View source for 5-cell honeycomb

{{short description|Geometric figure}}
{| class="wikitable" align="right" style="margin-left:10px"
!bgcolor=#e7dcc3 colspan=2|4-simplex honeycomb
|-
|bgcolor=#ffffff align=center colspan=2|(No image)
|-
|bgcolor=#e7dcc3|Type||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|Symmetry||<math>{\tilde{A}}_4</math>×2, {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|tessellation]] [[Honeycomb (geometry)|honeycomb]]. It is composed of [[5-cell|5-cell]]s and [[Rectified 5-cell|rectified 5-cell]]s facets in a ratio of 1:1.
== Structure==
Cells of the [[Vertex figure|vertex figure]] are ten [[Tetrahedron|tetrahedron]]s and 20 [[Triangular prism|triangular prism]]s, corresponding to the ten [[5-cell|5-cell]]s and 20 [[Rectified 5-cell|rectified 5-cell]]s 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.<ref>Olshevsky (2006), Model 134</ref>

== Alternate names==
* Cyclopentachoric tetracomb
* Pentachoric-dispentachoric tetracomb

== Projection by folding ==

The ''5-cell honeycomb'' can be projected into the 2-dimensional [[Square tiling|square tiling]] by a geometric folding operation that maps two pairs of mirrors into each other, sharing the same [[Vertex arrangement|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}}
|}

== A4 lattice ==
The [[Vertex arrangement|vertex arrangement]] of the ''5-cell honeycomb'' is called the '''A4 lattice''', or '''4-simplex lattice'''. The 20 vertices of its [[Vertex figure|vertex figure]], the [[Runcinated 5-cell|runcinated 5-cell]] represent the 20 roots of the <math>{\tilde{A}}_4</math> Coxeter group.<ref>{{Cite web|url=http://www.math.rwth-aachen.de/~Gabriele.Nebe/LATTICES/A4.html|title = The Lattice A4}}</ref><ref>{{Cite web|url=https://m.wolframalpha.com/input/?i=A4+root+lattice&lk=3|title = A4 root lattice - Wolfram&#124;Alpha}}</ref>  It is the 4-dimensional case of a [[Simplectic honeycomb|simplectic honeycomb]].

The A{{sup sub|*|4}} lattice<ref>{{Cite web|url=http://www.math.rwth-aachen.de/~Gabriele.Nebe/LATTICES/As4.html|title=The Lattice A4}}</ref> is the union of five A<sub>4</sub> lattices, and is the dual to the [[Omnitruncated 5-simplex honeycomb|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_10lur|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}}

== 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 [[Earth:Lamination (geology)|laminae]] (or layers); 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|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>


=== Rectified 5-cell honeycomb ===
{| class="wikitable" align="right" style="margin-left:10px" width="320"
!bgcolor=#e7dcc3 colspan=2|Rectified 5-cell honeycomb
|-
|bgcolor=#ffffff align=center colspan=2|(No image)
|-
|bgcolor=#e7dcc3|Type||Uniform 4-honeycomb
|-
|bgcolor=#e7dcc3|[[Schläfli symbol]]||t<sub>0,2</sub>{3<sup>[5]</sup>} or r{3<sup>[5]</sup>}
|-
|bgcolor=#e7dcc3|Coxeter diagram||{{CDD|node|split1|nodes_11|3ab|branch}}
|-
|bgcolor=#e7dcc3|4-face types||[[Rectified 5-cell|t<sub>1</sub>{3<sup>3</sup>}]] [[File:Schlegel half-solid rectified 5-cell.png|40px]]<BR>[[Cantellated 5-cell|t<sub>0,2</sub>{3<sup>3</sup>}]] 40px<BR>[[Runcinated 5-cell|t<sub>0,3</sub>{3<sup>3</sup>}]] [[File:Schlegel half-solid runcinated 5-cell.png|40px]]
|-
|bgcolor=#e7dcc3|Cell types||[[Tetrahedron]] [[File:Tetrahedron.png|20px]]<BR>[[Octahedron|Octahedron]] 20px<BR>[[Cuboctahedron|Cuboctahedron]] 20px<BR>[[Triangular prism|Triangular prism]] [[File:Triangular prism.png|20px]]
|-
|bgcolor=#e7dcc3|Vertex figure||triangular elongated-antiprismatic prism
|-
|bgcolor=#e7dcc3|Symmetry||<math>{\tilde{A}}_4</math>×2, {3<sup>[5]</sup>}
|-
|bgcolor=#e7dcc3|Properties||vertex-transitive
|}
The '''rectified 4-simplex honeycomb''' or '''rectified 5-cell honeycomb''' is a space-filling [[Tessellation|tessellation]] [[Honeycomb (geometry)|honeycomb]].

==== Alternate names====
* small cyclorhombated pentachoric tetracomb
* small prismatodispentachoric tetracomb

{{-}}

=== Cyclotruncated 5-cell honeycomb===
{| class="wikitable" align="right" style="margin-left:10px" width="320"
!bgcolor=#e7dcc3 colspan=2|Cyclotruncated 5-cell honeycomb
|-
|bgcolor=#ffffff align=center colspan=2|(No image)
|-
|bgcolor=#e7dcc3|Type||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}]] 40px<BR>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>Tetrahedral antiprism<BR>[3,4,2<sup>+</sup>], order 48
|-
|bgcolor=#e7dcc3|Symmetry||<math>{\tilde{A}}_4</math>×2, {3<sup>[5]</sup>}
|-
|bgcolor=#e7dcc3|Properties||vertex-transitive
|}
The '''cyclotruncated 4-simplex honeycomb''' or '''cyclotruncated 5-cell honeycomb''' is a space-filling [[Tessellation|tessellation]] [[Honeycomb (geometry)|honeycomb]]. It can also be seen as a '''birectified 5-cell honeycomb'''.

It is composed of [[5-cell|5-cell]]s, [[Truncated 5-cell|truncated 5-cell]]s,  and bitruncated 5-cells facets in a ratio of 2:2:1. Its [[Vertex figure|vertex figure]] is a tetrahedral antiprism, with 2 regular tetrahedron, 8 triangular pyramid, and 6 tetragonal disphenoid cells, defining 2 [[5-cell|5-cell]], 8 [[Truncated 5-cell|truncated 5-cell]], and 6 bitruncated 5-cell facets around a vertex.

It can be constructed as five sets of parallel [[Hyperplane|hyperplane]]s that divide space into two half-spaces. The 3-space hyperplanes contain [[Quarter cubic honeycomb|quarter cubic honeycomb]]s as a collection facets.<ref>Olshevsky, (2006) Model 135</ref>

==== Alternate names====
* Cyclotruncated pentachoric tetracomb
* Small truncated-pentachoric tetracomb
{{-}}

=== Truncated 5-cell honeycomb===
{| class="wikitable" align="right" style="margin-left:10px" width="360"
!bgcolor=#e7dcc3 colspan=2|Truncated 4-simplex honeycomb
|-
|bgcolor=#ffffff align=center colspan=2|(No image)
|-
|bgcolor=#e7dcc3|Type||Uniform 4-honeycomb
|-
|bgcolor=#e7dcc3|[[Schläfli symbol]]||t<sub>0,1,2</sub>{3<sup>[5]</sup>} or t{3<sup>[5]</sup>}
|-
|bgcolor=#e7dcc3|Coxeter diagram||{{CDD|node_1|split1|nodes_11|3ab|branch}}
|-
|bgcolor=#e7dcc3|4-face types||[[Truncated 5-cell|t<sub>0,1</sub>{3<sup>3</sup>}]] [[File:Schlegel half-solid truncated pentachoron.png|40px]]<BR>t<sub>0,1,2</sub>{3<sup>3</sup>} 40px<BR>[[Runcinated 5-cell|t<sub>0,3</sub>{3<sup>3</sup>}]] [[File:Schlegel half-solid runcinated 5-cell.png|40px]]
|-
|bgcolor=#e7dcc3|Cell types||[[Tetrahedron]] [[File:Tetrahedron.png|20px]]<BR>[[Truncated tetrahedron|Truncated tetrahedron]] 20px<BR>[[Truncated octahedron|Truncated octahedron]] 20px<BR>[[Triangular prism|Triangular prism]] [[File:Triangular prism.png|20px]]
|-
|bgcolor=#e7dcc3|Vertex figure||triangular elongated-antiprismatic pyramid
|-
|bgcolor=#e7dcc3|Symmetry||<math>{\tilde{A}}_4</math>×2, {3<sup>[5]</sup>}
|-
|bgcolor=#e7dcc3|Properties||vertex-transitive
|}
The '''truncated 4-simplex honeycomb''' or '''truncated 5-cell honeycomb''' is a space-filling [[Tessellation|tessellation]] [[Honeycomb (geometry)|honeycomb]]. It can also be called a '''cyclocantitruncated 5-cell honeycomb'''.

==== Alaternate names ====
* Great cyclorhombated pentachoric tetracomb
* Great truncated-pentachoric tetracomb
{{-}}

=== Cantellated 5-cell honeycomb ===
{| class="wikitable" align="right" style="margin-left:10px" width="320"
!bgcolor=#e7dcc3 colspan=2|Cantellated 5-cell honeycomb
|-
|bgcolor=#ffffff align=center colspan=2|(No image)
|-
|bgcolor=#e7dcc3|Type||Uniform 4-honeycomb
|-
|bgcolor=#e7dcc3|[[Schläfli symbol]]||t<sub>0,1,3</sub>{3<sup>[5]</sup>} or rr{3<sup>[5]</sup>}
|-
|bgcolor=#e7dcc3|Coxeter diagram||{{CDD|node_1|split1|nodes|3ab|branch_11}}
|-
|bgcolor=#e7dcc3|4-face types||[[Cantellated 5-cell|t<sub>0,2</sub>{3<sup>3</sup>}]] [[File:Schlegel half-solid cantellated 5-cell.png|40px]]<BR>t<sub>1,2</sub>{3<sup>3</sup>} 40px<BR>t<sub>0,1,3</sub>{3<sup>3</sup>} [[File:Schlegel half-solid runcitruncated 5-cell.png|40px]]
|-
|bgcolor=#e7dcc3|Cell types||[[Truncated tetrahedron]] [[File:Truncated tetrahedron.png|20px]]<BR>[[Octahedron|Octahedron]] 20px<BR>[[Cuboctahedron|Cuboctahedron]] 20px<BR>[[Triangular prism|Triangular prism]] 20px<BR>[[Hexagonal prism|Hexagonal prism]] [[File:Hexagonal prism.png|20px]]
|-
|bgcolor=#e7dcc3|Vertex figure||Bidiminished rectified pentachoron
|-
|bgcolor=#e7dcc3|Symmetry||<math>{\tilde{A}}_4</math>×2, {3<sup>[5]</sup>}
|-
|bgcolor=#e7dcc3|Properties||vertex-transitive
|}
The '''cantellated 4-simplex honeycomb''' or '''cantellated 5-cell honeycomb''' is a space-filling [[Tessellation|tessellation]] [[Honeycomb (geometry)|honeycomb]]. It can also be called a '''cycloruncitruncated 5-cell honeycomb'''.


==== Alternate names====
* Cycloprismatorhombated pentachoric tetracomb
* Great prismatodispentachoric tetracomb

{{-}}

=== Bitruncated 5-cell honeycomb===
{| class="wikitable" align="right" style="margin-left:10px"
!bgcolor=#e7dcc3 colspan=2|Bitruncated 5-cell honeycomb
|-
|bgcolor=#ffffff align=center colspan=2|(No image)
|-
|bgcolor=#e7dcc3|Type||Uniform 4-honeycomb
|-
|bgcolor=#e7dcc3|[[Schläfli symbol]]||t<sub>0,1,2,3</sub>{3<sup>[5]</sup>} or 2t{3<sup>[5]</sup>}
|-
|bgcolor=#e7dcc3|Coxeter diagram||{{CDD|node|split1|nodes_11|3ab|branch_11}}
|-
|bgcolor=#e7dcc3|4-face types||t<sub>0,1,3</sub>{3<sup>3</sup>} [[File:Schlegel half-solid runcitruncated 5-cell.png|40px]]<BR>t<sub>0,1,2</sub>{3<sup>3</sup>} 40px<BR>t<sub>0,1,2,3</sub>{3<sup>3</sup>} [[File:Schlegel half-solid omnitruncated 5-cell.png|40px]]
|-
|bgcolor=#e7dcc3|Cell types||[[Cuboctahedron]] [[File:Cuboctahedron.png|20px]]<br>
[[Truncated octahedron]] [[File:Truncated octahedron.png|20px]]<BR>
[[Truncated tetrahedron]] [[File:truncated tetrahedron.png|20px]]<BR>
[[Hexagonal prism]] [[File:Hexagonal prism.png|20px]]<BR>
[[Triangular prism]] [[File:Triangular prism.png|20px]]
|-
|bgcolor=#e7dcc3|Vertex figure||tilted rectangular [[Duopyramid|duopyramid]]
|-
|bgcolor=#e7dcc3|Symmetry||<math>{\tilde{A}}_4</math>×2, {3<sup>[5]</sup>}
|-
|bgcolor=#e7dcc3|Properties||vertex-transitive
|}
The '''bitruncated 4-simplex honeycomb''' or '''bitruncated 5-cell honeycomb''' is a space-filling [[Tessellation|tessellation]] [[Honeycomb (geometry)|honeycomb]]. It can also be called a '''cycloruncicantitruncated 5-cell honeycomb'''.

==== Alternate names====
* Great cycloprismated pentachoric tetracomb
* Grand prismatodispentachoric tetracomb

{{-}}

=== Omnitruncated 5-cell honeycomb===
{| class="wikitable" align="right"
!bgcolor=#e7dcc3 colspan=2|Omnitruncated 4-simplex honeycomb
|-
|bgcolor=#ffffff align=center colspan=2|(No image)
|-
|bgcolor=#e7dcc3|Type||Uniform 4-honeycomb
|-
|bgcolor=#e7dcc3|Family||[[Omnitruncated simplectic honeycomb]]
|-
|bgcolor=#e7dcc3|[[Schläfli symbol]]||t<sub>0,1,2,3,4</sub>{3<sup>[5]</sup>} or tr{3<sup>[5]</sup>}
|-
|bgcolor=#e7dcc3|Coxeter diagram||{{CDD|node_1|split1|nodes_11|3ab|branch_11}}
|-
|bgcolor=#e7dcc3|4-face types||t<sub>0,1,2,3</sub>{3,3,3} [[File:Schlegel half-solid omnitruncated 5-cell.png|40px]]
|-
|bgcolor=#e7dcc3|Cell types||[[Truncated octahedron|t<sub>0,1,2</sub>{3,3}]] [[File:Uniform polyhedron-33-t012.png|20px]]<BR>[[Hexagonal prism|{6}x{}]] [[File:Hexagonal prism.png|20px]]
|-
|bgcolor=#e7dcc3|Face types||{4}<BR>[[Hexagon|{6}]]
|-
|bgcolor=#e7dcc3|Vertex figure||[[File:Omnitruncated 4-simplex honeycomb verf.png|80px]]<BR>Irr. [[5-cell|5-cell]]
|-
|bgcolor=#e7dcc3|Symmetry||<math>{\tilde{A}}_4</math>×10, <nowiki>[5[</nowiki>3<sup>[5]</sup>]]
|-
|bgcolor=#e7dcc3|Properties||vertex-transitive, cell-transitive
|}

The '''omnitruncated 4-simplex honeycomb''' or '''omnitruncated 5-cell honeycomb''' is a space-filling [[Tessellation|tessellation]] [[Honeycomb (geometry)|honeycomb]].  It can also be seen as a '''cyclosteriruncicantitruncated 5-cell honeycomb'''.
.

It is composed entirely of omnitruncated 5-cell (omnitruncated 4-simplex) facets.

Coxeter calls this '''Hinton's honeycomb''' after C. H. Hinton,  who described it in his book ''The Fourth Dimension'' in 1906.<ref name=cox>{{cite book|title=The Beauty of Geometry: Twelve Essays|year= 1999|publisher= Dover Publications|lccn=99035678|isbn= 0-486-40919-8 }} (The classification of Zonohededra, page 73)</ref>

The facets of all  [[Omnitruncated simplectic honeycomb|omnitruncated simplectic honeycomb]]s are called [[Permutohedron|permutohedra]] and can be positioned in ''n+1'' space with integral coordinates, permutations of the whole numbers (0,1,..,n).

==== Alternate names====
* Omnitruncated cyclopentachoric tetracomb
* Great-prismatodecachoric tetracomb

==== A<sub>4</sub><sup>*</sup> lattice ====

The A{{sup sub|*|4}} lattice is the union of five A<sub>4</sub> lattices, and is the dual to the omnitruncated 5-cell honeycomb, and therefore the Voronoi cell of this lattice is an omnitruncated 5-cell.<ref>[http://www.math.rwth-aachen.de/~Gabriele.Nebe/LATTICES/As4.html The Lattice A4*]</ref>
: {{CDD|node_1|split1|nodes|3ab|branch}} ∪ {{CDD|node|split1|nodes_10lur|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}}
{{-}}

== Alternated form ==

This honeycomb can be [[Alternation (geometry)|alternated]], creating [[Runcinated 5-cell#Full snub 5-cell|omnisnub 5-cells]] with irregular [[5-cell|5-cell]]s created at the deleted vertices. Although it is not uniform, the 5-cells have a symmetry of order 10.

==See also==
Regular and uniform honeycombs in 4-space:
*[[Tesseractic honeycomb]]
*[[16-cell honeycomb|16-cell honeycomb]]
*[[24-cell honeycomb|24-cell honeycomb]]
*[[Truncated 24-cell honeycomb]]
*[[Snub 24-cell honeycomb]]

==Notes==
{{reflist}}

== References ==
* [[Biography: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, x3x3x3x3x3*a - otcypit - 135, x3x3x3o3o3*a - gocyropit - O137, x3x3o3x3o3*a - cypropit  - O138, x3x3x3x3o3*a - gocypapit - O139, x3x3x3x3x3*a - otcypit - 140
* Affine Coxeter group Wa(A4), Quaternions, and Decagonal Quasicrystals, Mehmet Koca, Nazife O. Koca, Ramazan Koc (2013) {{ArXiv|1209.1878}}

{{Honeycombs}}

[[Category:Honeycombs (geometry)]]
[[Category:5-polytopes]]

{{Sourceattribution|5-cell honeycomb}}