| 释义 |
- Truncated 5-simplex Alternate names Coordinates Images
- Bitruncated 5-simplex Alternate names Coordinates Images
- Related uniform 5-polytopes
- Notes
- References
- External links
5-simplex
{{CDD>node_1|3|node|3|node|3|node|3|node}} | Truncated 5-simplex
{{CDD>node_1|3|node_1|3|node|3|node|3|node}} | Bitruncated 5-simplex
{{CDD>node|3|node_1|3|node_1|3|node|3|node}} | | Orthogonal projections in A5 Coxeter plane |
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In five-dimensional geometry, a truncated 5-simplex is a convex uniform 5-polytope, being a truncation of the regular 5-simplex. There are unique 2 degrees of truncation. Vertices of the truncation 5-simplex are located as pairs on the edge of the 5-simplex. Vertices of the bitruncation 5-simplex are located on the triangular faces of the 5-simplex. Truncated 5-simplex | Truncated 5-simplex | | Type | Uniform 5-polytope | | Schläfli symbol | t{3,3,3,3} | | Coxeter-Dynkin diagram | {{CDD | 3|node_1|3|node | node|3|node}}
{{CDD | 3b|nodeb|3b|nodeb|3b|nodeb} | | 4-faces | 12 | 6 {3,3,3}
6 t{3,3,3} | | Cells | 45 | 30 {3,3}
15 t{3,3} | | Faces | 80 | 60 {3}
20 {6} | | Edges | 75 | | Vertices | 30 | | Vertex figure |
( )v{3,3} | | Coxeter group | A5 [3,3,3,3], order 720 | | Properties | convex |
The truncated 5-simplex has 30 vertices, 75 edges, 80 triangular faces, 45 cells (15 tetrahedral, and 30 truncated tetrahedron), and 12 4-faces (6 5-cell and 6 truncated 5-cells). Alternate names - Truncated hexateron (Acronym: tix) (Jonathan Bowers)[1]
Coordinates The vertices of the truncated 5-simplex can be most simply constructed on a hyperplane in 6-space as permutations of (0,0,0,0,1,2) or of (0,1,2,2,2,2). These coordinates come from facets of the truncated 6-orthoplex and bitruncated 6-cube respectively. Images {{5-simplex Coxeter plane graphs|t01|150}} Bitruncated 5-simplex | bitruncated 5-simplex | | Type | Uniform 5-polytope | | Schläfli symbol | 2t{3,3,3,3} | | Coxeter-Dynkin diagram | {{CDD | 3|node_1|3|node_1|3|node|3|node}}
{{CDD | 3ab|nodes|3b|nodeb} | | 4-faces | 12 | 6 2t{3,3,3}
6 t{3,3,3} | | Cells | 60 | 45 {3,3}
15 t{3,3} | | Faces | 140 | 80 {3}
60 {6} | | Edges | 150 | | Vertices | 60 | | Vertex figure |
{ }v{3} | | Coxeter group | A5 [3,3,3,3], order 720 | | Properties | convex |
Alternate names - Bitruncated hexateron (Acronym: bittix) (Jonathan Bowers)[2]
Coordinates The vertices of the bitruncated 5-simplex can be most simply constructed on a hyperplane in 6-space as permutations of (0,0,0,1,2,2) or of (0,0,1,2,2,2). These represent positive orthant facets of the bitruncated 6-orthoplex, and the tritruncated 6-cube respectively. Images {{5-simplex Coxeter plane graphs|t12|150}} Related uniform 5-polytopes The truncated 5-simplex is one 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) {{Hexateron family}}Notes1. ^Klitizing, (x3x3o3o3o - tix)
2. ^Klitizing, (o3x3x3o3o - bittix)
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}}
- (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.
- {{KlitzingPolytopes|polytera.htm|5D|uniform polytopes (polytera)}} x3x3o3o3o - tix, o3x3x3o3o - bittix
External links - {{PolyCell | urlname = glossary.html#simplex| title = Glossary for hyperspace}}
- Polytopes of Various Dimensions, Jonathan Bowers
- Truncated uniform polytera (tix), Jonathan Bowers
- Multi-dimensional Glossary
{{Polytopes}} 1 : 5-polytopes |