View source for 24-cell honeycomb

{| class="wikitable" align="right" style="margin-left:10px" width="320"
!bgcolor=#e7dcc3 colspan=2|24-cell honeycomb
|-
|bgcolor=#ffffff align=center colspan=2|[[Image:Icositetrachoronic tetracomb.png|220px]]<br> A [[24-cell]] and first layer of its adjacent 4-faces.
|-
|bgcolor=#e7dcc3|Type||[[List of regular polytopes#Tessellations of Euclidean 4-space|Regular 4-honeycomb]]<BR>[[Uniform 4-honeycomb]]
|-
|bgcolor=#e7dcc3|[[Schläfli symbol]]||{3,4,3,3}<BR>r{3,3,4,3}<BR>2r{4,3,3,4}<BR>2r{4,3,3<sup>1,1</sup>}<BR>{3<sup>1,1,1,1</sup>}
|-
|bgcolor=#e7dcc3|[[Coxeter-Dynkin diagram]]s||{{CDD|node_1|3|node|4|node|3|node|3|node}}<BR>{{CDD|node|3|node|4|node|3|node_1|3|node}}<BR>{{CDD|node|4|node|3|node_1|3|node|4|node}}<BR>{{CDD|nodes|split2|node_1|3|node|4|node}}<BR>{{CDD|nodes|split2|node_1|split1|nodes}}
|-
|bgcolor=#e7dcc3|4-face type||[[24-cell|{3,4,3}]] [[File:Schlegel wireframe 24-cell.png|40px]]
|-
|bgcolor=#e7dcc3|Cell type||[[octahedron|{3,4}]] [[File:Uniform polyhedron-43-t2.svg|20px]]
|-
|bgcolor=#e7dcc3|Face type||[[triangle|{3}]]
|-
|bgcolor=#e7dcc3|[[Edge figure]]||[[tetrahedron|{3,3}]] 
|-
|bgcolor=#e7dcc3|[[Vertex figure]]||[[tesseract|{4,3,3}]] 
|-
|bgcolor=#e7dcc3|Dual||[[demitesseractic honeycomb|{3,3,4,3}]]
|-
|bgcolor=#e7dcc3|[[Coxeter group]]s||<math>{\tilde{F}}_4</math>, [3,4,3,3]<BR><math>{\tilde{C}}_4</math>, [4,3,3,4]<BR><math>{\tilde{B}}_4</math>, [4,3,3<sup>1,1</sup>]<BR><math>{\tilde{D}}_4</math>, [3<sup>1,1,1,1</sup>]
|-
|bgcolor=#e7dcc3|Properties||regular
|}
In [[Four-dimensional space|four-dimensional]] [[Euclidean geometry]], the '''24-cell honeycomb''', or '''icositetrachoric honeycomb''' is a [[regular polytope|regular]] space-filling [[tessellation]] (or [[honeycomb (geometry)|honeycomb]]) of 4-dimensional [[Euclidean space]] by regular [[24-cell]]s. It can be represented by [[Schläfli symbol]] {3,4,3,3}.

The [[dual polytope|dual]] tessellation by regular [[16-cell honeycomb]] has Schläfli symbol {3,3,4,3}. Together with the [[tesseractic honeycomb]] (or 4-cubic honeycomb) these are the only regular tessellations of Euclidean 4-space.

== Coordinates ==

The 24-cell honeycomb can be constructed as the [[Voronoi tessellation]] of the D<sub>4</sub> or [[F4 lattice|F<sub>4</sub> root lattice]]. Each 24-cell is then centered at a D<sub>4</sub> lattice point, i.e. one of

:<math>\left\{(x_i)\in\mathbb Z^4 : {\textstyle\sum_i} x_i \equiv 0\;(\mbox{mod }2)\right\}.</math>

These points can also be described as [[Hurwitz quaternion]]s with even square norm.

The vertices of the honeycomb lie at the deep holes of the D<sub>4</sub> lattice. These are the Hurwitz quaternions with odd square norm.

It can be constructed as a [[#Symmetry constructions|birectified tesseractic honeycomb]], by taking a [[tesseractic honeycomb]] and placing vertices at the centers of all the square faces. The [[24-cell]] facets exist between these vertices as ''rectified 16-cells''. If the coordinates of the tesseractic honeycomb are integers (i,j,k,l), the ''birectified tesseractic honeycomb'' vertices can be placed at all permutations of half-unit shifts in two of the four dimensions, thus: (i+½,j+½,k,l), (i+½,j,k+½,l), (i+½,j,k,l+½), (i,j+½,k+½,l), (i,j+½,k,l+½), (i,j,k+½,l+½).

== Configuration ==

Each 24-cell in the 24-cell honeycomb has 24 neighboring 24-cells. With each neighbor it shares exactly one octahedral cell.

It has 24 more neighbors such that with each of these it shares a single vertex.

It has no neighbors with which it shares only an edge or only a face.

The [[vertex figure]] of the 24-cell honeycomb is a [[tesseract]] (4-dimensional cube). So there are 16 edges, 32 triangles, 24 octahedra, and 8 24-cells meeting at every vertex. The [[edge figure]] is a [[tetrahedron]], so there are 4 triangles, 6 octahedra, and 4 24-cells surrounding every edge. Finally, the [[face figure]] is a triangle, so there are 3 octahedra and 3 24-cells meeting at every face.

== Cross-sections ==

One way to visualize a 4-dimensional figure is to consider various 3-dimensional [[cross section (geometry)|cross-sections]]. That is, the intersection of various hyperplanes with the figure in question. Applying this technique to the 24-cell honeycomb gives rise to various 3-dimensional honeycombs with varying degrees of regularity.

{| class="wikitable" width="300" style="text-align:center; float:right; margin:0.5em; background:white;"
! colspan="2" |Vertex-first sections
|-
|[[Image:Rhombic dodecahedra.png|150px]]
|[[Image:Partial cubic honeycomb.png|150px]]
|-
|[[Rhombic dodecahedral honeycomb]]
|[[Cubic honeycomb]]
|-
! colspan="2" |Cell-first sections
|-
|[[Image:Rectified cubic honeycomb.png|150px]]
|[[Image:Bitruncated cubic honeycomb.png|150px]]
|-
|[[Rectified cubic honeycomb]]
|[[Bitruncated cubic honeycomb]]
|}
A ''vertex-first'' cross-section uses some hyperplane [[orthogonal]] to a line joining opposite vertices of one of the 24-cells. For instance, one could take any of the coordinate hyperplanes in the coordinate system given above (i.e. the planes determined by ''x''<sub>''i''</sub> = 0). The cross-section of {3,4,3,3} by one of these hyperplanes gives a [[rhombic dodecahedral honeycomb]]. Each of the rhombic dodecahedra corresponds to a maximal cross-section of one of the 24-cells intersecting the hyperplane (the center of each such (4-dimensional) 24-cell lies in the hyperplane). Accordingly, the rhombic dodecahedral honeycomb is the [[Voronoi tessellation]] of the D<sub>3</sub> root lattice (a [[face-centered cubic]] lattice). Shifting this hyperplane halfway to one of the vertices (e.g. ''x''<sub>''i''</sub> = ½) gives rise to a regular [[cubic honeycomb]]. In this case the center of each 24-cell lies off the hyperplane. Shifting again, so the hyperplane intersects the vertex, gives another rhombic dodecahedral honeycomb but with new 24-cells (the former ones having shrunk to points). In general, for any integer ''n'', the cross-section through ''x''<sub>''i''</sub> = ''n'' is a rhombic dodecahedral honeycomb, and the cross-section through ''x''<sub>''i''</sub> = ''n'' + ½ is a cubic honeycomb. As the hyperplane moves through 4-space, the cross-section morphs between the two periodically.

A ''cell-first'' cross-section uses some hyperplane parallel to one of the octahedral cells of a 24-cell. Consider, for instance, some hyperplane orthogonal to the vector (1,1,0,0). The cross-section of {3,4,3,3} by this hyperplane is a [[rectified cubic honeycomb]]. Each [[cuboctahedron]] in this honeycomb is a maximal cross-section of a 24-cell whose center lies in the plane. Meanwhile, each [[octahedron]] is a boundary cell of a (4-dimensional) 24-cell whose center lies off the plane. Shifting this hyperplane till it lies halfway between the center of a 24-cell and the boundary, one obtains a [[bitruncated cubic honeycomb]]. The cuboctahedra have shrunk, and the octahedra have grown until they are both [[truncated octahedron|truncated octahedra]]. Shifting again, so the hyperplane intersects the boundary of the central 24-cell gives a rectified cubic honeycomb again, the cuboctahedra and octahedra having swapped positions. As the hyperplane sweeps through 4-space, the cross-section morphs between these two honeycombs periodically.{{clear}}

== Kissing number ==
If a [[3-sphere]] is [[inscribed sphere|inscribed]] in each hypercell of this tessellation, the resulting arrangement is the densest known{{NoteTag|The sphere packing problem and the kissing number problem are remarkably difficult and optimal solutions are only known in 1, 2, 3, 8, and 24 dimensions (plus dimension 4 for the kissing number problem).}} regular [[sphere packing]] in four dimensions, with the [[kissing number]] 24. The packing density of this arrangement is
: <math>\frac{\pi^2}{16}\cong0.61685.</math>

Each inscribed [[3-sphere]] kisses 24 others at the centers of the octahedral facets of its 24-cell, since each such octahedral cell is shared with an adjacent 24-cell. In a unit-edge-length tessellation, the diameter of the spheres (the distance between the centers of kissing spheres) is {{radic|2}}.

Just outside this surrounding shell of 24 kissing 3-spheres is another less dense shell of 24 3-spheres which do not kiss each other or the central 3-sphere; they are inscribed in 24-cells with which the central 24-cell shares only a single vertex (rather than an octahedral cell). The center-to-center distance between one of these spheres and any of its shell neighbors or the central sphere is 2.

Alternatively, the same sphere packing arrangement with kissing number 24 can be carried out with smaller 3-spheres of edge-length-diameter, by locating them at the centers and the vertices of the 24-cells. (This is equivalent to locating them at [[16-cell honeycomb#D4 lattice|the vertices of a 16-cell honeycomb]] of unit-edge-length.) In this case the central 3-sphere kisses 24 others at the centers of the cubical facets of [[24-cell#Tesseract|the three tesseracts inscribed in the 24-cell]]. (This is the [[Tesseractic honeycomb#Sphere packing|unique body-centered cubic packing of edge-length spheres]] of the tesseractic honeycomb.)

Just outside this shell of kissing 3-spheres of diameter 1 is another less dense shell of 24 non-kissing 3-spheres of diameter 1; they are centered in the adjacent 24-cells with which the central 24-cell shares an octahedral facet. The center-to-center distance between one of these spheres and any of its shell neighbors or the central sphere is {{radic|2}}.

== Symmetry constructions ==

There are five different [[Wythoff construction]]s of this tessellation as a [[uniform polytope]]. They are geometrically identical to the regular form, but the symmetry differences can be represented by colored 24-cell facets. In all cases, eight 24-cells meet at each vertex, but the vertex figures have different symmetry generators.

{| class="wikitable"
![[Coxeter group]]
!colspan=2|[[Schläfli symbol]]s
![[Coxeter diagram]]
![[Facet (geometry)|Facets]]<BR>([[24-cell]]s)
![[Vertex figure]]<BR>([[8-cell]])
!Vertex<BR>figure<BR>symmetry<BR>order
|- align=center
!rowspan=2|<math>{\tilde{F}}_4</math> = [[Template:F4_honeycombs|[3,4,3,3]]]
|<math>\begin{Bmatrix} 3, 4, 3, 3 \end{Bmatrix}</math>||{3,4,3,3}
|{{CDD|node_1|3|node|4|node|3|node|3|node}}
|'''8:''' {{CDD|node_1|3|node|4|node|3|node}}
|{{CDD|node_1|4|node|3|node|3|node}}
|384
|- align=center
|<math>\left\{\begin{array}{l}3\\3,4,3\end{array}\right\}</math>||r{3,3,4,3}
|{{CDD|node|3|node_1|3|node|4|node|3|node}}
|'''6:''' {{CDD|node|3|node|4|node|3|node_1}}<BR>'''2:''' {{CDD|node|4|node|3|node_1|3|node}}
|{{CDD|node|3|node|4|node_1|2|node_1}}
|96
|- align=center
!<math>{\tilde{C}}_4</math> = [[Template:C4_honeycombs|[4,3,3,4]]]
|<math>\left\{\begin{array}{l}3,4\\3,4\end{array}\right\}</math>||2r{4,3,3,4}
|{{CDD|node|4|node|3|node_1|3|node|4|node}}
|'''4,4:''' {{CDD|node|3|node_1|3|node|4|node}}
|{{CDD|node|4|node_1|2|node_1|4|node}}
|64
|- align=center
!<math>{\tilde{B}}_4</math> = [[Template:B4_honeycombs|[4,3,3<sup>1,1</sup>]]]
|<math>\left\{\begin{array}{l}3\\3\\3,4\end{array}\right\}</math>||2r{4,3,3<sup>1,1</sup>}
|{{CDD|nodes|split2|node_1|3|node|4|node}}
|'''2,2:''' {{CDD|node|3|node_1|3|node|4|node}}<BR>'''4:''' {{CDD|nodes|split2|node_1|3|node}}
|{{CDD|node_1|2|node_1|2|node_1|4|node}}
|32
|- align=center
!<math>{\tilde{D}}_4</math> = [[Template:D4_honeycombs|[3<sup>1,1,1,1</sup>]]]
|<math>\left\{\begin{array}{l}3\\3\\3\\3\end{array}\right\}</math>||{3<sup>1,1,1,1</sup>}
|{{CDD|nodes|split2|node_1|split1|nodes}}
|'''2,2,2,2:'''<BR>{{CDD|nodes|split2|node_1|3|node}}
|{{CDD|node_1|2|node_1|2|node_1|2|node_1}}
|16
|}

== See also ==
Other uniform honeycombs in 4-space:
*[[Truncated 5-cell honeycomb]]
*[[Omnitruncated 5-cell honeycomb]]
*[[Truncated 24-cell honeycomb]]
*[[Rectified 24-cell honeycomb]]
*[[Snub 24-cell honeycomb]]

== Notes ==
{{NoteFoot}}

== References ==
{{Reflist}}
{{refbegin}}
* [[Coxeter|Coxeter, H.S.M.]] ''[[Regular Polytopes (book)|Regular Polytopes]]'', (3rd edition, 1973), Dover edition, {{ISBN|0-486-61480-8}} p.&nbsp;296, Table II: Regular honeycombs
* '''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 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 88
* {{KlitzingPolytopes|flat.htm|4D|Euclidean tesselations}} o4o3x3o4o, o3x3o *b3o4o, o3x3o *b3o4o, o3x3o4o3o, o3o3o4o3x - icot - O88
{{refend}}

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{{Honeycombs}}

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