Runcinated 5-simplexes
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 | 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.png | |
| 4-faces | 47 | 6 t0,3{3,3,3} File:4-simplex t03.svg 20 {3}×{3} 15 { }×r{3,3} 6 r{3,3,3} File:4-simplex t1.svg |
| Cells | 255 | 45 {3,3} File:3-simplex t0.svg 180 { }×{3} 30 r{3,3} File:3-simplex t1.svg |
| Faces | 420 | 240 {3} File:2-simplex t0.svg 180 {4} |
| Edges | 270 | |
| Vertices | 60 | |
| Vertex figure | File:Runcinated 5-simplex verf.png | |
| 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 | File:5-simplex t03.svg | File:5-simplex t03 A4.svg |
| Dihedral symmetry | [6] | [5] |
| Ak Coxeter plane |
A3 | A2 |
| Graph | File:5-simplex t03 A3.svg | File:5-simplex t03 A2.svg |
| 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 | 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.png | |
| 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 | File:Runcitruncated 5-simplex verf.png | |
| 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 | File:5-simplex t013.svg | File:5-simplex t013 A4.svg |
| Dihedral symmetry | [6] | [5] |
| Ak Coxeter plane |
A3 | A2 |
| Graph | File:5-simplex t013 A3.svg | File:5-simplex t013 A2.svg |
| 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 | 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.png | |
| 4-faces | 47 | |
| Cells | 255 | |
| Faces | 570 | |
| Edges | 540 | |
| Vertices | 180 | |
| Vertex figure | File:Runcicantellated 5-simplex verf.png | |
| 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 | File:5-simplex t023.svg | File:5-simplex t023 A4.svg |
| Dihedral symmetry | [6] | [5] |
| Ak Coxeter plane |
A3 | A2 |
| Graph | File:5-simplex t023 A3.svg | File:5-simplex t023 A2.svg |
| 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 | 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.png | |
| 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 | File:Runcicantitruncated 5-simplex verf.png 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 | File:5-simplex t0123.svg | File:5-simplex t0123 A4.svg |
| Dihedral symmetry | [6] | [5] |
| Ak Coxeter plane |
A3 | A2 |
| Graph | File:5-simplex t0123 A3.svg | File:5-simplex t0123 A2.svg |
| 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)
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