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| {| class=wikitable width=280 align=right
| | This is a preview for the new '''MathML rendering mode''' (with SVG fallback), which is availble in production for registered users. |
| !<math>{\tilde{A}}_2</math>
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| !<math>{\tilde{A}}_3</math>
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| ![[Triangular tiling]]
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| ![[Tetrahedral-octahedral honeycomb]]
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| |[[File:Uniform_tiling_333-t1.png|120px]]<BR>With red and yellow equilateral triangles
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| |[[File:Tetrahedral-octahedral honeycomb2.png|160px]]<BR>With cyan and yellow [[tetrahedron|tetrahedra]], and red rectified tetrahedra ([[octahedron]])
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| !{{CDD|node_1|split1|branch}}
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| !{{CDD|node_1|split1|nodes|split2|node}}
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| |}
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| In [[geometry]], the '''simplectic honeycomb''' (or '''n-simplex honeycomb''') is a dimensional infinite series of [[Honeycomb (geometry)|honeycomb]]s, based on the <math>{\tilde{A}}_n</math> affine [[Coxeter group]] symmetry. It is given a [[Schläfli symbol]] {3<sup>[n+1]</sup>}, and is represented by a [[Coxeter-Dynkin diagram]] as a cyclic graph of ''n+1'' nodes with one node ringed. It is composed of n-[[simplex]] facets, along with all [[Rectification (geometry)|rectified]] n-simplices. The [[vertex figure]] of an ''n-simplex honeycomb'' is an [[Expansion (geometry)|expanded]] n-[[simplex]].
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| In 2 dimensions, the honeycomb represents the [[triangular tiling]], with Coxeter graph {{CDD|node_1|split1|branch}} filling the plane with alternately colored triangles. In 3 dimensions it represents the [[tetrahedral-octahedral honeycomb]], with Coxeter graph {{CDD|node_1|split1|nodes|split2|node}} filling space with alternately tetrahedral and octahedral cells. In 4 dimensions it is called the [[5-cell honeycomb]], with Coxeter graph {{CDD|node_1|split1|nodes|3ab|branch}}, with [[5-cell]] and [[rectified 5-cell]] facets. In 5 dimensions it is called the [[5-simplex honeycomb]], with Coxeter graph {{CDD|node_1|split1|nodes|3ab|nodes|split2|node}}, filling space by [[5-simplex]], [[rectified 5-simplex]], and [[birectified 5-simplex]] facets. In 6 dimensions it is called the [[6-simplex honeycomb]], with Coxeter graph {{CDD|node_1|split1|nodes|3ab|nodes|3ab|branch}}, filling space by [[6-simplex]], [[rectified 6-simplex]], and [[birectified 6-simplex]] facets.
| | If you would like use the '''MathML''' rendering mode, you need a wikipedia user account that can be registered here [[https://en.wikipedia.org/wiki/Special:UserLogin/signup]] |
| | * Only registered users will be able to execute this rendering mode. |
| | * Note: you need not enter a email address (nor any other private information). Please do not use a password that you use elsewhere. |
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| == By dimension ==
| | Registered users will be able to choose between the following three rendering modes: |
| {| class="wikitable"
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| !height=30|n
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| !<math>{\tilde{A}}_{2+}</math>
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| !Tessellation
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| !Vertex figure
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| !Facets per vertex figure
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| !Vertices per vertex figure
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| !Edge figure
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| |1
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| |<math>{\tilde{A}}_1</math>
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| |[[File:Regular_apeirogon.png|80px]]<BR>[[Apeirogon]]<BR>{{CDD|node_1|infin|node}}
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| |{{CDD|node_1}}
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| |1
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| |2
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| |2
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| |<math>{\tilde{A}}_2</math>
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| |[[Image:Uniform tiling 333-t1.png|80px]]<BR>[[Triangular tiling]]<BR>2-simplex honeycomb<BR>{{CDD|node_1|split1|branch}}
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| |[[Image:Truncated triangle.png|80px]]<BR>[[Hexagon]]<BR>(Truncated triangle)<BR>{{CDD|node_1|3|node_1}}
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| |3 [[triangle]]s<BR>3 [[hexagon|rectified triangles]]
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| |6
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| |[[Line segment]]<BR>{{CDD|node_1}}
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| |3
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| |<math>{\tilde{A}}_3</math>
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| |[[File:Tetrahedral-octahedral honeycomb2.png|80px]]<BR>[[Tetrahedral-octahedral honeycomb]]<BR>3-simplex honeycomb<BR>{{CDD|node_1|split1|nodes|split2|node}}
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| |[[Image:Uniform polyhedron-33-t02.png|80px]]<BR>[[Cuboctahedron]]<BR>(Cantellated tetrahedron)<BR>{{CDD|node_1|3|node|3|node_1}}
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| |4+4 [[tetrahedron]]<BR>6 [[octahedron|rectified tetrahedra]]
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| |12
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| |[[Rectangle]]<BR>{{CDD|node_1|2|node_1}}
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| |-
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| |4
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| |<math>{\tilde{A}}_4</math>
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| |[[4-simplex honeycomb]]<BR>{{CDD|node_1|split1|nodes|3ab|branch}}
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| |[[Image:Schlegel half-solid runcinated 5-cell.png|80px]]<BR>[[Runcinated 5-cell]]<BR>{{CDD|node_1|3|node|3|node|3|node_1}}
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| |5+5 [[5-cell]]s<BR>10+10 [[rectified 5-cell]]s
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| |20
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| |[[File:Runcinated_5-cell_verf.png|60px]]<BR>Triangular antiprism<BR>{{CDD|node_h|3|node_h|2|node_h}}
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| |-
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| |5
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| |<math>{\tilde{A}}_5</math>
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| |[[5-simplex honeycomb]]<BR>{{CDD|node_1|split1|nodes|3ab|nodes|split2|node}}
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| |[[File:5-simplex_t04.svg|80px]]<BR>[[Stericated 5-simplex]]<BR>{{CDD|node_1|3|node|3|node|3|node|3|node_1}}
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| |6+6 [[5-simplex]]<BR>15+15 [[rectified 5-simplex]]<BR>20 [[birectified 5-simplex]]
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| |30
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| |[[File:Stericated_hexateron_verf.png|60px]]<BR>Tetrahedral antiprism<BR>{{CDD|node_h|4|node|3|node|2|node_h}}
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| |-
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| |6
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| |<math>{\tilde{A}}_6</math>
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| |[[6-simplex honeycomb]]<BR>{{CDD|node_1|split1|nodes|3ab|nodes|3ab|branch}}
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| |[[File:6-simplex_t05.svg|80px]]<BR>[[Pentellated 6-simplex]]<BR>{{CDD|node_1|3|node|3|node|3|node|3|node|3|node_1}}
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| |7+7 [[6-simplex]]<BR>21+21 [[rectified 6-simplex]]<BR>35+35 [[birectified 6-simplex]]
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| |42
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| |4-simplex antiprism
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| |-
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| |7
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| |<math>{\tilde{A}}_7</math>
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| |[[7-simplex honeycomb]]<BR>{{CDD|node_1|split1|nodes|3ab|nodes|3ab|nodes|split2|node}}
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| |[[File:7-simplex_t06.svg|80px]]<BR>[[Hexicated 7-simplex]]<BR>{{CDD|node_1|3|node|3|node|3|node|3|node|3|node|3|node_1}}
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| |8+8 [[7-simplex]]<BR>28+28 [[rectified 7-simplex]]<BR>56+56 [[birectified 7-simplex]]<BR>70 [[trirectified 7-simplex]]
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| |56
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| |5-simplex antiprism
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| |8
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| |<math>{\tilde{A}}_8</math>
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| |[[8-simplex honeycomb]]<BR>{{CDD|node_1|split1|nodes|3ab|nodes|3ab|nodes|3ab|branch}}
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| |[[File:8-simplex_t07.svg|80px]]<BR>[[Heptellated 8-simplex]]<BR>{{CDD|node_1|3|node|3|node|3|node|3|node|3|node|3|node|3|node_1}}
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| |9+9 [[8-simplex]]<BR>36+36 [[rectified 8-simplex]]<BR>84+84 [[birectified 8-simplex]]<BR>126+126 [[trirectified 8-simplex]]
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| |72
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| |6-simplex antiprism
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| |-
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| |9
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| |<math>{\tilde{A}}_9</math>
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| |[[9-simplex honeycomb]]<BR>{{CDD|node_1|split1|nodes|3ab|nodes|3ab|nodes|3ab|nodes|split2|node}}
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| |[[File:9-simplex_t08.svg|80px]]<BR>[[Octellated 9-simplex]]<BR>{{CDD|node_1|3|node|3|node|3|node|3|node|3|node|3|node|3|node|3|node_1}}
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| |10+10 [[9-simplex]]<BR>45+45 [[rectified 9-simplex]]<BR>120+120 [[birectified 9-simplex]]<br>210+210 [[trirectified 9-simplex]]<br>252 [[quadrirectified 9-simplex]]
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| |90
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| |7-simplex antiprism
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| |10
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| |<math>{\tilde{A}}_{10}</math>
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| |[[10-simplex honeycomb]]<BR>{{CDD|node_1|split1|nodes|3ab|nodes|3ab|nodes|3ab|nodes|3ab|branch}}
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| |[[File:10-simplex_t09.svg|80px]]<BR>[[Ennecated 10-simplex]]<BR>{{CDD|node_1|3|node|3|node|3|node|3|node|3|node|3|node|3|node|3|node|3|node_1}}
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| |11+11 [[10-simplex]]<BR>55+55 [[rectified 10-simplex]]<BR>165+165 [[birectified 10-simplex]]<BR>330+330 [[trirectified 10-simplex]]<BR>462+462 [[quadrirectified 10-simplex]]
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| |110
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| |8-simplex antiprism
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| |11
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| |<math>{\tilde{A}}_{11}</math>
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| |11-simplex honeycomb
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| |...
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| |...
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| |...
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| |}
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| == Projection by folding == | | '''MathML''' |
| | :<math forcemathmode="mathml">E=mc^2</math> |
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| The (2n-1)-simplex honeycombs and 2n-simplex honeycombs can be projected into the n-dimensional [[hypercubic honeycomb]] by a [[Coxeter–Dynkin diagram#Geometric folding|geometric folding]] operation that maps two pairs of mirrors into each other, sharing the same [[vertex arrangement]]:
| | <!--'''PNG''' (currently default in production) |
| | :<math forcemathmode="png">E=mc^2</math> |
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| {|class=wikitable
| | '''source''' |
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| | :<math forcemathmode="source">E=mc^2</math> --> |
| !<math>{\tilde{A}}_2</math>
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| |{{CDD|node_1|split1|branch}}
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| !<math>{\tilde{A}}_4</math>
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| |{{CDD|node_1|split1|nodes|3ab|branch}}
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| !<math>{\tilde{A}}_6</math>
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| |{{CDD|node_1|split1|nodes|3ab|nodes|3ab|branch}}
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| !<math>{\tilde{A}}_8</math>
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| |{{CDD|node_1|split1|nodes|3ab|nodes|3ab|nodes|3ab|branch}}
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| !<math>{\tilde{A}}_{10}</math>
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| |{{CDD|node_1|split1|nodes|3ab|nodes|3ab|nodes|3ab|nodes|3ab|branch}}
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| |...
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| | <span style="color: red">Follow this [https://en.wikipedia.org/wiki/Special:Preferences#mw-prefsection-rendering link] to change your Math rendering settings.</span> You can also add a [https://en.wikipedia.org/wiki/Special:Preferences#mw-prefsection-rendering-skin Custom CSS] to force the MathML/SVG rendering or select different font families. See [https://www.mediawiki.org/wiki/Extension:Math#CSS_for_the_MathML_with_SVG_fallback_mode these examples]. |
| !<math>{\tilde{A}}_3</math>
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| |{{CDD|nodes_10r|splitcross|nodes}}
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| !<math>{\tilde{A}}_3</math>
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| |{{CDD|node_1|split1|nodes|split2|node}}
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| !<math>{\tilde{A}}_5</math>
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| |{{CDD|node_1|split1|nodes|3ab|nodes|split2|node}}
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| !<math>{\tilde{A}}_7</math>
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| |{{CDD|node_1|split1|nodes|3ab|nodes|3ab|nodes|split2|node}}
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| !<math>{\tilde{A}}_9</math>
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| |{{CDD|node_1|split1|nodes|3ab|nodes|3ab|nodes|3ab|nodes|split2|node}}
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| |...
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| |-
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| !<math>{\tilde{C}}_1</math>
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| |{{CDD|node_1|infin|node}}
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| !<math>{\tilde{C}}_2</math>
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| |{{CDD|node_1|4|node|4|node}}
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| !<math>{\tilde{C}}_3</math>
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| |{{CDD|node_1|4|node|3|node|4|node}}
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| !<math>{\tilde{C}}_4</math>
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| |{{CDD|node_1|4|node|3|node|3|node|4|node}}
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| !<math>{\tilde{C}}_5</math>
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| |{{CDD|node_1|4|node|3|node|3|node|3|node|4|node}}
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| |...
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| |}
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| == Kissing number == | | ==Demos== |
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| These honeycombs, seen as tangent n-spheres located at the center of each honeycomb vertex have a fixed number of contacting spheres and correspond to the number of vertices in the [[vertex figure]]. For 2 and 3 dimensions, this represents the highest [[kissing number]] for 2 and 3 dimensions, but fall short on higher dimensions. In 2-dimensions, the triangular tiling defines a circle packing of 6 tangent spheres arranged in a regular hexagon, and for 3 dimensions there are 12 tangent spheres arranged in an [[cuboctahedron|cuboctahedral]] configuration. For 4 to 8 dimensions, the kissing numbers are [[Expanded 4-simplex|20]], [[Expanded 5-simplex|30]], [[Expanded 5-simplex|42]], [[Expanded 6-simplex|56]], and [[Expanded 7-simplex|72]] spheres, while the greatest solutions are 24, 40, 72, 126, and 240 spheres respectively.
| | Here are some [https://commons.wikimedia.org/w/index.php?title=Special:ListFiles/Frederic.wang demos]: |
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| == See also==
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| * [[Truncated simplectic honeycomb]]
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| * [[Omnitruncated simplectic honeycomb]]
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| == References ==
| | * accessibility: |
| * [[George Olshevsky]], ''Uniform Panoploid Tetracombs'', Manuscript (2006) ''(Complete list of 11 convex uniform tilings, 28 convex uniform honeycombs, and 143 convex uniform tetracombs)'' | | ** Safari + VoiceOver: [https://commons.wikimedia.org/wiki/File:VoiceOver-Mac-Safari.ogv video only], [[File:Voiceover-mathml-example-1.wav|thumb|Voiceover-mathml-example-1]], [[File:Voiceover-mathml-example-2.wav|thumb|Voiceover-mathml-example-2]], [[File:Voiceover-mathml-example-3.wav|thumb|Voiceover-mathml-example-3]], [[File:Voiceover-mathml-example-4.wav|thumb|Voiceover-mathml-example-4]], [[File:Voiceover-mathml-example-5.wav|thumb|Voiceover-mathml-example-5]], [[File:Voiceover-mathml-example-6.wav|thumb|Voiceover-mathml-example-6]], [[File:Voiceover-mathml-example-7.wav|thumb|Voiceover-mathml-example-7]] |
| * [[Branko Grünbaum]], Uniform tilings of 3-space. [[Geombinatorics]] 4(1994), 49 - 56.
| | ** [https://commons.wikimedia.org/wiki/File:MathPlayer-Audio-Windows7-InternetExplorer.ogg Internet Explorer + MathPlayer (audio)] |
| * [[Norman Johnson (mathematician)|Norman Johnson]] ''Uniform Polytopes'', Manuscript (1991)
| | ** [https://commons.wikimedia.org/wiki/File:MathPlayer-SynchronizedHighlighting-WIndows7-InternetExplorer.png Internet Explorer + MathPlayer (synchronized highlighting)] |
| * [[Coxeter|Coxeter, H.S.M.]] ''[[Regular Polytopes (book)|Regular Polytopes]]'', (3rd edition, 1973), Dover edition, ISBN 0-486-61480-8
| | ** [https://commons.wikimedia.org/wiki/File:MathPlayer-Braille-Windows7-InternetExplorer.png Internet Explorer + MathPlayer (braille)] |
| * '''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] | | ** NVDA+MathPlayer: [[File:Nvda-mathml-example-1.wav|thumb|Nvda-mathml-example-1]], [[File:Nvda-mathml-example-2.wav|thumb|Nvda-mathml-example-2]], [[File:Nvda-mathml-example-3.wav|thumb|Nvda-mathml-example-3]], [[File:Nvda-mathml-example-4.wav|thumb|Nvda-mathml-example-4]], [[File:Nvda-mathml-example-5.wav|thumb|Nvda-mathml-example-5]], [[File:Nvda-mathml-example-6.wav|thumb|Nvda-mathml-example-6]], [[File:Nvda-mathml-example-7.wav|thumb|Nvda-mathml-example-7]]. |
| ** (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) | | ** Orca: There is ongoing work, but no support at all at the moment [[File:Orca-mathml-example-1.wav|thumb|Orca-mathml-example-1]], [[File:Orca-mathml-example-2.wav|thumb|Orca-mathml-example-2]], [[File:Orca-mathml-example-3.wav|thumb|Orca-mathml-example-3]], [[File:Orca-mathml-example-4.wav|thumb|Orca-mathml-example-4]], [[File:Orca-mathml-example-5.wav|thumb|Orca-mathml-example-5]], [[File:Orca-mathml-example-6.wav|thumb|Orca-mathml-example-6]], [[File:Orca-mathml-example-7.wav|thumb|Orca-mathml-example-7]]. |
| ** (Paper 24) H.S.M. Coxeter, ''Regular and Semi-Regular Polytopes III'', [Math. Zeit. 200 (1988) 3-45] | | ** From our testing, ChromeVox and JAWS are not able to read the formulas generated by the MathML mode. |
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| {{Honeycombs}}
| | ==Test pages == |
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| [[Category:Honeycombs (geometry)]] | | To test the '''MathML''', '''PNG''', and '''source''' rendering modes, please go to one of the following test pages: |
| [[Category:Polytopes]] | | *[[Displaystyle]] |
| | *[[MathAxisAlignment]] |
| | *[[Styling]] |
| | *[[Linebreaking]] |
| | *[[Unique Ids]] |
| | *[[Help:Formula]] |
| | |
| | *[[Inputtypes|Inputtypes (private Wikis only)]] |
| | *[[Url2Image|Url2Image (private Wikis only)]] |
| | ==Bug reporting== |
| | If you find any bugs, please report them at [https://bugzilla.wikimedia.org/enter_bug.cgi?product=MediaWiki%20extensions&component=Math&version=master&short_desc=Math-preview%20rendering%20problem Bugzilla], or write an email to math_bugs (at) ckurs (dot) de . |
