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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
Uniform 6-polytope
constructed from 14 snub 5-simplexes, 42 snub 5-cell antiprisms, 70 3-s{3,4} duoantiprisms, and 2520 irregular 5-simplexes filling the gaps at the deleted
Pentellated_6-simplexes
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
seen as facets of the rectified 6-orthoplex or birectified 6-cube respectively. This configuration matrix represents the rectified 5-simplex. The rows and
Rectified_5-simplexes
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
5-cell centers of the 10-simplex. The rectified 10-simplex is the vertex figure of the 11-demicube. Rectified hendecaxennon (acronym: ru) (Jonathan Bowers)
Rectified_10-simplexes
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
7-demicube. A uniform 7-polytope is vertex-transitive and constructed from uniform 6-polytope facets, and can be represented a coxeter diagram with ringed nodes
Cantic_7-cube
Polytope in 8-dimensional geometry
containing all regular polytope facets, containing all simplexes and orthoplexes. The rectified 421 can be seen as a rectification of the 421 polytope
4_21_polytope
Uniform 8-polytope
vertices of a truncated 8-demicube centered at the origin and edge length 6√2 are coordinate permutations: (±1,±1,±3,±3,±3,±3,±3,±3) with an odd number
Cantic_8-cube
Uniform Polytope
4-faces (4-simplexes), 4788 5-faces (756 pentacrosses, and 4032 5-simplexes), 632 6-faces (576 6-simplexes and 56 221). Its vertex figure is a 6-demicube
2_31_polytope
Uniform 6-polytope
simplexes and orthoplexes. The 221 polytope is fourth in dimensional series 2k1. The 221 polytope is second in dimensional series 22k. The rectified 221
2_21_polytope
Uniform 7-dimensional polytope
uniform polytope. It has 56 vertices, and 702 facets: 126 311 and 576 6-simplexes. For visualization this 7-dimensional polytope is often displayed in
3_21_polytope
Type of 7-polytope
8-orthoplex, . A second construction in 8-space, from the center of a rectified 8-orthoplex is given by coordinate permutations of: (1,-1,0,0,0,0,0,0)
Hexicated_7-simplexes
Polytope constructed from alternation of a hypercube
vertices of {4,3,...,3}. The vertex figures of demihypercubes are rectified n-simplexes. They are represented by Coxeter-Dynkin diagrams of three constructive
Demihypercube
Uniform polychoron
containing all simplexes and orthoplexes (tetrahedrons and octahedrons in the case of the rectified 5-cell). The Coxeter symbol for the rectified 5-cell is
Rectified_5-cell
Four-dimensional analogue of the tetrahedron
face-to-face) with the two opposite vertices centered. In the case of simplexes such as the 5-cell, certain irregular forms are in some sense more fundamental
5-cell
pentacomb). Each vertex is shared by 12 5-simplexes, 30 rectified 5-simplexes, and 20 birectified 5-simplexes. These facet types occur in proportions of
5-simplex_honeycomb
Geometric object
vertices at infinity.) The family starts uniquely as 6-polytopes. The triangular prism and rectified 5-cell are included at the beginning for completeness
Uniform_k_21_polytope
Weiss, Wiley-Interscience Publication, 1995, wiley.com, ISBN 978-0-471-01003-6 (Paper 22) H.S.M. Coxeter, Regular and Semi-Regular Polytopes I, [Math. Zeit
Truncated_8-simplexes
Regular 5-polytope
hyperplane in 6-space as permutations of (0,0,0,0,0,1) or (0,1,1,1,1,1). These constructions can be seen as facets of the 6-orthoplex or rectified 6-cube respectively
5-simplex
heptellated 9-orthoplex. A second construction in 9-space, from the center of a rectified 9-orthoplex is given by coordinate permutations of: (1,-1,0,0,0,0,0,0
Heptellated_8-simplexes
positive orthant facet of the stericated 6-orthoplex. A second construction in 6-space, from the center of a rectified 6-orthoplex is given by coordinate permutations
Stericated_5-simplexes
15 cuboctahedra and 60 triangular prisms), and 27 4-faces (6 cantellated 5-cell, 6 rectified 5-cells, and 15 tetrahedral prisms). Cantellated hexateron
Cantellated_5-simplexes
Seven-dimensional geometric object
a 7-polytope is a polytope contained by 6-polytope facets. Each 5-polytope ridge being shared by exactly two 6-polytope facets. A uniform 7-polytope is
Uniform_7-polytope
vertex figure. All facets of these polytopes are regular polytopes, namely simplexes and orthoplexes. The 261 honeycomb is composed of 251 9-honeycomb and
E9_honeycomb
Prism with a 3-sided base
series in 1900 as containing all regular polytope facets, containing all simplexes and orthoplexes (equilateral triangles and squares in the case of the
Triangular_prism
Four-dimensional analog of the dodecahedron
unless it is all 5 vertices. It is impossible to rotate two concentric 4-simplexes with respect to each other such that some, but not all, of their vertices
120-cell
Regular 5-polytope
series in 1900 as containing all regular polytope facets, containing all simplexes and orthoplexes (5-simplices and 5-orthoplexes in the case of the 5-demicube)
5-demicube
Generalization of a polytope in real space
construction as and quasiregular form as . All elements are simplexes. Real {3,4}, or , with 6 vertices, 12 edges, and 8 faces 2{3}2{4}3, or , with 9 vertices
Complex_polytope
Weiss, Wiley-Interscience Publication, 1995, wiley.com, ISBN 978-0-471-01003-6 (Paper 22) H.S.M. Coxeter, Regular and Semi-Regular Polytopes I, [Math. Zeit
Truncated_5-cell
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