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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
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
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
a rectified 10-orthoplex is a convex uniform 10-polytope, being a rectification of the regular 10-orthoplex. There are 10 rectifications of the 10-orthoplex
Rectified_10-orthoplexes
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
ten-dimensional geometry, a rectified 10-simplex is a convex uniform 10-polytope, being a rectification of the regular 10-simplex. These polytopes are
Rectified_10-simplexes
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
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
5-demicube 5-cube, Rectified 5-cube, 5-cube, Truncated 5-cube, Cantellated 5-cube, Runcinated 5-cube, Stericated 5-cube 5-orthoplex, Rectified 5-orthoplex
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
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
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 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 tesseract, rectified 8-cell is a uniform 4-polytope (4-dimensional polytope) bounded by 24 cells: 8 cuboctahedra, and 16 tetrahedra
Rectified_tesseract
the 6-orthoplex. The rectified 6-orthoplex is the vertex figure for the demihexeractic honeycomb. or Rectified hexacross Rectified hexacontatetrapeton
Rectified_6-orthoplexes
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
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
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
Uniform 6-dimensional polytope
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 (hexacross) {3,3,3,3,4}. Regular
Uniform_6-polytope
constructed from this honeycomb. or rectified heptacross rectified hecatonicosaoctaexon (Acronym: rez) (Jonathan Bowers) - rectified 128-faceted polyexon There
Rectified_7-orthoplexes
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
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
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
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
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
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
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
as a first rectification of a 5-dimensional cross polytope. Rectified pentacross Rectified triacontaditeron (32-faceted 5-polytope) Acronym: rat (Jonathan
Rectified_5-orthoplexes
of tetrahedron and icosahedron cells. (The other two are the rectified 5-cell and rectified 600-cell.) Snub icositetrachoron Snub demitesseract Semi-snub
Snub_24-cell
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
Isogonal polyhedron with regular faces
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
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
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
Uniform 10-polytope
In geometry, a 10-demicube or demidekeract is a uniform 10-polytope, constructed from the 10-cube with alternated vertices removed. It is part of a dimensionally
10-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 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
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
Five-dimensional_space
Uniform 8 dimensional polytope
polytope Quadrirectified 421 polytope Rectified diacositetraconta-dischiliahectohexaconta-zetton as a rectified 240-2160 facetted polyzetton (acronym:
1_42_polytope
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
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
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
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
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 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
generated with 2 or more rings of the Coxeter group : , , , , . The rectified cubic-octahedral honeycomb is a compact uniform honeycomb, constructed
Cubic-octahedral_honeycomb
Uniform 6-polytope
in 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
2_21_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
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 polytope with uniform facets
and so on. An extended Schläfli symbol can be used for representing rectified forms, with a single subscript: k-th rectification = tk{p1, p2, ..., pn−1}
Uniform_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
Map projection
265–270. doi:10.1080/00050326.1990.10438681. ISSN 0005-0326. Grafarend, E. W.; Engels, J. (2001). Benciolini, Battista (ed.). "The Hotine Rectified Skew Orthomorphic
Oblique_Mercator_projection
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
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
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
D4 / A2 [6] A5 [6] D6 / A4 [10] D3 / A3 [4] 6 122 Pentacontatetrapeton (mo) 7 Rectified 122 / Birectified 221 Rectified pentacontatetrapeton (ram) 8
E6_polytope
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
Type of uniform space-filling tessellation
densest known sphere packing in 5 dimensions. The 40 vertices of the rectified 5-orthoplex vertex figure of the 5-demicubic honeycomb reflect the kissing
5-demicubic_honeycomb
Regular 5-polytope
called the Clebsch graph, though that name sometimes refers to the folded cube graph of order five instead. Cartesian coordinates for the vertices of a
5-demicube
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
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
simple Lie groups. The rectified 8-orthoplex is the vertex figure for the demiocteractic honeycomb. or Rectified octacross Rectified diacosipentacontahexazetton
Rectified_8-orthoplexes
5-dimensional geometric object
5-simplex honeycomb, . The 5-demicube honeycomb, , vertex figure is a rectified 5-orthoplex and facets are the 5-orthoplex and 5-demicube. Pyramidal 5-polytopes
5-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
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
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
Trirectified 421 (torfy) 5 Rectified 142 (buffy) 6 Rectified 241 (robay) 7 241 (bay) 8 Truncated 241 9 Truncated 421 (tiffy) 10 142 (bif) 11 Truncated 142
E8_polytope
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
forms, the 5-orthoplex, and 5-cube with 10 and 32 vertices respectively. The 5-demicube is added as an alternation of the 5-cube. They can be visualized as
B5_polytope
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
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
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
Type of uniform 4-polytope in four-dimensional geography
prismatic prism - - 4 cubes and 4 cubes - (same as 4-4 duoprism and same as tesseract) Pentagonal prismatic prism - - 5 cubes and 4 pentagonal prisms
Prismatic_uniform_4-polytope
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)
9-polytopes with BC9 symmetry. The rectified 9-orthoplex is the vertex figure for the demienneractic honeycomb. or Rectified enneacross (Acronym: riv) (Jonathan
Rectified_9-orthoplexes
the square faces. The 24-cell facets exist between these vertices as rectified 16-cells. If the coordinates of the tesseractic honeycomb are integers
24-cell_honeycomb
German remote sensing cubesat
GPS tracking. On 10 August 2008, the satellite developed a problem and switched into "emergency mode". Initial attempts to rectify this problem failed;
Compass-1
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
Type of geometric object
the 5-cell centers of the 9-simplex. The rectified 9-simplex is the vertex figure of the 10-demicube. Rectified decayotton (reday) (Jonathan Bowers) The
Rectified_9-simplexes
Uniform 4-polytope
120-cell Truncated 120-cell Rectified 120-cell Bitruncated 120-cell Bitruncated 600-cell 600-cell Truncated 600-cell Rectified 600-cell Orthogonal projections
Truncated_120-cells
Octahedron centered Dual octahedron centered 1 24-cell (rectified 16-cell) = {3,4,3} = r{3,3,4} 2 rectified 24-cell (cantellated 16-cell) = r{3,4,3} = rr{3,3
List_of_F4_polytopes
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
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
24-cell honeycomb, containing tesseract and rectified 24-cell cells. Rectified icositetrachoric tetracomb Rectified icositetrachoric honeycomb Cantellated
Rectified_24-cell_honeycomb
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
A3 [4] Cube centered Tetrahedron centered 13 *rectified 16-cell (Same as 24-cell) = r{3,3,4} = {3,4,3} 14 *cantellated 16-cell (Same as rectified 24-cell)
B4_polytope
Group of polytopes
with half symmetry). There are two regular forms, the 8-orthoplex and 8-cube, with 16 and 256 vertices respectively. They can be visualized as symmetric
B8_polytope
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
Tessellation of convex uniform polyhedron cells
LCCN 99-35678, ISBN 0-486-40919-8 (Chapter 10, Regular Honeycombs in Hyperbolic Space Archived 2016-06-10 at the Wayback Machine) Coxeter, Regular Polytopes
Paracompact uniform honeycombs
Paracompact_uniform_honeycombs
Five-dimensional geometric shape
such regular polytopes, 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
Uniform_5-polytope
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
Type of geometric object
Coxeter-Dynkin diagrams with one or more rings. Eleven cases are shown below: Nine rectified forms and 2 truncations. Bowers-style acronym names are given in parentheses
Uniform_9-polytope
4D geometry item
Cantellated tetrahedron (Cuboctahedron) 1 1200 None (Degenerate triangular prism) 2 720 Pentagonal prism 3 120 Rectified dodecahedron (Icosidodecahedron)
Cantellated_120-cell
vertex figures. The aforementioned honeycombs are also quasiregular: The rectified order-4 hexagonal tiling honeycomb, t1{6,3,4}, has octahedral and trihexagonal
Order-4 hexagonal tiling honeycomb
Order-4_hexagonal_tiling_honeycomb
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
then be alternated to produce another nonuniform polychoron with 48 snub cubes, 144 square antiprisms, 192 octahedra (as triangular antiprisms), three
Runcinated_24-cells
Solid with eight equal triangular faces
equilateral triangles. The regular octahedron can also be considered a rectified tetrahedron, sometimes called a tetratetrahedron (by analogy to the cuboctahedron
Regular_octahedron
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