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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
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
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
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
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
Convex polyhedron with 38 faces
creates two more polyhedra: Rectified truncated tetrahedron Rectified truncated octahedron Rectified truncated dodecahedron Rectified truncated icosahedron
Rectified_truncated_cube
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
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
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
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
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
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)
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
Uniform 6-polytope
(dual of E6 lattice). Birectified 221 polytope Rectified pentacontatetrapeton (Acronym: ram) - rectified 54-facetted polypeton (Jonathan Bowers) Vertices
1_22_polytope
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
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
constructed from this honeycomb. or rectified heptacross rectified hecatonicosaoctaexon (Acronym: rez) (Jonathan Bowers) - rectified 128-faceted polyexon There
Rectified_7-orthoplexes
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
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
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
Type of polyhedron
also be constructed as a rectified rhombicuboctahedron. Expanded rhombic dodecahedron Rectified rhombicuboctahedron Rectified small rhombicuboctahedron
Expanded_cuboctahedron
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
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)
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
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
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
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
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
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
the 6-orthoplex. The rectified 6-orthoplex is the vertex figure for the demihexeractic honeycomb. or Rectified hexacross Rectified hexacontatetrapeton
Rectified_6-orthoplexes
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
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
Uniform 8 dimensional polytope
polytope Quadrirectified 421 polytope Rectified diacositetraconta-dischiliahectohexaconta-zetton as a rectified 240-2160 facetted polyzetton (acronym:
1_42_polytope
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
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
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
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
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
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
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
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)
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
A5 [6] D3 / A3 [4] 1 221 Icosihepta-heptacontadipeton (jak) 2 Rectified 221 Rectified icosihepta-heptacontadipeton (rojak) 3 Trirectified 221 Trirectified
E6_polytope
Concept in euclidean geometry
centered cubic (a checkerboard in which the red 4-cubes have a central vertex but the black 4-cubes do not). The tesseract can make a regular tessellation
Tesseractic_honeycomb
Uniform 9-polytope
demienneract or 9-demicube is a uniform 9-polytope, constructed from the 9-cube, with alternated vertices removed. It is part of a dimensionally infinite
9-demicube
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
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)
Cantellated pentachoron Cantellated 4-simplex (small) prismatodispentachoron Rectified dispentachoron Small rhombated pentachoron (Acronym: Srip) (Jonathan Bowers)
Cantellated_5-cell
square tiling honeycomb, is the same thing as the rectified square tiling honeycomb, . It has cube and square tiling facets, with a triangular prism vertex
Order-4 square tiling honeycomb
Order-4_square_tiling_honeycomb
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
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
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
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
Regular 5-polytope
6-orthoplex or rectified 6-cube respectively. A lower symmetry form is a 5-cell pyramid {3,3,3}∨( ), with [3,3,3] symmetry order 120, constructed as a 5-cell base
5-simplex
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
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
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
Type of uniform 4-polytope in four-dimensional geography
- 4 cubes and 4 cubes - (same as 4-4 duoprism and same as tesseract) Pentagonal prismatic prism - - 5 cubes and 4 pentagonal prisms - (same as 4-5 duoprism)
Prismatic_uniform_4-polytope
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
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
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
simple Lie groups. The rectified 8-orthoplex is the vertex figure for the demiocteractic honeycomb. or Rectified octacross Rectified diacosipentacontahexazetton
Rectified_8-orthoplexes
[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
In seven-dimensional geometry, a cantic 7-cube or truncated 7-demicube as a uniform 7-polytope, being a truncation of the 7-demicube. A uniform 7-polytope
Cantic_7-cube
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