. There are 8 unique
Stericated_6-cubes
geometry, a truncated 5-cube is a convex uniform 5-polytope, being a truncation of the regular 5-cube. There are four unique truncations of the 5-cube. Vertices
Truncated_5-cubes
Shape in six-dimensional geometry
geometry, a cantic 6-cube (or a truncated 6-demicube) is a uniform 6-polytope. Truncated 6-demicube Truncaced demihexeract Truncated hemihexeract (Acronym:
Cantic_6-cube
a truncated 6-orthoplex is a convex uniform 6-polytope, being a truncation of the regular 6-orthoplex. There are 5 degrees of truncation for the 6-orthoplex
Truncated_6-orthoplexes
which is bounded by 48 cells: 24 cubes, and 24 truncated octahedra. Each vertex joins three truncated octahedra and one cube, in an equilateral triangular
Truncated_24-cells
Archimedean solid Truncated tetrahedron Cuboctahedron Truncated cube Truncated octahedron Rhombicuboctahedron Truncated cuboctahedron Snub cube Icosidodecahedron
List_of_mathematical_shapes
Geometric object
The truncated tetrakis cube, or more precisely an order-6 truncated tetrakis cube or hexatruncated tetrakis cube, is a convex polyhedron with 32 faces:
Truncated_tetrakis_cube
Archimedean solid with 26 faces
to Truncated cuboctahedron. Cube Cuboctahedron Octahedron Truncated icosidodecahedron Truncated octahedron – truncated tetratetrahedron Snub cube Wenninger
Truncated_cuboctahedron
geometry, a truncated 6-simplex is a convex uniform 6-polytope, being a truncation of the regular 6-simplex. There are unique 3 degrees of truncation. Vertices
Truncated_6-simplexes
Archimedean solid with 14 faces
(1986) constructs a truncated octahedron by attaching eight cubes to each face. In this hinge, eight segments on each half-cube form hexagons, and the
Truncated_octahedron
Polygon shape with eight sides
also be constructed as a quasiregular truncated square, t{4}, which alternates two types of edges. A truncated octagon, t{8} is a hexadecagon, {16}. A
Octagon
Operation that cuts polytope vertices, creating a new facet in place of each vertex
regular hexagonal tiling {6,3}. A truncated n-sided polygon will have 2n sides (edges). A regular polygon uniformly truncated will become another regular polygon:
Truncation_(geometry)
Solid with six equal square faces
cube. Joining one or more cubes face-to-face yields a polycube, a three-dimensional version of polyominoes. Two or more cubes can have the same centre
Cube
Archimedean solid Truncated tetrahedron, Cuboctahedron, Truncated cube, Truncated octahedron, Rhombicuboctahedron, Truncated cuboctahedron, Snub cube, Icosidodecahedron
List of polygons, polyhedra and polytopes
List_of_polygons,_polyhedra_and_polytopes
Geometric operation which truncates the edges of polyhedra
alternated truncated cube: from a regular tetrahedron, this replaces its six edges with congruent flattened hexagons; or alternately truncating a cube, replacing
Chamfer_(geometry)
Uniform 6-dimensional polytope
uniform 6-polytope. The simplest uniform polypeta are regular polypeta: the 6-simplex {3,3,3,3,3}, the 6-cube (hexeract) {4,3,3,3,3}, and the 6-orthoplex
Uniform_6-polytope
Shape with six sides
the truncated tetrahedron, truncated octahedron, truncated icosahedron (of soccer ball and fullerene fame), truncated cuboctahedron and the truncated icosidodecahedron
Hexagon
Only regular space-filling tessellation of the cube
honeycomb) in Euclidean 3-space made up of cubic cells. It has 4 cubes around every edge, and 8 cubes around each vertex. Its vertex figure is a regular octahedron
Cubic_honeycomb
cantellated 6-cube is a convex uniform 6-polytope, being a cantellation of the regular 6-cube. There are 8 cantellations for the 6-cube, including truncations. Half
Cantellated_6-cubes
3D combination puzzle
Cubes. Rubik's Cubes continued to be marketed and sold throughout the 1980s and 1990s, but it was not until the early 2000s that interest in the Cube
Rubik's_Cube
Convex polyhedron with 38 faces
truncated cube is a polyhedron, constructed as a rectified, truncated cube. It has 38 faces: 8 equilateral triangles, 24 isosceles triangles, and 6 octagons
Rectified_truncated_cube
disprismatotesseractihexadecachoron has 16 tetrahedra, 32 cubes, and 32 triangular prisms. Each vertex is shared by 4 cubes, 3 triangular prisms and one tetrahedron.
Runcinated_tesseracts
truncated forms, column C shows quasi-truncated forms, column D shows a different method of truncation. These truncated forms all have a vertex figure p.q
List of uniform polyhedra by vertex figure
List_of_uniform_polyhedra_by_vertex_figure
7-polytope
the 7-orthoplex. The final three truncations are best expressed relative to the 7-cube. Truncated heptacross Truncated hecatonicosaoctaexon (Jonathan Bowers)
Truncated_7-orthoplexes
Regular 5-polytope
5-cell cells. These are seen as vertex figures of truncated regular 6-polytopes, like a truncated 6-cube. Another form is {3,3}∨{ }, with [3,3,2,1] symmetry
5-simplex
4-faces (6 5-cell and 6 truncated 5-cells). Truncated hexateron (Acronym: tix) (Jonathan Bowers) The vertices of the truncated 5-simplex can be most simply
Truncated_5-simplexes
Polyhedra in which all vertices are the same
cuboctahedron, truncated octahedron, truncated cube, rhombicuboctahedron, icosidodecahedron, truncated cuboctahedron, truncated icosahedron, truncated dodecahedron
Archimedean_solid
Removal of alternate vertices
faces of the original cube. A snub (in Coxeter's terminology) can be seen as an alternation of a truncated regular or truncated quasiregular polyhedron
Alternation_(geometry)
6×6×6 Rubik's Cube
The V-Cube 6 is a 6×6×6 version of the original Rubik's Cube. The first mass-produced 6×6×6 was invented by Panagiotis Verdes and is produced by the Greek
V-Cube_6
Uniform 8-polytope
a cantic 8-cube or truncated 8-demicube is a uniform 8-polytope, being a truncation of the 8-demicube. Truncated demiocteract Truncated hemiocteract;
Cantic_8-cube
Archimedean solid with 38 faces
triangles. The snub cube can also be derived from the truncated cuboctahedron by the process of alternation. 24 vertices of the truncated cuboctahedron form
Snub_cube
Euclidean 3-space. It is composed of cubes and triangular prisms in a ratio of 1:2. It is created by alternating layers of cubes and triangular prisms, with the
Triangular prismatic honeycomb
Triangular_prismatic_honeycomb
British puzzle designer
including his Cylinder Cube, Golden Cube, Hexagonal Prism, Truncated Octaminx and Truncated Octahedron. Slocum, Jerry (2009). The Cube. The Ultimate Guide
Tony_Fisher_(puzzle_designer)
Polyhedron with 14 faces
Polyhedron". MathWorld. Maeder, Roman. "19: stellated truncated hexahedron". MathConsult. Weisstein, Eric W. "Stellated truncated hexahedron". MathWorld. v t e
Stellated truncated hexahedron
Stellated_truncated_hexahedron
Uniform 5-polytope
dimensions or higher, a cantic 5-cube, cantihalf 5-cube, truncated 5-demicube is a uniform 5-polytope, being a truncation of the 5-demicube. It has half
Cantic_5-cube
Convex uniform 4-polytope
eight cubes becomes a rhombicuboctahedron in the described way. In addition however, since each cube's edge was previously shared with two other cubes, the
Cantellated_tesseract
analogue would be a truncated tetrahedron (truncated 3-demicube), and Coxeter diagram or as a cantic cube. Truncated demihepteract Truncated hemihepteract (acronym:
Cantic_7-cube
Shape between a square and a circle
example, one wishes to create nested squircles. Another similar shape is a truncated circle, the boundary of the intersection of the regions enclosed by a
Squircle
Geometrical Shape
brox) (Jonathan Bowers) Rectified 6-demicube The birectified 6-cube may be constructed from the 6-cube by truncating its vertices at the midpoints of its
Rectified_6-cubes
disphenoidal vertex figure: two hexagonal prisms, and two truncated cuboctahedra. The 48 truncated cuboctahedral cells are joined to each other via their
Runcinated_24-cells
Tiling of hyperbolic 3-space by uniform polyhedra
made from 8 cubes and 12 p-gonal prisms at a vertex for any integer p. In the case p = 4 (a regular icosahedron), all cells are cubes and the result
Uniform honeycombs in hyperbolic space
Uniform_honeycombs_in_hyperbolic_space
Class of 4-dimensional polytopes
cantitruncated 16-cell or truncated 24-cell, with the half symmetry group [(3,3)+,4]. The truncated octahedral cells become icosahedra. The cubes becomes tetrahedra
Uniform_4-polytope
Cube that fits through hole in smaller cube
icosahedron, truncated dodecahedron, and the truncated tetrahedron, as well as the truncated icosidodecahedron. Cubes and all rectangular solids have Rupert
Prince_Rupert's_cube
There are two degrees of truncations, including a bitruncation. The truncated 5-cell, truncated pentachoron or truncated 4-simplex is bounded by 10
Truncated_5-cell
third and fourth truncations are more easily constructed as second and first truncations of the 5-cube. Truncated pentacross Truncated triacontaditeron
Truncated_5-orthoplexes
Tuttminx (/ˈtʊtmɪŋks/ or /ˈtʌtmɪŋks/) is a Rubik's Cube-like twisty puzzle, in the shape of a truncated icosahedron. It was invented by Lee Tutt in 2005
Tuttminx
Polyhedron resulting from the snub operation
icosahedron, snub cube and snub dodecahedron are the only three convex ones. They are obtained by snubification of the truncated octahedron, truncated cuboctahedron
Snub_polyhedron
7-cube is a convex uniform 7-polytope with 5th order truncations (pentellation) of the regular 7-cube. There are 32 unique pentellations of the 7-cube with
Pentellated_7-cubes
mirror as . The truncated cubic-octahedral honeycomb is a compact uniform honeycomb, constructed from truncated octahedron, truncated cube, rhombicuboctahedron
Cubic-octahedral_honeycomb
Variant of Rubik's Cube
very different from it Pocket Cube Rubik's Cube Rubik's Revenge Rubik's Triamid Professor's Cube V-Cube 6 V-Cube 7 V-Cube 8 Skewb Skewb Diamond Megaminx
Pyraminx
runcinations of the 6-orthoplex with permutations of truncations, and cantellations. 7 are expressed relative to the dual 6-cube. Small prismatohexacontatetrapeton
Runcinated_6-orthoplexes
American speedcuber (born 2001)
the single 6×6×6 world record, at 57.69 seconds, achieved at Burbank Big Cubes 2025. He formerly held the world record mean of three 6×6×6 solves: 1:05
Max_Park
Near-miss Johnson solid with 16 faces
truncating the order-6 vertices. The truncated vertices become four hexagon faces, and the obtuse triangles of the triakis tetrahedron are truncated at
Truncated_triakis_tetrahedron
the faces. The DaYan Gem 6 is a twisty puzzle in this shape. Truncated triakis tetrahedron Truncated tetrakis cube Truncated triakis icosahedron "TwistyPuzzles
Truncated_triakis_octahedron
2x2x2 combination puzzle
calculating the permutations of N×N×N cubes where N is odd, since those puzzles have fixed centers which identify the cube's spatial orientation. The number
Pocket_Cube
Method of describing higher-order polyhedra
operators, like truncation as defined by Kepler, to build related polyhedra of the same symmetry. For example, tC represents a truncated cube, and taC, parsed
Conway_polyhedron_notation
Rubik's cube variant
mechanism is nearly identical to that of the Rubik's Cube, although it differs from normal 3×3 cubes in that all pieces are the same color (typically reflective
Mirror_Blocks
Larger variant of the Rubik's cube
original. Like the 5×5×5, the V-Cube 7 has both fixed and movable center facets. The puzzle consists of 218 unique miniature cubes ("cubies") on the surface
V-Cube_7
cantellated 5-cube is a convex uniform 5-polytope, being a cantellation of the regular 5-cube. There are 6 unique cantellation for the 5-cube, including
Cantellated_5-cubes
Type of polyhedron
The truncated rhombicuboctahedron—alternatively known as truncated small rhombicuboctahedron or beveled cuboctahedron—is a polyhedron, constructed from
Truncated_rhombicuboctahedron
Quasiregular space-filling tesselation
colored truncated tetrahedra. It is related to the cantellated cubic honeycomb. Rhombicuboctahedra are reduced to truncated octahedra, and cubes are reduced
Tetrahedral-octahedral honeycomb
Tetrahedral-octahedral_honeycomb
geometry, a stericated 5-cube is a convex uniform 5-polytope with fourth-order truncations (sterication) of the regular 5-cube. There are eight degrees
Stericated_5-cubes
Archimedean solid with 8 faces
(of two types). It can be constructed by truncating all 4 vertices of a regular tetrahedron. The truncated tetrahedron can be constructed from a regular
Truncated_tetrahedron
7-cube is a convex uniform 7-polytope with 4th-order truncations (sterication) of the regular 7-cube. There are 24 unique sterication for the 7-cube with
Stericated_7-cubes
Isogonal polytope with uniform facets
Faces are truncated, doubling their edges. (The term, coined by Kepler, comes from Latin truncare 'to cut off'.) There are higher truncations also: bitruncation
Uniform_polytope
Seven-dimensional geometric object
Regular 6-cube honeycomb, represented by symbols {4,34,4}, B ~ 6 {\displaystyle {\tilde {B}}_{6}} , [31,1,33,4], 95 forms, 64 shared with C ~ 6 {\displaystyle
Uniform_7-polytope
5x5x5 version of the Rubik's Cube
versions of the 5×5×5 cube sold at Barnes & Noble were marketed under the name "Professor's Cube", but currently, Barnes and Noble sells cubes that are simply
Professor's_Cube
Puzzle
complex cubes with marked centres. The properties of Rubik’s family cubes of any size together with some special attention to software cubes is the main
Rubik's family cubes of varying sizes
Rubik's_family_cubes_of_varying_sizes
Truncated trapezohedron with a 3-sided base
In geometry, the truncated triangular trapezohedron is the first in an infinite series of truncated trapezohedra. It has 6 pentagon and 2 triangle faces
Truncated triangular trapezohedron
Truncated_triangular_trapezohedron
7-cube is a convex uniform 7-polytope with 3rd order truncations (runcination) of the regular 7-cube. There are 16 unique runcinations of the 7-cube with
Runcinated_7-cubes
Isogonal polyhedron with regular faces
the five truncations of the Platonic solids—truncated tetrahedron, truncated octahedron, truncated cube, truncated dodecahedron, and truncated icosahedron—and
Uniform_polyhedron
Geometric operation applied to a polyhedron
alternated truncated 24-cell. Coxeter's snub terminology is slightly different, meaning an alternated truncation, deriving the snub cube as a snub cuboctahedron
Snub_(geometry)
Convex polyhedron whose faces are almost regular polygons
hexagonal pyramid Triangular cupola Truncated tetrahedron Truncated octahedron 4.4.4.4 Square icositetrahedron (Cube) 3.4.6.4: Hexagonal cupola (Degenerate)
Near-miss_Johnson_solid
by 48 cells: 24 cubes, and 24 cuboctahedra. It can be obtained by rectification of the 24-cell, reducing its octahedral cells to cubes and cuboctahedra
Rectified_24-cell
Hungarian inventor (born 1944)
over 350 million Rubik's Cubes have been sold, making it one of the best selling toys of all time. In addition to Rubik's Cube, Rubik is also the inventor
Ernő_Rubik
geometry, a truncated 8-orthoplex is a convex uniform 8-polytope, being a truncation of the regular 8-orthoplex. There are 7 truncation for the 8-orthoplex
Truncated_8-orthoplexes
Greek inventor
Rubik's cubes; his mechanism is based on concentric, right-angle conical surfaces whose axes of rotation coincide with the semi-axes of the cube. The patents
Panagiotis_Verdes
World Championship for the Rubik's cube held in 1982
participated in the contest, criticised the cubes for being "really hard to turn and were not prepared for serious speed cubing". This competition was the first
1982 Rubik's Cube World Championship
1982_Rubik's_Cube_World_Championship
8×8×8 version of Rubik's Cube
The V-Cube 8 is an 8×8×8 version of the Rubik's Cube. Unlike the original puzzle (but like the 4×4×4 and 6×6×6 cubes), it has no fixed centers: the center
V-Cube_8
Nonconvex uniform polyhedron with 20 faces
aforementioned directions. It shares the vertex arrangement with the convex truncated cube and two other nonconvex uniform polyhedra. It additionally shares its
Great_cubicuboctahedron
Field of mathematics dealing with three-dimensional Euclidean spaces
various solids, including pyramids, prisms, cubes (and other polyhedrons), cylinders, cones (including truncated) and other solids of revolution. The Pythagoreans
Solid_geometry
American puzzle designer (1939–2022)
the “Rubik's Cube”, Dr. Nichols conceived of a twist cube puzzle with six colored faces. It was a 2×2×2 cube assembled from eight unit cubes with magnets
Larry_D._Nichols
Competitive solving of combination puzzles
score is calculated by subtracting the number of unsolved cubes from the number of solved cubes. Official competitions are currently being held in several
Speedcubing
Polyhedron with 6 faces
MR 1368518 Kolpakov, Alexander; Murakami, Jun (2013), "Volume of a doubly truncated hyperbolic tetrahedron", Aequationes Mathematicae, 85 (3): 449–463, arXiv:1203
Hexahedron
Convex polyhedron projected from hypercube
the Minkowski sum of a cube and a truncated octahedron forms the truncated cuboctahedron, while the Minkowski sum of the cube and the rhombic dodecahedron
Zonohedron
tiling honeycomb, ↔ , is composed of trihexagonal tiling, truncated cuboctahedron, truncated cube, and triangular prism cells, with a rectangular pyramid
Order-4 hexagonal tiling honeycomb
Order-4_hexagonal_tiling_honeycomb
cantellated 7-cube is a convex uniform 7-polytope, being a cantellation of the regular 7-cube. There are 10 degrees of cantellation for the 7-cube, including
Cantellated_7-cubes
Puzzles solved by mechanical manipulation
been many different shapes of Rubik type puzzles constructed. As well as cubes, all of the regular polyhedra and many of the semi-regular and stellated
Combination_puzzle
during truncation. The snub 24-cell is related to the truncated 24-cell by an alternation operation. Half the vertices are deleted, the 24 truncated octahedron
Snub_24-cell
Goldberg polyhedron with 42 faces
triacontahedron because only the order-5 vertices are truncated. These 12 order-5 vertices can be truncated such that all edges are of equal length. The original
Chamfered_dodecahedron
The snub rhombicuboctahedron is a polyhedron, constructed as a truncated rhombicuboctahedron. It has 74 faces: 18 squares, and 56 triangles. It can also
Snub_rhombicuboctahedron
geometry, a runcinated 5-cube is a convex uniform 5-polytope that is a runcination (a 3rd order truncation) of the regular 5-cube. There are 8 unique degrees
Runcinated_5-cubes
Generalization of the Rubik's Cube in n-dimensions
archived 25 December 2008. David Vanderschel, "Lower-dimensional cubes", 4D Cubing Forum, 21 August 2006. "MC2D's (reflecting) moves would require a
N-dimensional sequential move puzzle
N-dimensional_sequential_move_puzzle
Catalan solid with 30 faces
the arrangement of four Platonic solids. It contains ten tetrahedra, five cubes, an icosahedron and a dodecahedron. The centers of the faces contain five
Rhombic_triacontahedron
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