 5-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 | |
Alternate names
- Runcinated hexateron
- Small prismated hexateron (Acronym: spix) (Jonathan Bowers)[1]
Coordinates
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.
Images
| 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 | |
Alternate names
- Runcitruncated hexateron
- Prismatotruncated hexateron (Acronym: pattix) (Jonathan Bowers)[2]
Coordinates
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.
Images
| 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 | |
Alternate names
- Runcicantellated hexateron
- Biruncitruncated 5-simplex/hexateron
- Prismatorhombated hexateron (Acronym: pirx) (Jonathan Bowers)[3]
Coordinates
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.
Images
| 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 | |
Alternate names
- Runcicantitruncated hexateron
- Great prismated hexateron (Acronym: gippix) (Jonathan Bowers)[4]
Coordinates
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.
Images
| Ak Coxeter plane | A5 | A4 |
|---|---|---|
| Graph | |  |
| Dihedral symmetry | [6] | [5] |
| Ak Coxeter plane | A3 | A2 |
| Graph |  |  |
| Dihedral symmetry | [4] | [3] |
Related uniform 5-polytopes
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 | |||||||
Notes
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, 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
External links
- Glossary for hyperspace, George Olshevsky.
- Polytopes of Various Dimensions, Jonathan Bowers
- Runcinated uniform polytera (spid), Jonathan Bowers
- Multi-dimensional Glossary