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geometry, a truncated 8-simplex is a convex uniform 8-polytope, being a truncation of the regular 8-simplex. There are four unique degrees of truncation. Vertices
Truncated_8-simplexes
Class of eight-dimensional polytopes
eight-dimensional geometry, a stericated 8-simplex is a convex uniform 8-polytope with 4th order truncations (sterication) of the regular 8-simplex. There are 16 unique
Stericated_8-simplexes
geometry, a heptellated 8-simplex is a convex uniform 8-polytope, including 7th-order truncations (heptellation) from the regular 8-simplex. There are 35
Heptellated_8-simplexes
pentellated 8-simplex is a convex uniform 8-polytope with 5th order truncations of the regular 8-simplex. There are two unique pentellations of the 8-simplex
Pentellated_8-simplexes
eight-dimensional geometry, a hexicated 8-simplex is a uniform 8-polytope, being a hexication (6th order truncation) of the regular 8-simplex. Acronym: supane (Jonathan
Hexicated_8-simplexes
permutations of truncation. Small rhombated enneazetton (acronym: srene) (Jonathan Bowers) The Cartesian coordinates of the vertices of the cantellated 8-simplex
Cantellated_8-simplexes
4-faces (6 5-cell and 6 truncated 5-cells). Truncated hexateron (Acronym: tix) (Jonathan Bowers) The vertices of the truncated 5-simplex can be most simply
Truncated_5-simplexes
Uniform 7-polytope
geometry, a truncated 7-simplex is a convex uniform 7-polytope, being a truncation of the regular 7-simplex. There are unique 3 degrees of truncation. Vertices
Truncated_7-simplexes
geometry, a truncated 6-simplex is a convex uniform 6-polytope, being a truncation of the regular 6-simplex. There are unique 3 degrees of truncation. Vertices
Truncated_6-simplexes
eight-dimensional geometry, a runcinated 8-simplex is a convex uniform 8-polytope with 3rd order truncations (runcination) of the regular 8-simplex. There are eleven
Runcinated_8-simplexes
Uniform 8-polytope
geometry, a cantic 8-cube or truncated 8-demicube is a uniform 8-polytope, being a truncation of the 8-demicube. Truncated demiocteract Truncated hemiocteract;
Cantic_8-cube
Type of 7-polytope
6th-order truncations (hexication) from the regular 7-simplex. There are 20 unique hexications for the 7-simplex, including all permutations of truncations, cantellations
Hexicated_7-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
analogue would be a truncated tetrahedron (truncated 3-demicube), and Coxeter diagram or as a cantic cube. Truncated demihepteract Truncated hemihepteract (acronym:
Cantic_7-cube
Uniform 6-polytope
Its vertex figure is a rectified 5-cell pyramid. Truncated icosihepta-heptacontadi-peton as a truncated 27-72 facetted polypeton (Acronym: tojak) Vertices
2_21_polytope
facets of the steriruncicantitruncated 7-orthoplex. The stericated 6-simplexes are in a set of 35 uniform 6-polytopes based on the [3,3,3,3,3] Coxeter
Stericated_6-simplexes
Convex uniform 7-polytope in seven-dimensional geometry
8-orthoplex. The trirectified 7-simplex is the intersection of two regular 7-simplexes in dual configuration. E. L. Elte identified it in 1912 as a semiregular
Rectified_7-simplexes
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
3rd order truncations (runcination) of the regular 7-simplex. There are 8 unique runcinations of the 7-simplex with permutations of truncations, and cantellations
Runcinated_7-simplexes
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
4th order truncations (sterication) of the regular 7-simplex. There are 14 unique sterication for the 7-simplex with permutations of truncations, cantellations
Stericated_7-simplexes
Regular 5-polytope
5-cell cells. These are seen as vertex figures of truncated regular 6-polytopes, like a truncated 6-cube. Another form is {3,3}∨{ }, with [3,3,2,1] symmetry
5-simplex
There are two degrees of truncations, including a bitruncation. The truncated 5-cell, truncated pentachoron or truncated 4-simplex is bounded by 10
Truncated_5-cell
3rd order truncations (Runcination) of the regular 5-simplex. There are 4 unique runcinations of the 5-simplex with permutations of truncations, and cantellations
Runcinated_5-simplexes
due to symmetrically-ringed Coxeter-Dynkin diagram. The runcinated 6-simplexes are in a set of 35 uniform 6-polytopes based on the [3,3,3,3,3] Coxeter
Runcinated_6-simplexes
Uniform polychoron
series in 1900 as containing all regular polytope facets, containing all simplexes and orthoplexes (tetrahedrons and octahedrons in the case of the rectified
Rectified_5-cell
based on facets of the bicantitruncated 7-orthoplex. The cantellated 6-simplexes are in a set of 35 uniform 6-polytopes based on the [3,3,3,3,3] Coxeter
Cantellated_6-simplexes
squares), 540 cells (360 truncated octahedra, 90 cubes, and 90 hexagonal prisms), and 62 4-faces (12 omnitruncated 5-cells, 30 truncated octahedral prisms,
Stericated_5-simplexes
Cube. The birectified 5-simplex is the intersection of two regular 5-simplexes in dual configuration. The vertices of a birectification exist at the
Rectified_5-simplexes
There are unique 4 degrees of cantellation for the 5-simplex, including truncations. The cantellated 5-simplex has 60 vertices, 240 edges, 290 faces (200
Cantellated_5-simplexes
64 snub 5-simplexes, 60 snub 24-cell antiprisms, 160 3-s{3,4} duoantiprisms, 240 2-sr{3,3} duoantiprisms, and 11520 irregular 5-simplexes filling the
Stericated_6-orthoplexes
truncations. Small rhombated octaexon (acronym: saro) (Jonathan Bowers) The vertices of the cantellated 7-simplex can be most simply positioned in 8-space
Cantellated_7-simplexes
Seven-dimensional geometric object
with one or more rings. All 71 are enumerated below. Norman Johnson's truncation names are given. Bowers names and acronym are also given for cross-referencing
Uniform_7-polytope
Topological invariant in mathematics
\chi =k_{0}-k_{1}+k_{2}-k_{3}+\cdots ,} where kn denotes the number of n-simplexes in the complex. More generally still, for any topological space, we can
Euler_characteristic
contained in an n-simplex and n−1 simplex is contained in exactly two n-simplexes, then the underlying space of X is a topological pseudomanifold. If X
Intersection_homology
Shape with three equal sides
types of a triangle belong to the infinite family of n {\displaystyle n} -simplexes, with n = 2 {\displaystyle n=2} . Clifton Cathedral Dirichlet distribution
Equilateral_triangle
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
Generalization of a polytope in real space
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
Complex_polytope
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