eight-dimensional geometry, a rectified 8-simplex is a convex uniform 8-polytope, being a rectification of the regular 8-simplex. There are unique 3 degrees...

it as S1 7. Rectified octaexon (Acronym: roc) (Jonathan Bowers) The vertices of the rectified 7-simplex can be most simply positioned in 8-space as permutations...

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...

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...

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)...

eight-dimensional geometry, a truncated 8-simplex is a convex uniform 8-polytope, being a truncation of the regular 8-simplex. There are four unique degrees...

Coxeter-Dynkin diagram, shown as . Rectified heptapeton (Acronym: ril) (Jonathan Bowers) The vertices of the rectified 6-simplex can be most simply positioned...

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...

that are facets of 7-simplexes. Since every 7-orthoplex has 128 (27) 6-simplex facets, half of which are not incident to 7-simplexes, the 421 polytope has...

(tetrahedra), 16128 4-faces (3-simplexes), 4788 5-faces (756 pentacrosses, and 4032 5-simplexes), 632 6-faces (576 6-simplexes and 56 221). Its vertex figure...

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...

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...

3-sphere tiling, a tetrahedral hosohedron.) Rectified hecatonicosihexa-pentacosiheptacontihexa-exon as a rectified 126-576 facetted polyexon (acronym ranq)...

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...

simplexes and orthoplexes. The 221 polytope is fourth in dimensional series 2k2. The 221 polytope is second in dimensional series 22k. The rectified 221...

vertices of {4,3,...,3}. The vertex figures of demihypercubes are rectified n-simplexes. They are represented by Coxeter-Dynkin diagrams of three constructive...

stericated 6-orthoplex. A second construction in 6-space, from the center of a rectified 6-orthoplex is given by coordinate permutations of: (1,-1,0,0,0,0) The...

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)...

Dih6 or simple rotation group [6,2]+, order 12. The compound of two 5-simplexes in dual configurations can be seen in this A6 Coxeter plane projection...

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...

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...

vertex figure. All facets of these polytopes are regular polytopes, namely simplexes and orthoplexes. The 261 honeycomb is composed of 251 9-honeycomb and...

) The family starts uniquely as 6-polytopes. The triangular prism and rectified 5-cell are included at the beginning for completeness. The demipenteract...

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)...

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...

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, 27 edges...

Beauty of Geometry: Twelve Essays, Dover Publications, 1999, ISBN 0-486-40919-8 p. 88 (Chapter 5: Regular Skew Polyhedra in three and four dimensions and...