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| !bgcolor=#e7dcc3 colspan=2|24-cell honeycomb
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| |bgcolor=#ffffff align=center colspan=2|[[Image:Icositetrachoronic tetracomb.png|220px]]<br> A [[24-cell]] and first layer of its adjacent 4-faces.
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| |bgcolor=#e7dcc3|Type||[[List_of_regular_polytopes#Five_Dimensions_2|Regular 4-space honeycomb]]<BR>[[Uniform_polyteron#Regular_and_uniform_honeycombs|Uniform 4-honeycomb]]
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| |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>}
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| |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}}
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| |bgcolor=#e7dcc3|4-face type||[[24-cell|{3,4,3}]] [[File:Schlegel wireframe 24-cell.png|40px]]
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| |bgcolor=#e7dcc3|Cell type||[[octahedron|{3,4}]] [[File:Octahedron.png|20px]]
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| |bgcolor=#e7dcc3|Face type||[[triangle|{3}]]
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| |bgcolor=#e7dcc3|[[Edge figure]]||[[tetrahedron|{3,3}]]
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| |bgcolor=#e7dcc3|[[Vertex figure]]||[[tesseract|{4,3,3}]]
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| |bgcolor=#e7dcc3|Dual||[[demitesseractic honeycomb|{3,3,4,3}]]
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| |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>]
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| |bgcolor=#e7dcc3|Properties||regular
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| |}
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| 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}.
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| 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.
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| == Kissing number ==
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| If a [[3-sphere]] is [[inscribed sphere|inscribed]] in each hypercell of this tessellation, the resulting arrangement is the densest possible regular [[sphere packing]] in four dimensions, with the [[kissing number]] 24. The packing density of this arrangement is
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| :<math>\frac{\pi^2}{16}\cong0.61685.</math>
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| == Coordinates ==
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| The 24-cell honeycomb can be constructed as the [[Voronoi tessellation]] of the [[D4 root lattice|D<sub>4</sub> root lattice]] or [[F4 lattice]]. Each 24-cell is then centered at a D<sub>4</sub> lattice point, i.e. one of
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| :<math>\left\{(x_i)\in\mathbb Z^4 : {\textstyle\sum_i} x_i \equiv 0\;(\mbox{mod }2)\right\}.</math>
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| These points can also be described as [[Hurwitz quaternion]]s with even square norm.
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| 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.
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| 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).
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| == Configuration ==
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| Each 24-cell in the 24-cell honeycomb has 24 neighboring 24-cells. With each neighbor it shares exactly one octahedral cell.
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| It has 24 more neighbors such that with each of these it shares a single vertex.
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| It has no neighbors with which it shares only an edge or only a face.
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| 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.
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| == Cross-sections ==
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| One way to visualize 4-dimensional figures is to consider various 3-dimensional [[cross section (geometry)|cross-sections]]. Applying this technique to the 24-cell honeycomb gives rise to various 3-dimensional honeycombs with varying degrees of regularity.
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| {|class='wikitable' style="text-align:center; float:right; margin:0.5em; background:white;"
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| !colspan=2|Vertex-first sections
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| |[[Image:Rhombic dodecahedra.png|220px]]
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| |[[Image:Partial cubic honeycomb.png|220px]]
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| |[[Rhombic dodecahedral honeycomb]]
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| |[[Cubic honeycomb]]
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| !colspan=2|Cell-first sections
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| |[[Image:Rectified cubic honeycomb.png|220px]]
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| |[[Image:Bitruncated cubic honeycomb.png|220px]]
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| |[[Rectified cubic honeycomb]]
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| |[[Bitruncated cubic honeycomb]]
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| |}
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| A ''vertex-first'' cross-section is one [[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 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.
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| A ''cell-first'' cross-section is one parallel to one of the octahedral cells of a 24-cell. Consider, for instance, the hyperplane orthogonal to (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 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.<br style="clear:both">
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| == Symmetry constructions ==
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| There are five different symmetry constructions of this tessellation. Each symmetry can be represented by different arrangements of colored 24-cell facets. In all cases, eight 24-cells meet at each vertex, but the vertex figures have different symmetry generators.
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| {| class="wikitable"
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| ![[Coxeter group]]
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| ![[Coxeter diagram]]
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| ![[Facet (geometry)|Facets]]<BR>([[24-cell]]s)
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| ![[Vertex figure]]<BR>([[8-cell]])
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| !Vertex<BR>figure<BR>symmetry<BR>order
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| |- align=center
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| !<math>{\tilde{F}}_4</math>
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| |{{CDD|node_1|3|node|4|node|3|node|3|node}}
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| |'''8:''' {{CDD|node_1|3|node|4|node|3|node}}
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| |{{CDD|node_1|4|node|3|node|3|node}}
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| |384
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| |- align=center
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| !<math>{\tilde{F}}_4</math>
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| |{{CDD|node|3|node_1|3|node|4|node|3|node}}
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| |'''6:''' {{CDD|node|3|node|4|node|3|node_1}}<BR>'''2:''' {{CDD|node|4|node|3|node_1|3|node}}
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| |{{CDD|node|3|node|4|node_1|2|node_1}}
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| |96
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| |- align=center
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| !<math>{\tilde{C}}_4</math>
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| |{{CDD|node|4|node|3|node_1|3|node|4|node}}
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| |'''4,4:''' {{CDD|node|3|node_1|3|node|4|node}}
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| |{{CDD|node|4|node_1|2|node_1|4|node}}
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| |64
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| |- align=center
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| !<math>{\tilde{B}}_4</math>
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| |{{CDD|nodes|split2|node_1|3|node|4|node}}
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| |'''2,2:''' {{CDD|node|3|node_1|3|node|4|node}}<BR>'''4:''' {{CDD|nodes|split2|node_1|3|node}}
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| |{{CDD|node_1|2|node_1|2|node_1|4|node}}
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| |32
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| |- align=center
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| !<math>{\tilde{D}}_4</math>
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| |{{CDD|nodes|split2|node_1|split1|nodes}}
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| |'''2,2,2,2:'''<BR>{{CDD|nodes|split2|node_1|3|node}}
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| |{{CDD|node_1|2|node_1|2|node_1|2|node_1}}
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| |16
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| |}
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| == Related honeycombs==
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| {{F4 honeycombs}}
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| {{C4_honeycombs}}
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| {{B4_honeycombs}}
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| {{D4 honeycombs}}
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| == See also ==
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| Other uniform honeycombs in 4-space:
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| *[[Truncated 5-cell honeycomb]]
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| *[[Omnitruncated 5-cell honeycomb]]
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| *[[Truncated 24-cell honeycomb]]
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| *[[Rectified 24-cell honeycomb]]
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| *[[Snub 24-cell honeycomb]]
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| == References ==
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| * [[Coxeter|Coxeter, H.S.M.]] ''[[Regular Polytopes (book)|Regular Polytopes]]'', (3rd edition, 1973), Dover edition, ISBN 0-486-61480-8 p. 296, Table II: Regular honeycombs
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| * '''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]
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| ** (Paper 24) H.S.M. Coxeter, ''Regular and Semi-Regular Polytopes III'', [Math. Zeit. 200 (1988) 3-45]
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| * [[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
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| * {{KlitzingPolytopes|flat.htm|4D|Euclidean tesselations}} o4o3x3o4o, o3x3o *b3o4o, o3x3o *b3o4o, o3x3o4o3o, o3o3o4o3x - icot - O88
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| {{Honeycombs}}
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| [[Category:5-polytopes]]
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| [[Category:Honeycombs (geometry)]]
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