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birectified 5-cube are located in the square face centers of the 5-cube. Rectified penteract (acronym: rin) (Jonathan Bowers) The rectified 5-cube may be constructed
Rectified_5-cubes
centers of the 7-cube. rectified hepteract (acronym: rasa) (Jonathan Bowers) Cartesian coordinates for the vertices of a rectified 7-cube, centered at the origin
Rectified_7-cubes
as a first rectification of a 5-dimensional cross polytope. Rectified pentacross Rectified triacontaditeron (32-faceted 5-polytope) Acronym: rat (Jonathan
Rectified_5-orthoplexes
semiregular polytope, labeling it as S1 5. Rectified hexateron (Acronym: rix) (Jonathan Bowers) The vertices of the rectified 5-simplex can be more simply positioned
Rectified_5-simplexes
7-cube cell centers of the 8-cube. Rectified octeract Acronym: recto (Jonathan Bowers) Birectified octeract Rectified 8-demicube Acronym: bro (Jonathan
Rectified_8-cubes
Geometrical Shape
birectified 6-cube are located in the square face centers of the 6-cube. Rectified hexeract (acronym: rax) (Jonathan Bowers) The rectified 6-cube may be constructed
Rectified_6-cubes
geometry, a rectified 9-cube is a convex uniform 9-polytope, being a rectification of the regular 9-cube. There are 9 rectifications of the 9-cube. The zeroth
Rectified_9-cubes
stellated 120-cell Uniform 4-polytope Rectified 5-cell, Truncated 5-cell, Cantellated 5-cell, Runcinated 5-cell Rectified tesseract, Truncated tesseract, Cantellated
List_of_mathematical_shapes
geometry, a rectified 10-cube is a convex uniform 10-polytope, being a rectification of the regular 10-cube. There are 10 rectifications of the 10-cube, with
Rectified_10-cubes
rectified 24-cell or rectified icositetrachoron is a uniform 4-dimensional polytope (or uniform 4-polytope), which is bounded by 48 cells: 24 cubes,
Rectified_24-cell
6-demicube 6-cube, Rectified 6-cube, Truncated 6-cube, Cantellated 6-cube, Runcinated 6-cube, Stericated 6-cube, Pentellated 6-cube 6-orthoplex, Rectified 6-orthoplex
List of polygons, polyhedra and polytopes
List_of_polygons,_polyhedra_and_polytopes
Polyhedron with 8 triangles and 6 squares
A cuboctahedron, rectified cube, or rectified octahedron is a polyhedron with 8 triangular faces and 6 square faces. A cuboctahedron has 12 identical vertices
Cuboctahedron
Convex polyhedron with 38 faces
creates two more polyhedra: Rectified truncated tetrahedron Rectified truncated octahedron Rectified truncated dodecahedron Rectified truncated icosahedron
Rectified_truncated_cube
Four-dimensional shape
as the cells of the dual of the uniform rectified 5-simplex, and rectified 5-cube or the dual of any uniform 5-polytope with a tetrahedral prism vertex
Tetrahedral_bipyramid
Uniform polychoron
(Thorold Gosset) Dispentachoron Rectified 5-cell (Norman W. Johnson) Rectified 4-simplex Fully truncated 4-simplex Rectified pentachoron (Acronym: rap) (Jonathan
Rectified_5-cell
Operation in Euclidean geometry
rectified form. New vertices are placed at the center of the edges of the original polygon. Each platonic solid and its dual have the same rectified polyhedron
Rectification_(geometry)
Five-dimensional geometric shape
all convex: {3,3,3,3} - 5-simplex {4,3,3,3} - 5-cube {3,3,3,4} - 5-orthoplex There are no nonconvex regular polytopes in 5 dimensions or above. Unsolved
Uniform_5-polytope
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
Polyhedron with eight triangular faces
vertices at which four sides meet, and twelve edges. Its dual polyhedron is a cube. It can be formed as the convex hull of the six axis-parallel unit vectors
Octahedron
Polytope
makes the dual of rectified 5-orthoplex. The intersection of the 5-cube and 5-orthoplex compound is the uniform birectified 5-cube: = ∩ . The compound
Compound of 5-cube and 5-orthoplex
Compound_of_5-cube_and_5-orthoplex
Polytope in 8-dimensional geometry
orthoplexes. The rectified 421 can be seen as a rectification of the 421 polytope, creating new vertices on the center of edges of the 421. Rectified
4_21_polytope
In geometry, the rectified 600-cell or rectified hexacosichoron is a convex uniform 4-polytope composed of 600 regular octahedra and 120 icosahedra cells
Rectified_600-cell
Uniform 6-dimensional polytope
hypercube honeycomb of Euclidean 5-space, the 5-cube honeycomb, with symbols {4,33,4}, = B ~ 5 {\displaystyle {\tilde {B}}_{5}} There are 47 uniform honeycombs
Uniform_6-polytope
Geometric space with five dimensions
kissing number of the lattice, 30, is represented in its vertices. The rectified 5-orthoplex is the vertex figure of the D5 lattice, . Its 40 vertices represent
Five-dimensional_space
In geometry, the rectified tesseract, rectified 8-cell is a uniform 4-polytope (4-dimensional polytope) bounded by 24 cells: 8 cuboctahedra, and 16 tetrahedra
Rectified_tesseract
Class of 4-dimensional polytopes
Space of n Dimensions. In four dimensions, this gives the rectified 5-cell, the rectified 600-cell, and the snub 24-cell. 1910: Alicia Boole Stott, in
Uniform_4-polytope
Polytope constructed from alternation of a hypercube
the vertices of {4,3,...,3}. The vertex figures of demihypercubes are rectified n-simplexes. They are represented by Coxeter-Dynkin diagrams of three
Demihypercube
Regular tiling of hyperbolic 3-space
hyperbolic 3-space. With Schläfli symbol {4,3,5}, it has five cubes {4,3} around each edge, and 20 cubes around each vertex. It is dual with the order-4 dodecahedral
Order-5_cubic_honeycomb
Uniform 6-polytope
(dual of E6 lattice). Birectified 221 polytope Rectified pentacontatetrapeton (Acronym: ram) - rectified 54-facetted polypeton (Jonathan Bowers) Vertices
1_22_polytope
constructed from this honeycomb. or rectified heptacross rectified hecatonicosaoctaexon (Acronym: rez) (Jonathan Bowers) - rectified 128-faceted polyexon There
Rectified_7-orthoplexes
13k series. The final is a noncompact hyperbolic honeycomb, 134. The rectified 133 or 0331, Coxeter diagram has facets and , and vertex figure .
1_33_honeycomb
Uniform 8 dimensional polytope
polytope Quadrirectified 421 polytope Rectified diacositetraconta-dischiliahectohexaconta-zetton as a rectified 240-2160 facetted polyzetton (acronym:
1_42_polytope
5-dimensional geometric object
honeycomb, , vertex figure is a rectified 5-orthoplex and facets are the 5-orthoplex and 5-demicube. Pyramidal 5-polytopes, or 5-pyramids, can be generated
5-polytope
Notation for polytopes and tessellations
edge is represented by {p,q,r}. For example, a tesseract, {4,3,3}, has 3 cubes, {4,3}, around an edge. In general, a regular polytope {p,q,r,...,y,z} has
Schläfli_symbol
Uniform Polytope
group orders. The rectified 231 is a rectification of the 231 polytope, creating new vertices on the center of edge of the 231. Rectified
2_31_polytope
Type of uniform space-filling tessellation
of the 5-demicubic honeycomb is the D5 lattice which is the densest known sphere packing in 5 dimensions. The 40 vertices of the rectified 5-orthoplex
5-demicubic_honeycomb
2002 video game
before being bought by HP. This was rectified in the Microsoft Windows and Game Boy Advance versions. The GameCube version is a port of F1 2001, with only
F1_2002_(video_game)
are 5 related uniform honeycombs generated within the same family, generated with 2 or more rings of the Coxeter group : , , , , . The rectified cubic-octahedral
Cubic-octahedral_honeycomb
Near-miss Johnson solid with 92 faces
polyhedra: Near-miss Johnson solid Rectified truncated tetrahedron Rectified truncated octahedron Rectified truncated cube Rectified truncated dodecahedron "PolyHédronisme"
Rectified truncated icosahedron
Rectified_truncated_icosahedron
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
Regular tiling of hyperbolic 3-space
of a sequence of polychora and honeycombs with dodecahedral cells: The rectified order-4 dodecahedral honeycomb, , has alternating octahedron and icosidodecahedron
Order-4 dodecahedral honeycomb
Order-4_dodecahedral_honeycomb
Type of polyhedron
also be constructed as a rectified rhombicuboctahedron. Expanded rhombic dodecahedron Rectified rhombicuboctahedron Rectified small rhombicuboctahedron
Expanded_cuboctahedron
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
Seven-dimensional geometric object
convex regular 7-polytopes: {3,3,3,3,3,3} - 7-simplex {4,3,3,3,3,3} - 7-cube {3,3,3,3,3,4} - 7-orthoplex There are no nonconvex regular 7-polytopes. The
Uniform_7-polytope
sequence of honeycombs with square tiling cells: The rectified square tiling honeycomb, t1{4,4,3}, has cube and square tiling facets, with a triangular prism
Square_tiling_honeycomb
Archimedean solid with 26 faces
constructed from a cube by drawing a smaller one in the middle of each face, parallel to the cube's edges. After removing the edges of a cube, the squares may
Rhombicuboctahedron
Isogonal polyhedron with regular faces
compounds, which can be split as a union of polyhedra, such as the compound of 5 cubes. If we drop the condition that the realization of the polyhedron is non-degenerate
Uniform_polyhedron
Regular 5-polytope
five-dimensional geometry, a demipenteract or 5-demicube is a semiregular 5-polytope, constructed from a 5-hypercube (penteract) with alternated vertices
5-demicube
from the center of a rectified 6-orthoplex is given by coordinate permutations of: (1,-1,0,0,0,0) The Cartesian coordinates in 5-space for the normalized
Stericated_5-simplexes
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
Uniform 6-polytope
dimensional series 22k. The rectified 221 has 216 vertices, and 126 facets: 72 rectified 5-simplices, and 27 rectified 5-orthoplexes and 27 5-demicubes . Its vertex
2_21_polytope
is cell-transitive, consisting of 48 truncated cubes, and also edge-transitive, with 3 truncated cubes cells per edge and with one triangle and two octagons
Truncated_24-cells
the 6-orthoplex. The rectified 6-orthoplex is the vertex figure for the demihexeractic honeycomb. or Rectified hexacross Rectified hexacontatetrapeton
Rectified_6-orthoplexes
Uniform polytope
triangle, doubled into a prism: {3,3}×{3}×{}. Rectified pentacontahexa-hecatonicosihexa-exon for rectified 56-126 facetted polyexon (acronym: lanq) (Jonathan
1_32_polytope
Uniform polytope in 8 dimensional geometry
also shown. The rectified 241 is a rectification of the 241 polytope, with vertices positioned at the mid-edges of the 241. Rectified
2_41_polytope
Convex polyhedron with 20 faces
In geometry, the rectified truncated tetrahedron is a polyhedron, constructed as a rectified, truncated tetrahedron. It has 20 faces: 4 equilateral triangles
Rectified truncated tetrahedron
Rectified_truncated_tetrahedron
Convex polyhedron with 38 faces
In geometry, the rectified truncated octahedron is a convex polyhedron, constructed as a rectified, truncated octahedron. It has 38 faces: 24 isosceles
Rectified truncated octahedron
Rectified_truncated_octahedron
Isogonal polytope with uniform facets
quintirectification reduced 5-faces to vertices, and so on. An extended Schläfli symbol can be used for representing rectified forms, with a single subscript:
Uniform_polytope
Removal of alternate vertices
be snubbed if its truncation has only even-sided faces. All truncated rectified polyhedra can be snubbed, not just from regular polyhedra. The snub square
Alternation_(geometry)
Anise-flavored liquor
aperitif that is widely consumed in Cyprus and Greece. It is made from rectified spirits that have undergone a process of distillation and flavoring. Its
Ouzo
Four-dimensional analogue of the tetrahedron
a cube face (not of the illustrated 3-cube, but of another of the tesseract's eight 3-cubes). The third additional edge is a √3 diagonal of a 3-cube (again
5-cell
Archimedean solid with 32 faces
contributing two radii and an edge. The icosidodecahedron is a rectified dodecahedron and also a rectified icosahedron, existing as the full-edge truncation between
Icosidodecahedron
6-dimensional geometric object
6-simplex honeycomb, . The 6-demicube honeycomb, , vertex figure is a rectified 6-orthoplex and facets are the 6-orthoplex and 6-demicube. The uniform
6-polytope
and edges, called a square bipyramidal honeycomb, or the dual of the rectified cubic honeycomb. It is analogous to the 2-dimensional tetrakis square
Tetragonal disphenoid honeycomb
Tetragonal_disphenoid_honeycomb
of the 24-cell include the rectified 24-cell (which cuts each edge at its midpoint, producing a polytope bounded by 24 cubes and 24 cuboctahedra), and
Snub_24-cell
A5 [6] D3 / A3 [4] 1 221 Icosihepta-heptacontadipeton (jak) 2 Rectified 221 Rectified icosihepta-heptacontadipeton (rojak) 3 Trirectified 221 Trirectified
E6_polytope
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
Convex uniform 7-polytope in seven-dimensional geometry
semiregular polytope, labeling it as S1 7. Rectified octaexon (Acronym: roc) (Jonathan Bowers) The vertices of the rectified 7-simplex can be most simply positioned
Rectified_7-simplexes
Geometric operation applied to a polyhedron
, r } {\displaystyle s{\begin{Bmatrix}p,q,r\end{Bmatrix}}} , and . A rectified polychoron { p q , r } {\displaystyle {\begin{Bmatrix}p\\q,r\end{Bmatrix}}}
Snub_(geometry)
The other is the bitruncated 24-cell, which is composed of 48 truncated cubes. This 4-polytope has a higher extended pentachoric symmetry (2×A4, [[3,3
Truncated_5-cell
Polyhedron made by truncating one end of a trapezohedron
faces. It can also be seen as a tetrahedron with 3 of 4 of its vertices rectified. The three rhombic faces fold out flat to form half of a hexagram. Elongated
Diminished_trapezohedron
Uniform 7-dimensional polytope
3-sphere tiling, a tetrahedral hosohedron.) Rectified hecatonicosihexa-pentacosiheptacontahexa-exon as a rectified 126-576 facetted polyexon (acronym: ranq)
3_21_polytope
cube (), and octahedron (). The Hessian is analogous to the tetrahedron, like the cube is a double tetrahedron, and the octahedron as a rectified tetrahedron
Hessian_polyhedron
Cube capped by two square pyramids
antiprisms), formed by superimposing octahedra into the cuboctahedra of the rectified cubic honeycomb. Both honeycombs have a symmetry of [ 4 , 3 , 4 ] {\displaystyle
Elongated_square_bipyramid
Tessellation of convex uniform polyhedron cells
9 forms, generated by ring permutations of the Coxeter group: . There are 5 forms, 1 unique, generated by ring permutations of the Coxeter group: . Repeat
Paracompact uniform honeycombs
Paracompact_uniform_honeycombs
Cantellated pentachoron Cantellated 4-simplex (small) prismatodispentachoron Rectified dispentachoron Small rhombated pentachoron (Acronym: Srip) (Jonathan Bowers)
Cantellated_5-cell
Uniform 10-polytope
10-demicube or demidekeract is a uniform 10-polytope, constructed from the 10-cube with alternated vertices removed. It is part of a dimensionally infinite
10-demicube
Convex polyhedron whose faces are almost regular polygons
Truncated tetrahedron Truncated octahedron 4.4.4.4 Square icositetrahedron (Cube) 3.4.6.4: Hexagonal cupola (Degenerate) Geodesic polyhedron Goldberg polyhedron
Near-miss_Johnson_solid
Type of geometrical object
with one or more rings. Twelve cases are shown below: ten single-ring (rectified) forms, and two truncations. Bowers-style acronym names are given in parentheses
Uniform_10-polytope
Uniform 8-polytope
In eight-dimensional geometry, a cantic 8-cube or truncated 8-demicube is a uniform 8-polytope, being a truncation of the 8-demicube. Truncated demiocteract
Cantic_8-cube
Group of irregular uniform polytopes
and all rectified (single-ring) Coxeter–Dynkin diagrams. The family of n-simplices contain Gosset–Elte figures of the form 0ij as all rectified forms of
Gosset–Elte_figures
In eight-dimensional geometry, a rectified 8-simplex is a convex uniform 8-polytope, being a rectification of the regular 8-simplex. There are unique
Rectified_8-simplexes
Geometric object
The family starts uniquely as 6-polytopes. The triangular prism and rectified 5-cell are included at the beginning for completeness. The demipenteract
Uniform_k_21_polytope
cuboctahedra and 60 triangular prisms), and 27 4-faces (6 cantellated 5-cell, 6 rectified 5-cells, and 15 tetrahedral prisms). Cantellated hexateron Small rhombated
Cantellated_5-simplexes
Natural number
regular compounds. The cuboctahedron, on the other hand, is a rectified cube or rectified octahedron, and one of only two convex quasiregular polyhedra
8
Regular object in four dimensional geometry
polytopes constructed from the 24-cell. The 24-cell can also be derived as a rectified 16-cell: Octacube (sculpture) Uniform 4-polytope § The F4 family Coxeter
24-cell
In 5-dimensional geometry, there are 31 uniform polytopes with B5 symmetry. There are two regular forms, the 5-orthoplex, and 5-cube with 10 and 32 vertices
B5_polytope
[8] 1 421 (fy) 2 Rectified 421 (riffy) 3 Birectified 421 (borfy) 4 Trirectified 421 (torfy) 5 Rectified 142 (buffy) 6 Rectified 241 (robay) 7 241 (bay)
E8_polytope
2006 video game
made on their journey. The group are then escorted by the Space Police, rectified by former general Knucklestorm and his daughter Brie Pourri, before returning
Magical_Starsign
Type of polyhedron
triangle and squares of the rhombicuboctahedron can be independently rectified or truncated, creating four permutations of polyhedra. The partially truncated
Truncated_rhombicuboctahedron
Convex polyhedron with 92 faces
In geometry, the rectified truncated dodecahedron is a convex polyhedron, constructed as a rectified, truncated dodecahedron. It has 92 faces: 20 equilateral
Rectified truncated dodecahedron
Rectified_truncated_dodecahedron
simple Lie groups. The rectified 8-orthoplex is the vertex figure for the demiocteractic honeycomb. or Rectified octacross Rectified diacosipentacontahexazetton
Rectified_8-orthoplexes
H4 family polytopes 120-cell rectified 120-cell truncated 120-cell cantellated 120-cell runcinated 120-cell cantitruncated 120-cell runcitruncated 120-cell
Runcinated_120-cells
Self-intersecting uniform polyhedron
cubes sharing edges and vertices) Great complex rhombicosidodecahedron (looks like a great ditrigonal icosidodecahedron and a compound of five cubes sharing
Uniform_star_polyhedron
Polytope contained by 7-polytope facets
convex regular 8-polytopes: {3,3,3,3,3,3,3} - 8-simplex {4,3,3,3,3,3,3} - 8-cube {3,3,3,3,3,3,4} - 8-orthoplex There are no nonconvex regular 8-polytopes
Uniform_8-polytope
Series of artworks by Marcel Duchamp
punning object, it was auctioned in 2009 for $11.5 million. Why Not Sneeze, Rose Sélavy?, 1921. Marble cubes in the shape of sugar lumps with a thermometer
Readymades_of_Marcel_Duchamp
Coxeter-Dynkin diagram, shown as . Rectified heptapeton (Acronym: ril) (Jonathan Bowers) The vertices of the rectified 6-simplex can be most simply positioned
Rectified_6-simplexes
Regular tessellation in 4D Euclidean space
honeycombs in 4-space: Truncated 5-cell honeycomb Omnitruncated 5-cell honeycomb Truncated 24-cell honeycomb Rectified 24-cell honeycomb Snub 24-cell honeycomb
24-cell_honeycomb
ten-dimensional geometry, a rectified 10-orthoplex is a 10-polytope, being a rectification of the regular 10-orthoplex. The rectified 10-orthoplex is the vertex
Rectified_10-orthoplexes
Uniform 4-polytope
vertex figure is a pentagonal pyramid, with one icosahedron on the base, and 5 truncated tetrahedra around the sides. Truncated 600-cell (Norman W. Johnson)
Truncated_120-cells
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