Runcinated 7-simplexes

Orthogonal projections in A₇ Coxeter plane
File:7-simplex t0.svg 7-simplex File:CDel node 1.pngFile:CDel 3.pngFile:CDel node.pngFile:CDel 3.pngFile:CDel node.pngFile:CDel 3.pngFile:CDel node.pngFile:CDel 3.pngFile:CDel node.pngFile:CDel 3.pngFile:CDel node.pngFile:CDel 3.pngFile:CDel node.png	File:7-simplex t03.svg Runcinated 7-simplex File:CDel node 1.pngFile:CDel 3.pngFile:CDel node.pngFile:CDel 3.pngFile:CDel node.pngFile:CDel 3.pngFile:CDel node 1.pngFile:CDel 3.pngFile:CDel node.pngFile:CDel 3.pngFile:CDel node.pngFile:CDel 3.pngFile:CDel node.png	File:7-simplex t14.svg Biruncinated 7-simplex File:CDel node.pngFile:CDel 3.pngFile:CDel node 1.pngFile:CDel 3.pngFile:CDel node.pngFile:CDel 3.pngFile:CDel node.pngFile:CDel 3.pngFile:CDel node 1.pngFile:CDel 3.pngFile:CDel node.pngFile:CDel 3.pngFile:CDel node.png
File:7-simplex t013.svg Runcitruncated 7-simplex File:CDel node 1.pngFile:CDel 3.pngFile:CDel node 1.pngFile:CDel 3.pngFile:CDel node.pngFile:CDel 3.pngFile:CDel node 1.pngFile:CDel 3.pngFile:CDel node.pngFile:CDel 3.pngFile:CDel node.pngFile:CDel 3.pngFile:CDel node.png	File:7-simplex t124.svg Biruncitruncated 7-simplex File:CDel node.pngFile:CDel 3.pngFile:CDel node 1.pngFile:CDel 3.pngFile:CDel node 1.pngFile:CDel 3.pngFile:CDel node.pngFile:CDel 3.pngFile:CDel node 1.pngFile:CDel 3.pngFile:CDel node.pngFile:CDel 3.pngFile:CDel node.png	File:7-simplex t023.svg Runcicantellated 7-simplex File:CDel node 1.pngFile:CDel 3.pngFile:CDel node.pngFile:CDel 3.pngFile:CDel node 1.pngFile:CDel 3.pngFile:CDel node 1.pngFile:CDel 3.pngFile:CDel node.pngFile:CDel 3.pngFile:CDel node.pngFile:CDel 3.pngFile:CDel node.png
File:7-simplex t134.svg Biruncicantellated 7-simplex File:CDel node.pngFile:CDel 3.pngFile:CDel node 1.pngFile:CDel 3.pngFile:CDel node.pngFile:CDel 3.pngFile:CDel node 1.pngFile:CDel 3.pngFile:CDel node 1.pngFile:CDel 3.pngFile:CDel node.pngFile:CDel 3.pngFile:CDel node.png	File:7-simplex t0123.svg Runcicantitruncated 7-simplex File:CDel node 1.pngFile:CDel 3.pngFile:CDel node 1.pngFile:CDel 3.pngFile:CDel node 1.pngFile:CDel 3.pngFile:CDel node 1.pngFile:CDel 3.pngFile:CDel node.pngFile:CDel 3.pngFile:CDel node.pngFile:CDel 3.pngFile:CDel node.png	File:7-simplex t1234.svg Biruncicantitruncated 7-simplex File:CDel node.pngFile:CDel 3.pngFile:CDel node 1.pngFile:CDel 3.pngFile:CDel node 1.pngFile:CDel 3.pngFile:CDel node 1.pngFile:CDel 3.pngFile:CDel node 1.pngFile:CDel 3.pngFile:CDel node.pngFile:CDel 3.pngFile:CDel node.png

In seven-dimensional geometry, a runcinated 7-simplex is a convex uniform 7-polytope with 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-simplex

Runcinated 7-simplex
Type	uniform 7-polytope
Schläfli symbol	t_0,3{3,3,3,3,3,3}
Coxeter-Dynkin diagrams	File:CDel node 1.pngFile:CDel 3.pngFile:CDel node.pngFile:CDel 3.pngFile:CDel node.pngFile:CDel 3.pngFile:CDel node 1.pngFile:CDel 3.pngFile:CDel node.pngFile:CDel 3.pngFile:CDel node.pngFile:CDel 3.pngFile:CDel node.png
6-faces
5-faces
4-faces
Cells
Faces
Edges	2100
Vertices	280
Vertex figure
Coxeter group	A₇, [3⁶], order 40320
Properties	convex

Alternate names

Small prismated octaexon (acronym: spo) (Jonathan Bowers)^[1]

Coordinates

The vertices of the runcinated 7-simplex can be most simply positioned in 8-space as permutations of (0,0,0,0,1,1,1,2). This construction is based on facets of the runcinated 8-orthoplex.

Images

orthographic projections
A_k Coxeter plane	A₇	A₆	A₅
Graph	File:7-simplex t03.svg	File:7-simplex t03 A6.svg	File:7-simplex t03 A5.svg
Dihedral symmetry	[8]	[7]	[6]
A_k Coxeter plane	A₄	A₃	A₂
Graph	File:7-simplex t03 A4.svg	File:7-simplex t03 A3.svg	File:7-simplex t03 A2.svg
Dihedral symmetry	[5]	[4]	[3]

Biruncinated 7-simplex

Biruncinated 7-simplex
Type	uniform 7-polytope
Schläfli symbol	t_1,4{3,3,3,3,3,3}
Coxeter-Dynkin diagrams	File:CDel node.pngFile:CDel 3.pngFile:CDel node 1.pngFile:CDel 3.pngFile:CDel node.pngFile:CDel 3.pngFile:CDel node.pngFile:CDel 3.pngFile:CDel node 1.pngFile:CDel 3.pngFile:CDel node.pngFile:CDel 3.pngFile:CDel node.png
6-faces
5-faces
4-faces
Cells
Faces
Edges	4200
Vertices	560
Vertex figure
Coxeter group	A₇, [3⁶], order 40320
Properties	convex

Alternate names

Small biprismated octaexon (sibpo) (Jonathan Bowers)^[2]

Coordinates

The vertices of the biruncinated 7-simplex can be most simply positioned in 8-space as permutations of (0,0,0,1,1,1,2,2). This construction is based on facets of the biruncinated 8-orthoplex.

Images

orthographic projections
A_k Coxeter plane	A₇	A₆	A₅
Graph	File:7-simplex t14.svg	File:7-simplex t14 A6.svg	File:7-simplex t14 A5.svg
Dihedral symmetry	[8]	[7]	[6]
A_k Coxeter plane	A₄	A₃	A₂
Graph	File:7-simplex t14 A4.svg		File:7-simplex t14 A2.svg
Dihedral symmetry	[5]	[4]	[3]

Runcitruncated 7-simplex

runcitruncated 7-simplex
Type	uniform 7-polytope
Schläfli symbol	t_0,1,3{3,3,3,3,3,3}
Coxeter-Dynkin diagrams	File:CDel node 1.pngFile:CDel 3.pngFile:CDel node 1.pngFile:CDel 3.pngFile:CDel node.pngFile:CDel 3.pngFile:CDel node 1.pngFile:CDel 3.pngFile:CDel node.pngFile:CDel 3.pngFile:CDel node.pngFile:CDel 3.pngFile:CDel node.png
6-faces
5-faces
4-faces
Cells
Faces
Edges	4620
Vertices	840
Vertex figure
Coxeter group	A₇, [3⁶], order 40320
Properties	convex

Alternate names

Prismatotruncated octaexon (acronym: patto) (Jonathan Bowers)^[3]

Coordinates

The vertices of the runcitruncated 7-simplex can be most simply positioned in 8-space as permutations of (0,0,0,0,1,1,2,3). This construction is based on facets of the runcitruncated 8-orthoplex.

Images

orthographic projections
A_k Coxeter plane	A₇	A₆	A₅
Graph	File:7-simplex t013.svg	File:7-simplex t013 A6.svg	File:7-simplex t013 A5.svg
Dihedral symmetry	[8]	[7]	[6]
A_k Coxeter plane	A₄	A₃	A₂
Graph	File:7-simplex t013 A4.svg	File:7-simplex t013 A3.svg	File:7-simplex t013 A2.svg
Dihedral symmetry	[5]	[4]	[3]

Biruncitruncated 7-simplex

Biruncitruncated 7-simplex
Type	uniform 7-polytope
Schläfli symbol	t_1,2,4{3,3,3,3,3,3}
Coxeter-Dynkin diagrams	File:CDel node.pngFile:CDel 3.pngFile:CDel node 1.pngFile:CDel 3.pngFile:CDel node 1.pngFile:CDel 3.pngFile:CDel node.pngFile:CDel 3.pngFile:CDel node 1.pngFile:CDel 3.pngFile:CDel node.pngFile:CDel 3.pngFile:CDel node.png
6-faces
5-faces
4-faces
Cells
Faces
Edges	8400
Vertices	1680
Vertex figure
Coxeter group	A₇, [3⁶], order 40320
Properties	convex

Alternate names

Biprismatotruncated octaexon (acronym: bipto) (Jonathan Bowers)^[4]

Coordinates

The vertices of the biruncitruncated 7-simplex can be most simply positioned in 8-space as permutations of (0,0,0,1,1,2,3,3). This construction is based on facets of the biruncitruncated 8-orthoplex.

Images

orthographic projections
A_k Coxeter plane	A₇	A₆	A₅
Graph	File:7-simplex t124.svg	File:7-simplex t124 A6.svg	File:7-simplex t124 A5.svg
Dihedral symmetry	[8]	[7]	[6]
A_k Coxeter plane	A₄	A₃	A₂
Graph	File:7-simplex t124 A4.svg	File:7-simplex t124 A3.svg	File:7-simplex t124 A2.svg
Dihedral symmetry	[5]	[4]	[3]

Runcicantellated 7-simplex

runcicantellated 7-simplex
Type	uniform 7-polytope
Schläfli symbol	t_0,2,3{3,3,3,3,3,3}
Coxeter-Dynkin diagrams	File:CDel node 1.pngFile:CDel 3.pngFile:CDel node.pngFile:CDel 3.pngFile:CDel node 1.pngFile:CDel 3.pngFile:CDel node 1.pngFile:CDel 3.pngFile:CDel node.pngFile:CDel 3.pngFile:CDel node.pngFile:CDel 3.pngFile:CDel node.png
6-faces
5-faces
4-faces
Cells
Faces
Edges	3360
Vertices	840
Vertex figure
Coxeter group	A₇, [3⁶], order 40320
Properties	convex

Alternate names

Prismatorhombated octaexon (acronym: paro) (Jonathan Bowers)^[5]

Coordinates

The vertices of the runcicantellated 7-simplex can be most simply positioned in 8-space as permutations of (0,0,0,0,1,2,2,3). This construction is based on facets of the runcicantellated 8-orthoplex.

Images

orthographic projections
A_k Coxeter plane	A₇	A₆	A₅
Graph	File:7-simplex t023.svg	File:7-simplex t023 A6.svg	File:7-simplex t023 A5.svg
Dihedral symmetry	[8]	[7]	[6]
A_k Coxeter plane	A₄	A₃	A₂
Graph	File:7-simplex t023 A4.svg	File:7-simplex t023 A3.svg	File:7-simplex t023 A2.svg
Dihedral symmetry	[5]	[4]	[3]

Biruncicantellated 7-simplex

biruncicantellated 7-simplex
Type	uniform 7-polytope
Schläfli symbol	t_1,3,4{3,3,3,3,3,3}
Coxeter-Dynkin diagrams	File:CDel node.pngFile:CDel 3.pngFile:CDel node 1.pngFile:CDel 3.pngFile:CDel node.pngFile:CDel 3.pngFile:CDel node 1.pngFile:CDel 3.pngFile:CDel node 1.pngFile:CDel 3.pngFile:CDel node.pngFile:CDel 3.pngFile:CDel node.png
6-faces
5-faces
4-faces
Cells
Faces
Edges
Vertices
Vertex figure
Coxeter group	A₇, [3⁶], order 40320
Properties	convex

Alternate names

Biprismatorhombated octaexon (acronym: bipro) (Jonathan Bowers)

Coordinates

The vertices of the biruncicantellated 7-simplex can be most simply positioned in 8-space as permutations of (0,0,0,1,2,2,3,3). This construction is based on facets of the biruncicantellated 8-orthoplex.

Images

orthographic projections
A_k Coxeter plane	A₇	A₆	A₅
Graph	File:7-simplex t134.svg	File:7-simplex t134 A6.svg	File:7-simplex t134 A5.svg
Dihedral symmetry	[8]	[7]	[6]
A_k Coxeter plane	A₄	A₃	A₂
Graph	File:7-simplex t134 A4.svg	File:7-simplex t134 A3.svg	File:7-simplex t134 A2.svg
Dihedral symmetry	[5]	[4]	[3]

Runcicantitruncated 7-simplex

runcicantitruncated 7-simplex
Type	uniform 7-polytope
Schläfli symbol	t_0,1,2,3{3,3,3,3,3,3}
Coxeter-Dynkin diagrams	File:CDel node 1.pngFile:CDel 3.pngFile:CDel node 1.pngFile:CDel 3.pngFile:CDel node 1.pngFile:CDel 3.pngFile:CDel node 1.pngFile:CDel 3.pngFile:CDel node.pngFile:CDel 3.pngFile:CDel node.pngFile:CDel 3.pngFile:CDel node.png
6-faces
5-faces
4-faces
Cells
Faces
Edges	5880
Vertices	1680
Vertex figure
Coxeter group	A₇, [3⁶], order 40320
Properties	convex

Alternate names

Great prismated octaexon (acronym: gapo) (Jonathan Bowers)^[6]

Coordinates

The vertices of the runcicantitruncated 7-simplex can be most simply positioned in 8-space as permutations of (0,0,0,0,1,2,3,4). This construction is based on facets of the runcicantitruncated 8-orthoplex.

Images

orthographic projections
A_k Coxeter plane	A₇	A₆	A₅
Graph	File:7-simplex t0123.svg	File:7-simplex t0123 A6.svg	File:7-simplex t0123 A5.svg
Dihedral symmetry	[8]	[7]	[6]
A_k Coxeter plane	A₄	A₃	A₂
Graph	File:7-simplex t0123 A4.svg	File:7-simplex t0123 A3.svg	File:7-simplex t0123 A2.svg
Dihedral symmetry	[5]	[4]	[3]

Biruncicantitruncated 7-simplex

biruncicantitruncated 7-simplex
Type	uniform 7-polytope
Schläfli symbol	t_1,2,3,4{3,3,3,3,3,3}
Coxeter-Dynkin diagrams	File:CDel node.pngFile:CDel 3.pngFile:CDel node 1.pngFile:CDel 3.pngFile:CDel node 1.pngFile:CDel 3.pngFile:CDel node 1.pngFile:CDel 3.pngFile:CDel node 1.pngFile:CDel 3.pngFile:CDel node.pngFile:CDel 3.pngFile:CDel node.png
6-faces
5-faces
4-faces
Cells
Faces
Edges	11760
Vertices	3360
Vertex figure
Coxeter group	A₇, [3⁶], order 40320
Properties	convex

Alternate names

Great biprismated octaexon (acronym: gibpo) (Jonathan Bowers)^[7]

Coordinates

The vertices of the biruncicantitruncated 7-simplex can be most simply positioned in 8-space as permutations of (0,0,0,1,2,3,4,4). This construction is based on facets of the biruncicantitruncated 8-orthoplex.

Images

orthographic projections
A_k Coxeter plane	A₇	A₆	A₅
Graph	File:7-simplex t1234.svg	File:7-simplex t1234 A6.svg	File:7-simplex t1234 A5.svg
Dihedral symmetry	[8]	[7]	[6]
A_k Coxeter plane	A₄	A₃	A₂
Graph	File:7-simplex t1234 A4.svg	File:7-simplex t1234 A3.svg	File:7-simplex t1234 A2.svg
Dihedral symmetry	[5]	[4]	[3]

Related polytopes

These polytopes are among 71 uniform 7-polytopes with A₇ symmetry.

A7 polytopes
File:7-simplex t0.svg t₀	File:7-simplex t1.svg t₁	File:7-simplex t2.svg t₂	File:7-simplex t3.svg t₃	File:7-simplex t01.svg t_0,1	File:7-simplex t02.svg t_0,2	File:7-simplex t12.svg t_1,2	File:7-simplex t03.svg t_0,3
File:7-simplex t13.svg t_1,3	File:7-simplex t23.svg t_2,3	File:7-simplex t04.svg t_0,4	File:7-simplex t14.svg t_1,4	File:7-simplex t24.svg t_2,4	File:7-simplex t05.svg t_0,5	File:7-simplex t15.svg t_1,5	File:7-simplex t06.svg t_0,6
File:7-simplex t012.svg t_0,1,2	File:7-simplex t013.svg t_0,1,3	File:7-simplex t023.svg t_0,2,3	File:7-simplex t123.svg t_1,2,3	File:7-simplex t014.svg t_0,1,4	File:7-simplex t024.svg t_0,2,4	File:7-simplex t124.svg t_1,2,4	File:7-simplex t034.svg t_0,3,4
File:7-simplex t134.svg t_1,3,4	File:7-simplex t234.svg t_2,3,4	File:7-simplex t015.svg t_0,1,5	File:7-simplex t025.svg t_0,2,5	File:7-simplex t125.svg t_1,2,5	File:7-simplex t035.svg t_0,3,5	File:7-simplex t135.svg t_1,3,5	File:7-simplex t045.svg t_0,4,5
File:7-simplex t016.svg t_0,1,6	File:7-simplex t026.svg t_0,2,6	File:7-simplex t036.svg t_0,3,6	File:7-simplex t0123.svg t_0,1,2,3	File:7-simplex t0124.svg t_0,1,2,4	File:7-simplex t0134.svg t_0,1,3,4	File:7-simplex t0234.svg t_0,2,3,4	File:7-simplex t1234.svg t_1,2,3,4
File:7-simplex t0125.svg t_0,1,2,5	File:7-simplex t0135.svg t_0,1,3,5	File:7-simplex t0235.svg t_0,2,3,5	File:7-simplex t1235.svg t_1,2,3,5	File:7-simplex t0145.svg t_0,1,4,5	File:7-simplex t0245.svg t_0,2,4,5	File:7-simplex t1245.svg t_1,2,4,5	File:7-simplex t0345.svg t_0,3,4,5
File:7-simplex t0126.svg t_0,1,2,6	File:7-simplex t0136.svg t_0,1,3,6	File:7-simplex t0236.svg t_0,2,3,6	File:7-simplex t0146.svg t_0,1,4,6	File:7-simplex t0246.svg t_0,2,4,6	File:7-simplex t0156.svg t_0,1,5,6	File:7-simplex t01234.svg t_0,1,2,3,4	File:7-simplex t01235.svg t_0,1,2,3,5
File:7-simplex t01245.svg t_0,1,2,4,5	File:7-simplex t01345.svg t_0,1,3,4,5	File:7-simplex t02345.svg t_0,2,3,4,5	File:7-simplex t12345.svg t_1,2,3,4,5	File:7-simplex t01236.svg t_0,1,2,3,6	File:7-simplex t01246.svg t_0,1,2,4,6	File:7-simplex t01346.svg t_0,1,3,4,6	File:7-simplex t02346.svg t_0,2,3,4,6
File:7-simplex t01256.svg t_0,1,2,5,6	File:7-simplex t01356.svg t_0,1,3,5,6	File:7-simplex t012345.svg t_0,1,2,3,4,5	File:7-simplex t012346.svg t_0,1,2,3,4,6	File:7-simplex t012356.svg t_0,1,2,3,5,6	File:7-simplex t012456.svg t_0,1,2,4,5,6	File:7-simplex t0123456.svg t_{0,1,2,3,4,5,6}

Notes

^ Klitzing, (x3o3o3x3o3o3o - spo)
^ Klitzing, (o3x3o3o3x3o3o - sibpo)
^ Klitzing, (x3x3o3x3o3o3o - patto)
^ Klitzing, (o3x3x3o3x3o3o - bipto)
^ Klitzing, (x3o3x3x3o3o3o - paro)
^ Klitzing, (x3x3x3x3o3o3o - gapo)
^ Klitzing, (o3x3x3x3x3o3o- gibpo)

References

H.S.M. Coxeter:
- H.S.M. Coxeter, Regular Polytopes, 3rd Edition, Dover New York, 1973
- Kaleidoscopes: Selected Writings of H.S.M. Coxeter, edited by F. Arthur Sherk, Peter McMullen, Anthony C. Thompson, Asia Ivic Weiss, Wiley-Interscience Publication, 1995, wiley.com, Lua error in Module:Citation/CS1/Configuration at line 2172: attempt to index field '?' (a nil value).
  - (Paper 22) H.S.M. Coxeter, Regular and Semi Regular Polytopes I, [Math. Zeit. 46 (1940) 380-407, MR 2,10]
  - (Paper 23) H.S.M. Coxeter, Regular and Semi-Regular Polytopes II, [Math. Zeit. 188 (1985) 559-591]
  - (Paper 24) H.S.M. Coxeter, Regular and Semi-Regular Polytopes III, [Math. Zeit. 200 (1988) 3-45]
Norman Johnson Uniform Polytopes, Manuscript (1991)
- N.W. Johnson: The Theory of Uniform Polytopes and Honeycombs, Ph.D.
Lua error in Module:Citation/CS1/Configuration at line 2172: attempt to index field '?' (a nil value). x3o3o3x3o3o3o - spo, o3x3o3o3x3o3o - sibpo, x3x3o3x3o3o3o - patto, o3x3x3o3x3o3o - bipto, x3o3x3x3o3o3o - paro, x3x3x3x3o3o3o - gapo, o3x3x3x3x3o3o- gibpo

External links

v t e Fundamental convex regular and uniform polytopes in dimensions 2–10
Family	$A n$	$B n$	$I 2 (p)$ / $D n$	$E 6$ / $E 7$ / $E 8$ / $F 4$ / $G 2$	$H n$
Regular polygon	Triangle	Square	p-gon	Hexagon	Pentagon
Uniform polyhedron	Tetrahedron	Octahedron • Cube	Demicube		Dodecahedron • Icosahedron
Uniform polychoron	Pentachoron	16-cell • Tesseract	Demitesseract	24-cell	120-cell • 600-cell
Uniform 5-polytope	5-simplex	5-orthoplex • 5-cube	5-demicube
Uniform 6-polytope	6-simplex	6-orthoplex • 6-cube	6-demicube	1₂₂ • 2₂₁
Uniform 7-polytope	7-simplex	7-orthoplex • 7-cube	7-demicube	1₃₂ • 2₃₁ • 3₂₁
Uniform 8-polytope	8-simplex	8-orthoplex • 8-cube	8-demicube	1₄₂ • 2₄₁ • 4₂₁
Uniform 9-polytope	9-simplex	9-orthoplex • 9-cube	9-demicube
Uniform 10-polytope	10-simplex	10-orthoplex • 10-cube	10-demicube
Uniform n-polytope	n-simplex	n-orthoplex • n-cube	n-demicube	1_k2 • 2_k1 • k₂₁	n-pentagonal polytope
Topics: Polytope families • Regular polytope • List of regular polytopes and compounds • Polytope operations

[1] Klitzing, (x3o3o3x3o3o3o - spo)

[2] Klitzing, (o3x3o3o3x3o3o - sibpo)

[3] Klitzing, (x3x3o3x3o3o3o - patto)

[4] Klitzing, (o3x3x3o3x3o3o - bipto)

[5] Klitzing, (x3o3x3x3o3o3o - paro)

[6] Klitzing, (x3x3x3x3o3o3o - gapo)

[7] Klitzing, (o3x3x3x3x3o3o- gibpo)

[1]

[2]

[3]

[4]

[5]

[6]

[7]

Runcinated 7-simplexes

Contents