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{| class="wikitable" align="right" style="margin-left:10px" width="320"
!bgcolor=#e7dcc3 colspan=2|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||[[Simplectic honeycomb]]
|-
|bgcolor=#e7dcc3|[[Schläfli symbol]]||{3<sup>[5]</sup>}
|-
|bgcolor=#e7dcc3|[[Coxeter diagram]]||{{CDD|node_1|split1|nodes|3ab|branch}}
|-
|bgcolor=#e7dcc3|4-face types||[[5-cell|{3,3,3}]][[File:Schlegel wireframe 5-cell.png|40px]]<BR>[[Rectified 5-cell|t<sub>1</sub>{3,3,3}]] [[File:Schlegel half-solid rectified 5-cell.png|40px]]
|-
|bgcolor=#e7dcc3|Cell types||[[tetrahedron|{3,3}]] [[File:Uniform polyhedron-33-t0.png|20px]]<BR>[[Octahedron|t<sub>1</sub>{3,3}]] [[File:Uniform polyhedron-33-t1.png|20px]]
|-
|bgcolor=#e7dcc3|Face types||[[triangle|{3}]]
|-
|bgcolor=#e7dcc3|Vertex figure||[[File:4-simplex_honeycomb_verf.png|80px]]<BR>[[Runcinated 5-cell|t<sub>0,3</sub>{3,3,3}]]
|-
|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 '''4-simplex honeycomb''', '''5-cell honeycomb''' or '''pentachoric-dispentachoric honeycomb''' is a space-filling [[tessellation]] [[honeycomb (geometry)|honeycomb]]. It is composed of [[5-cell]]s and [[rectified 5-cell]]s facets in a ratio of 1:1.
 
Cells of the [[vertex figure]] are ten [[tetrahedron]]s and 20 [[triangular prism]]s, corresponding to the ten [[5-cell]]s and 20 [[rectified 5-cell]]s that meet at each vertex. All the vertices lie in parallel realms in which they form [[alternated cubic honeycomb]]s, the tetrahedra being either tops of the rectified 5-cell or the bases of the 5-cell, and the octahedra being the bottoms of the rectified 5-cell.<ref name=tetra>Olshevsky (2006), Model 134</ref>
 
== A4 lattice ==
This [[vertex arrangement]] is called the [[A4 lattice]], or '''4-simplex lattice'''. The 20 vertices of its [[vertex figure]], the [[runcinated 5-cell]] represent the 20 roots of the <math>{\tilde{A}}_4</math> Coxeter group.<ref>http://www.math.rwth-aachen.de/~Gabriele.Nebe/LATTICES/A4.html</ref> It is the 4-dimensional case of a [[simplectic honeycomb]].
 
The A{{sup sub|*|4}} lattice is the union of five A<sub>4</sub> lattices, and is the dual to the [[omnitruncated 5-simplex honeycomb]], and therefore the [[Voronoi cell]] of this lattice is an [[omnitruncated 5-cell]]
: {{CDD|node_1|split1|nodes|3ab|branch}} + {{CDD|node|split1|nodes_10lr|3ab|branch}} + {{CDD|node|split1|nodes_01lr|3ab|branch}} + {{CDD|node|split1|nodes|3ab|branch_10l}} + {{CDD|node|split1|nodes|3ab|branch_01l}} = dual of {{CDD|node_1|split1|nodes_11|3ab|branch_11}}
 
== Alternate names==
* Cyclopentachoric tetracomb
* Pentachoric-dispentachoric tetracomb
 
== Related polytopes and honeycombs ==
 
The ''tops'' of the 5-cells in this honeycomb adjoin the ''bases'' of the 5-cells, and vice versa, in adjacent [[laminae]]; but alternating laminae may be inverted so that the tops of the rectified 5-cells adjoin the tops of the rectified 5-cells and the bases of the 5-cells adjoin the bases of other 5-cells. This inversion results in another non-Wythoffian uniform convex honeycomb. [[Octahedral prism]]s and [[tetrahedral prism]]s may be inserted in between alternated laminae as well, resulting in two more non-Wythoffian elongated uniform honeycombs.<ref>Olshevsky (2006), Klitzing, elong( x3o3o3o3o3*a ) - ecypit  - O141, schmo( x3o3o3o3o3*a ) - zucypit - O142, elongschmo( x3o3o3o3o3*a ) - ezucypit - O143</ref>
 
{{4-simplex honeycomb family}}
 
== Projection by folding ==
 
The ''5-cell honeycomb'' can be projected into the 2-dimensional [[square tiling]] by a [[Coxeter–Dynkin diagram#Geometric folding|geometric folding]] operation that maps two pairs of mirrors into each other, sharing the same [[vertex arrangement]]:
 
{|class=wikitable
|-
!<math>{\tilde{A}}_3</math>
|{{CDD|node_1|split1|nodes|3ab|branch}}
|-
!<math>{\tilde{C}}_2</math>
|{{CDD|node_1|4|node|4|node}}
|}
 
==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]]
* [[Truncated 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 134
* {{KlitzingPolytopes|flat.htm|4D|Euclidean tesselations}}, x3o3o3o3o3*a - cypit - O134
* [http://arxiv-web3.library.cornell.edu/abs/1209.1878 Affine Coxeter group Wa(A4), Quaternions, and Decagonal Quasicrystals] Mehmet Koca, Nazife O. Koca, Ramazan Koc (2013) [http://arxiv.org/ftp/arxiv/papers/1209/1209.1878.pdf]
 
{{Honeycombs}}
 
[[Category:Honeycombs (geometry)]]
[[Category:5-polytopes]]

Revision as of 01:03, 26 July 2013

4-simplex honeycomb
(No image)
Type Uniform 4-honeycomb
Family Simplectic honeycomb
Schläfli symbol {3[5]}
Coxeter diagram Template:CDD
4-face types {3,3,3}
t1{3,3,3}
Cell types {3,3}
t1{3,3}
Face types {3}
Vertex figure
t0,3{3,3,3}
Symmetry ×2, [[3[5]]]
Properties vertex-transitive

In four-dimensional Euclidean geometry, the 4-simplex honeycomb, 5-cell honeycomb or pentachoric-dispentachoric honeycomb is a space-filling tessellation honeycomb. It is composed of 5-cells and rectified 5-cells facets in a ratio of 1:1.

Cells of the vertex figure are ten tetrahedrons and 20 triangular prisms, corresponding to the ten 5-cells and 20 rectified 5-cells that meet at each vertex. All the vertices lie in parallel realms in which they form alternated cubic honeycombs, the tetrahedra being either tops of the rectified 5-cell or the bases of the 5-cell, and the octahedra being the bottoms of the rectified 5-cell.[1]

A4 lattice

This vertex arrangement is called the A4 lattice, or 4-simplex lattice. The 20 vertices of its vertex figure, the runcinated 5-cell represent the 20 roots of the Coxeter group.[2] It is the 4-dimensional case of a simplectic honeycomb.

The ATemplate:Sup sub lattice is the union of five A4 lattices, and is the dual to the omnitruncated 5-simplex honeycomb, and therefore the Voronoi cell of this lattice is an omnitruncated 5-cell

Template:CDD + Template:CDD + Template:CDD + Template:CDD + Template:CDD = dual of Template:CDD

Alternate names

  • Cyclopentachoric tetracomb
  • Pentachoric-dispentachoric tetracomb

Related polytopes and honeycombs

The tops of the 5-cells in this honeycomb adjoin the bases of the 5-cells, and vice versa, in adjacent laminae; but alternating laminae may be inverted so that the tops of the rectified 5-cells adjoin the tops of the rectified 5-cells and the bases of the 5-cells adjoin the bases of other 5-cells. This inversion results in another non-Wythoffian uniform convex honeycomb. Octahedral prisms and tetrahedral prisms may be inserted in between alternated laminae as well, resulting in two more non-Wythoffian elongated uniform honeycombs.[3]

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Projection by folding

The 5-cell honeycomb can be projected into the 2-dimensional square tiling by a geometric folding operation that maps two pairs of mirrors into each other, sharing the same vertex arrangement:

Template:CDD
Template:CDD

See also

Regular and uniform honeycombs in 4-space:

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 134
  • Template:KlitzingPolytopes, x3o3o3o3o3*a - cypit - O134
  • Affine Coxeter group Wa(A4), Quaternions, and Decagonal Quasicrystals Mehmet Koca, Nazife O. Koca, Ramazan Koc (2013) [2]

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  1. ↑ Olshevsky (2006), Model 134
  2. ↑ http://www.math.rwth-aachen.de/~Gabriele.Nebe/LATTICES/A4.html
  3. ↑ Olshevsky (2006), Klitzing, elong( x3o3o3o3o3*a ) - ecypit - O141, schmo( x3o3o3o3o3*a ) - zucypit - O142, elongschmo( x3o3o3o3o3*a ) - ezucypit - O143
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