simplicial honeycomb

class=wikitable width=280 align=right style="margin-left:1em" ! ${\tilde{A}}_2$ ! ${\tilde{A}}_3$
Triangular tiling !Tetrahedral-octahedral honeycomb
120px With red and yellow equilateral triangles \|160px With cyan and yellow tetrahedra, and red rectified tetrahedra (octahedra)
{{CDD\|node_1\|split1\|branch}} !{{CDD\|node_1\|split1\|nodes\|split2\|node}}

class=wikitable width=280 align=right style="margin-left:1em"

! ${\tilde{A}}_2$

! ${\tilde{A}}_3$

120px
With red and yellow equilateral triangles

|160px
With cyan and yellow tetrahedra, and red rectified tetrahedra (octahedra)

!{{CDD|node_1|split1|nodes|split2|node}}

In geometry, the simplicial honeycomb (or {{mvar|n}}-simplex honeycomb) is a dimensional infinite series of honeycombs, based on the ${\tilde{A}}_n$ affine Coxeter group symmetry. It is represented by a Coxeter-Dynkin diagram as a cyclic graph of {{math|n + 1}} nodes with one node ringed. It is composed of {{mvar|n}}-simplex facets, along with all rectified {{mvar|n}}-simplices. It can be thought of as an {{mvar|n}}-dimensional hypercubic honeycomb that has been subdivided along all hyperplanes $x+y+\cdots\in\mathbb{Z}$ , then stretched along its main diagonal until the simplices on the ends of the hypercubes become regular. The vertex figure of an {{mvar|n}}-simplex honeycomb is an expanded {{mvar|n}}-simplex.

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.

By dimension

class="wikitable"

!height=30|n

! ${\tilde{A}}_{2+}$

!Tessellation

!Vertex figure

!Facets per vertex figure

!Vertices per vertex figure

!Edge figure

align=center

| ${\tilde{A}}_1$

|80px
Apeirogon
{{CDD|node_1|infin|node}}

|Line segment
{{CDD|node_1}}

|Point

align=center

| ${\tilde{A}}_2$

|80px
Triangular tiling
2-simplex honeycomb
{{CDD|node_1|split1|branch}}

|80px
Hexagon
(Truncated triangle)
{{CDD|node_1|3|node_1}}

|3+3 triangles

|Line segment

align=center

| ${\tilde{A}}_3$

|80px
Tetrahedral-octahedral honeycomb
3-simplex honeycomb
{{CDD|node_1|split1|nodes|split2|node}}

|80px
Cuboctahedron
(Cantellated tetrahedron)
{{CDD|node_1|3|node|3|node_1}}

|4+4 tetrahedron
6 rectified tetrahedra

|12

|60px
Rectangle

align=center

| ${\tilde{A}}_4$

|4-simplex honeycomb
{{CDD|node_1|split1|nodes|3ab|branch}}

|80px
Runcinated 5-cell
{{CDD|node_1|3|node|3|node|3|node_1}}

|5+5 5-cells
10+10 rectified 5-cells

|20

|60px
Triangular antiprism

align=center

| ${\tilde{A}}_5$

|5-simplex honeycomb
{{CDD|node_1|split1|nodes|3ab|nodes|split2|node}}

|80px
Stericated 5-simplex
{{CDD|node_1|3|node|3|node|3|node|3|node_1}}

|6+6 5-simplex
15+15 rectified 5-simplex
20 birectified 5-simplex

|30

|60px
Tetrahedral antiprism

align=center

| ${\tilde{A}}_6$

|6-simplex honeycomb
{{CDD|node_1|split1|nodes|3ab|nodes|3ab|branch}}

|80px
Pentellated 6-simplex
{{CDD|node_1|3|node|3|node|3|node|3|node|3|node_1}}

|7+7 6-simplex
21+21 rectified 6-simplex
35+35 birectified 6-simplex

|42

|4-simplex antiprism

align=center

| ${\tilde{A}}_7$

|7-simplex honeycomb
{{CDD|node_1|split1|nodes|3ab|nodes|3ab|nodes|split2|node}}

|80px
Hexicated 7-simplex
{{CDD|node_1|3|node|3|node|3|node|3|node|3|node|3|node_1}}

|8+8 7-simplex
28+28 rectified 7-simplex
56+56 birectified 7-simplex
70 trirectified 7-simplex

|56

|5-simplex antiprism

align=center

| ${\tilde{A}}_8$

|8-simplex honeycomb
{{CDD|node_1|split1|nodes|3ab|nodes|3ab|nodes|3ab|branch}}

|80px
Heptellated 8-simplex
{{CDD|node_1|3|node|3|node|3|node|3|node|3|node|3|node|3|node_1}}

|9+9 8-simplex
36+36 rectified 8-simplex
84+84 birectified 8-simplex
126+126 trirectified 8-simplex

|72

|6-simplex antiprism

align=center

| ${\tilde{A}}_9$

|9-simplex honeycomb
{{CDD|node_1|split1|nodes|3ab|nodes|3ab|nodes|3ab|nodes|split2|node}}

|80px
Octellated 9-simplex
{{CDD|node_1|3|node|3|node|3|node|3|node|3|node|3|node|3|node|3|node_1}}

|10+10 9-simplex
45+45 rectified 9-simplex
120+120 birectified 9-simplex
210+210 trirectified 9-simplex
252 quadrirectified 9-simplex

|90

|7-simplex antiprism

align=center

|10

| ${\tilde{A}}_{10}$

|10-simplex honeycomb
{{CDD|node_1|split1|nodes|3ab|nodes|3ab|nodes|3ab|nodes|3ab|branch}}

|80px
Ennecated 10-simplex
{{CDD|node_1|3|node|3|node|3|node|3|node|3|node|3|node|3|node|3|node|3|node_1}}

|11+11 10-simplex
55+55 rectified 10-simplex
165+165 birectified 10-simplex
330+330 trirectified 10-simplex
462+462 quadrirectified 10-simplex

|110

|8-simplex antiprism

align=center

|11

| ${\tilde{A}}_{11}$

|11-simplex honeycomb

|...

Projection by folding

The (2n−1)-simplex honeycombs and 2n-simplex honeycombs can be projected into the n-dimensional hypercubic honeycomb by a geometric folding operation that maps two pairs of mirrors into each other, sharing the same vertex arrangement:

class=wikitable
${\tilde{A}}_2$ \|{{CDD\|node_1\|split1\|branch}} ! ${\tilde{A}}_4$ \|{{CDD\|node_1\|split1\|nodes\|3ab\|branch}} ! ${\tilde{A}}_6$ \|{{CDD\|node_1\|split1\|nodes\|3ab\|nodes\|3ab\|branch}} ! ${\tilde{A}}_8$ \|{{CDD\|node_1\|split1\|nodes\|3ab\|nodes\|3ab\|nodes\|3ab\|branch}} ! ${\tilde{A}}_{10}$ \|{{CDD\|node_1\|split1\|nodes\|3ab\|nodes\|3ab\|nodes\|3ab\|nodes\|3ab\|branch}} \|...
${\tilde{A}}_3$ \|{{CDD\|nodes_10r\|splitcross\|nodes}} ! ${\tilde{A}}_3$ \|{{CDD\|node_1\|split1\|nodes\|split2\|node}} ! ${\tilde{A}}_5$ \|{{CDD\|node_1\|split1\|nodes\|3ab\|nodes\|split2\|node}} ! ${\tilde{A}}_7$ \|{{CDD\|node_1\|split1\|nodes\|3ab\|nodes\|3ab\|nodes\|split2\|node}} ! ${\tilde{A}}_9$ \|{{CDD\|node_1\|split1\|nodes\|3ab\|nodes\|3ab\|nodes\|3ab\|nodes\|split2\|node}} \|...
${\tilde{C}}_1$ \|{{CDD\|node_1\|infin\|node}} ! ${\tilde{C}}_2$ \|{{CDD\|node_1\|4\|node\|4\|node}} ! ${\tilde{C}}_3$ \|{{CDD\|node_1\|4\|node\|3\|node\|4\|node}} ! ${\tilde{C}}_4$ \|{{CDD\|node_1\|4\|node\|3\|node\|3\|node\|4\|node}} ! ${\tilde{C}}_5$ \|{{CDD\|node_1\|4\|node\|3\|node\|3\|node\|3\|node\|4\|node}} \|...

class=wikitable

{\tilde{A}}_2

|{{CDD|node_1|split1|branch}}

! ${\tilde{A}}_4$

|{{CDD|node_1|split1|nodes|3ab|branch}}

! ${\tilde{A}}_6$

|{{CDD|node_1|split1|nodes|3ab|nodes|3ab|branch}}

! ${\tilde{A}}_8$

|{{CDD|node_1|split1|nodes|3ab|nodes|3ab|nodes|3ab|branch}}

! ${\tilde{A}}_{10}$

|{{CDD|node_1|split1|nodes|3ab|nodes|3ab|nodes|3ab|nodes|3ab|branch}}

|...

{\tilde{A}}_3

|{{CDD|nodes_10r|splitcross|nodes}}

! ${\tilde{A}}_3$

|{{CDD|node_1|split1|nodes|split2|node}}

! ${\tilde{A}}_5$

|{{CDD|node_1|split1|nodes|3ab|nodes|split2|node}}

! ${\tilde{A}}_7$

|{{CDD|node_1|split1|nodes|3ab|nodes|3ab|nodes|split2|node}}

! ${\tilde{A}}_9$

|{{CDD|node_1|split1|nodes|3ab|nodes|3ab|nodes|3ab|nodes|split2|node}}

|...

{\tilde{C}}_1

|{{CDD|node_1|infin|node}}

! ${\tilde{C}}_2$

|{{CDD|node_1|4|node|4|node}}

! ${\tilde{C}}_3$

|{{CDD|node_1|4|node|3|node|4|node}}

! ${\tilde{C}}_4$

|{{CDD|node_1|4|node|3|node|3|node|4|node}}

! ${\tilde{C}}_5$

|{{CDD|node_1|4|node|3|node|3|node|3|node|4|node}}

|...

Kissing number

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. This represents the highest kissing number for 2 and 3 dimensions, but falls 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 a cuboctahedral configuration. For 4 to 8 dimensions, the kissing numbers are 20, 30, 42, 56, and 72 spheres, while the greatest solutions are 24, 40, 72, 126, and 240 spheres respectively.

References

George Olshevsky, Uniform Panoploid Tetracombs, Manuscript (2006) (Complete list of 11 convex uniform tilings, 28 convex uniform honeycombs, and 143 convex uniform tetracombs)
Branko Grünbaum, Uniform tilings of 3-space. Geombinatorics 4(1994), 49 - 56.
Norman Johnson Uniform Polytopes, Manuscript (1991)
Coxeter, H.S.M. Regular Polytopes, (3rd edition, 1973), Dover edition, {{isbn|0-486-61480-8}}
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]

Category:Honeycombs (geometry)

Category:Uniform polytopes

simplicial honeycomb

By dimension

Projection by folding

Kissing number

See also

References