Runcinated 5-simplexes
Contributors to Wikimedia projects
Article Images5-simplex |
Runcinated 5-simplex |
Runcitruncated 5-simplex |
Birectified 5-simplex |
Runcicantellated 5-simplex |
Runcicantitruncated 5-simplex |
Orthogonal projections in A5 Coxeter plane |
---|
In six-dimensional geometry, a runcinated 5-simplex is a convex uniform 5-polytope with 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-simplex
Runcinated 5-simplex | ||
Type | Uniform 5-polytope | |
Schläfli symbol | t0,3{3,3,3,3} | |
Coxeter-Dynkin diagram | ||
4-faces | 47 | 6 t0,3{3,3,3} 20 {3}×{3} 15 { }×r{3,3} 6 r{3,3,3} |
Cells | 255 | 45 {3,3} 180 { }×{3} 30 r{3,3} |
Faces | 420 | 240 {3} 180 {4} |
Edges | 270 | |
Vertices | 60 | |
Vertex figure | ||
Coxeter group | A5 [3,3,3,3], order 720 | |
Properties | convex |
- Runcinated hexateron
- Small prismated hexateron (Acronym: spix) (Jonathan Bowers)[1]
The vertices of the runcinated 5-simplex can be most simply constructed on a hyperplane in 6-space as permutations of (0,0,1,1,1,2) or of (0,1,1,1,2,2), seen as facets of a runcinated 6-orthoplex, or a biruncinated 6-cube respectively.
Ak Coxeter plane |
A5 | A4 |
---|---|---|
Graph | ||
Dihedral symmetry | [6] | [5] |
Ak Coxeter plane |
A3 | A2 |
Graph | ||
Dihedral symmetry | [4] | [3] |
Runcitruncated 5-simplex
Runcitruncated 5-simplex | ||
Type | Uniform 5-polytope | |
Schläfli symbol | t0,1,3{3,3,3,3} | |
Coxeter-Dynkin diagram | ||
4-faces | 47 | 6 t0,1,3{3,3,3} 20 {3}×{6} 15 { }×r{3,3} 6 rr{3,3,3} |
Cells | 315 | |
Faces | 720 | |
Edges | 630 | |
Vertices | 180 | |
Vertex figure | ||
Coxeter group | A5 [3,3,3,3], order 720 | |
Properties | convex, isogonal |
- Runcitruncated hexateron
- Prismatotruncated hexateron (Acronym: pattix) (Jonathan Bowers)[2]
The coordinates can be made in 6-space, as 180 permutations of:
- (0,0,1,1,2,3)
This construction exists as one of 64 orthant facets of the runcitruncated 6-orthoplex.
Ak Coxeter plane |
A5 | A4 |
---|---|---|
Graph | ||
Dihedral symmetry | [6] | [5] |
Ak Coxeter plane |
A3 | A2 |
Graph | ||
Dihedral symmetry | [4] | [3] |
Runcicantellated 5-simplex
Runcicantellated 5-simplex | ||
Type | Uniform 5-polytope | |
Schläfli symbol | t0,2,3{3,3,3,3} | |
Coxeter-Dynkin diagram | ||
4-faces | 47 | |
Cells | 255 | |
Faces | 570 | |
Edges | 540 | |
Vertices | 180 | |
Vertex figure | ||
Coxeter group | A5 [3,3,3,3], order 720 | |
Properties | convex, isogonal |
- Runcicantellated hexateron
- Biruncitruncated 5-simplex/hexateron
- Prismatorhombated hexateron (Acronym: pirx) (Jonathan Bowers)[3]
The coordinates can be made in 6-space, as 180 permutations of:
- (0,0,1,2,2,3)
This construction exists as one of 64 orthant facets of the runcicantellated 6-orthoplex.
Ak Coxeter plane |
A5 | A4 |
---|---|---|
Graph | ||
Dihedral symmetry | [6] | [5] |
Ak Coxeter plane |
A3 | A2 |
Graph | ||
Dihedral symmetry | [4] | [3] |
Runcicantitruncated 5-simplex
Runcicantitruncated 5-simplex | ||
Type | Uniform 5-polytope | |
Schläfli symbol | t0,1,2,3{3,3,3,3} | |
Coxeter-Dynkin diagram | ||
4-faces | 47 | 6 t0,1,2,3{3,3,3} 20 {3}×{6} 15 {}×t{3,3} 6 tr{3,3,3} |
Cells | 315 | 45 t0,1,2{3,3} 120 { }×{3} 120 { }×{6} 30 t{3,3} |
Faces | 810 | 120 {3} 450 {4} 240 {6} |
Edges | 900 | |
Vertices | 360 | |
Vertex figure | Irregular 5-cell | |
Coxeter group | A5 [3,3,3,3], order 720 | |
Properties | convex, isogonal |
- Runcicantitruncated hexateron
- Great prismated hexateron (Acronym: gippix) (Jonathan Bowers)[4]
The coordinates can be made in 6-space, as 360 permutations of:
- (0,0,1,2,3,4)
This construction exists as one of 64 orthant facets of the runcicantitruncated 6-orthoplex.
Ak Coxeter plane |
A5 | A4 |
---|---|---|
Graph | ||
Dihedral symmetry | [6] | [5] |
Ak Coxeter plane |
A3 | A2 |
Graph | ||
Dihedral symmetry | [4] | [3] |
These polytopes are in a set of 19 uniform 5-polytopes based on the [3,3,3,3] Coxeter group, all shown here in A5 Coxeter plane orthographic projections. (Vertices are colored by projection overlap order, red, orange, yellow, green, cyan, blue, purple having progressively more vertices)
A5 polytopes | |||||||||||
---|---|---|---|---|---|---|---|---|---|---|---|
t0 |
t1 |
t2 |
t0,1 |
t0,2 |
t1,2 |
t0,3 | |||||
t1,3 |
t0,4 |
t0,1,2 |
t0,1,3 |
t0,2,3 |
t1,2,3 |
t0,1,4 | |||||
t0,2,4 |
t0,1,2,3 |
t0,1,2,4 |
t0,1,3,4 |
t0,1,2,3,4 |
- 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, ISBN 978-0-471-01003-6 [1]
- (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.
- Klitzing, Richard. "5D uniform polytopes (polytera)". x3o3o3x3o - spidtix, x3x3o3x3o - pattix, x3o3x3x3o - pirx, x3x3x3x3o - gippix
- Glossary for hyperspace, George Olshevsky.
- Polytopes of Various Dimensions, Jonathan Bowers
- Runcinated uniform polytera (spid), Jonathan Bowers
- Multi-dimensional Glossary
Fundamental convex regular and uniform polytopes in dimensions 2–10 | ||||||||||||
---|---|---|---|---|---|---|---|---|---|---|---|---|
Family | An | Bn | I2(p) / Dn | E6 / E7 / E8 / F4 / G2 | Hn | |||||||
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 | 1k2 • 2k1 • k21 | n-pentagonal polytope | |||||||
Topics: Polytope families • Regular polytope • List of regular polytopes and compounds |