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!bgcolor=#e7dcc3 colspan=2|Truncated 4-simplex honeycomb | |||
|- | |||
|bgcolor=#ffffff align=center colspan=2|(No image) | |||
|- | |||
|bgcolor=#e7dcc3|Type||[[Uniform_polyteron#Regular_and_uniform_honeycombs|Uniform 4-honeycomb]] | |||
|- | |||
|bgcolor=#e7dcc3|Family||[[Truncated simplectic honeycomb]] | |||
|- | |||
|bgcolor=#e7dcc3|[[Schläfli symbol]]||t<sub>0,1</sub>{3<sup>[5]</sup>} | |||
|- | |||
|bgcolor=#e7dcc3|[[Coxeter diagram]]||{{CDD|branch_11|3ab|nodes|split2|node}} | |||
|- | |||
|bgcolor=#e7dcc3|4-face types||[[5-cell|{3,3,3}]] [[File:Schlegel wireframe 5-cell.png|40px]]<BR>[[Truncated 5-cell|t{3,3,3}]] [[File:Schlegel half-solid truncated pentachoron.png|40px]]<BR>[[Bitruncated 5-cell|2t{3,3,3}]] [[File:Schlegel half-solid bitruncated 5-cell.png|40px]] | |||
|- | |||
|bgcolor=#e7dcc3|Cell types||[[tetrahedron|{3,3}]] [[File:Tetrahedron.png|20px]]<BR>[[Truncated tetrahedron|t{3,3}]] [[File:Truncated tetrahedron.png|20px]] | |||
|- | |||
|bgcolor=#e7dcc3|Face types||[[Triangle]] {3}<BR>[[Hexagon]] {6} | |||
|- | |||
|bgcolor=#e7dcc3|Vertex figure||[[File:Truncated_5-cell_honeycomb_verf.png|100px]]<BR>Elongated [[tetrahedral antiprism]]<BR>[3,4,2<sup>+</sup>], order 48 | |||
|- | |||
|bgcolor=#e7dcc3|[[Coxeter notation|Symmetry]]||<math>{\tilde{A}}_4</math>×2, [<span/>[3<sup>[5]</sup>]] | |||
|- | |||
|bgcolor=#e7dcc3|Properties||[[vertex-transitive]] | |||
|} | |||
In [[Four-dimensional space|four-dimensional]] [[Euclidean geometry]], the '''truncated 4-simplex honeycomb''', '''truncated 5-cell honeycomb''' is a space-filling [[tessellation]] [[honeycomb (geometry)|honeycomb]]. It is composed of [[5-cell]]s, [[truncated 5-cell]]s, and [[bitruncated 5-cell]]s facets in a ratio of 2:2:1. Its [[vertex figure]] is an Elongated [[tetrahedral antiprism]], with 8 [[equilateral triangle]] and 24 [[isosceles triangle]] faces, defining 8 [[5-cell]] and 24 [[truncated 5-cell]] facets around a vertex. | |||
It can be constructed as five sets of parallel [[hyperplane]]s that divide space into two half-spaces. The 3-space hyperplanes contain [[quarter cubic honeycomb]]s as a collection facets.<ref name=tetra>Olshevsky, 2006 (Model 135)</ref> | |||
== Alternate names== | |||
* Cyclotruncated pentachoric tetracomb | |||
* Small truncated-pentachoric tetracomb | |||
== Related polytopes and honeycombs == | |||
{{A4 honeycombs}} | |||
==See also== | |||
Regular and uniform honeycombs in 4-space: | |||
*[[Tesseractic honeycomb]] | |||
*[[16-cell honeycomb]] | |||
*[[24-cell honeycomb]] | |||
*[[Truncated 24-cell honeycomb]] | |||
*[[Snub 24-cell honeycomb]] | |||
*[[5-cell honeycomb]] | |||
*[[Omnitruncated 5-cell honeycomb]] | |||
==Notes== | |||
{{reflist}} | |||
== References == | |||
* [[Norman Johnson (mathematician)|Norman Johnson]] ''Uniform Polytopes'', Manuscript (1991) | |||
* '''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 [http://www.wiley.com/WileyCDA/WileyTitle/productCd-0471010030.html] | |||
** (Paper 22) H.S.M. Coxeter, ''Regular and Semi Regular Polytopes I'', [Math. Zeit. 46 (1940) 380-407, MR 2,10] (1.9 Uniform space-fillings) | |||
** (Paper 24) H.S.M. Coxeter, ''Regular and Semi-Regular Polytopes III'', [Math. Zeit. 200 (1988) 3-45] | |||
* [[George Olshevsky]], ''Uniform Panoploid Tetracombs'', Manuscript (2006) ''(Complete list of 11 convex uniform tilings, 28 convex uniform honeycombs, and 143 convex uniform tetracombs)'' Model 135 | |||
* {{KlitzingPolytopes|flat.htm|4D|Euclidean tesselations}}, x3x3x3x3x3*a - otcypit - 135 | |||
{{Honeycombs}} | |||
[[Category:Honeycombs (geometry)]] | |||
[[Category:5-polytopes]] | |||
Latest revision as of 22:14, 10 December 2012
| Truncated 4-simplex honeycomb | |
|---|---|
| (No image) | |
| Type | Uniform 4-honeycomb |
| Family | Truncated simplectic honeycomb |
| Schläfli symbol | t0,1{3[5]} |
| Coxeter diagram | Template:CDD |
| 4-face types | {3,3,3}  t{3,3,3}  2t{3,3,3} 
|
| Cell types | {3,3}  t{3,3} 
|
| Face types | Triangle {3} Hexagon {6} |
| Vertex figure |  Elongated tetrahedral antiprism [3,4,2+], order 48 |
| Symmetry | ×2, [[3[5]]] |
| Properties | vertex-transitive |
In four-dimensional Euclidean geometry, the truncated 4-simplex honeycomb, truncated 5-cell honeycomb is a space-filling tessellation honeycomb. It is composed of 5-cells, truncated 5-cells, and bitruncated 5-cells facets in a ratio of 2:2:1. Its vertex figure is an Elongated tetrahedral antiprism, with 8 equilateral triangle and 24 isosceles triangle faces, defining 8 5-cell and 24 truncated 5-cell facets around a vertex.
It can be constructed as five sets of parallel hyperplanes that divide space into two half-spaces. The 3-space hyperplanes contain quarter cubic honeycombs as a collection facets.[1]
Alternate names
- Cyclotruncated pentachoric tetracomb
- Small truncated-pentachoric tetracomb
Related polytopes and honeycombs
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See also
Regular and uniform honeycombs in 4-space:
- Tesseractic honeycomb
- 16-cell honeycomb
- 24-cell honeycomb
- Truncated 24-cell honeycomb
- Snub 24-cell honeycomb
- 5-cell honeycomb
- Omnitruncated 5-cell honeycomb
Notes
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References
- Norman Johnson Uniform Polytopes, Manuscript (1991)
- 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] (1.9 Uniform space-fillings)
- (Paper 24) H.S.M. Coxeter, Regular and Semi-Regular Polytopes III, [Math. Zeit. 200 (1988) 3-45]
- George Olshevsky, Uniform Panoploid Tetracombs, Manuscript (2006) (Complete list of 11 convex uniform tilings, 28 convex uniform honeycombs, and 143 convex uniform tetracombs) Model 135
- Template:KlitzingPolytopes, x3x3x3x3x3*a - otcypit - 135
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- ↑ Olshevsky, 2006 (Model 135)