Simplectic honeycomb

${\tilde {A}}_{2}$	${\tilde {A}}_{3}$
Triangular tiling	Tetrahedral-octahedral honeycomb
With red and yellow equilateral triangles	With cyan and yellow tetrahedra, and red rectified tetrahedra (octahedron)

In geometry, the simplectic honeycomb (or n-simplex honeycomb) is a dimensional infinite series of honeycombs, based on the ${\tilde {A}}_{n}$ affine Coxeter group symmetry. It is given a Schläfli symbol {3^[n+1]}, 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 rectified n-simplices. It can be thought of as an n-dimensional hypercubic honeycomb that has been subdivided along all hyperplanes $x+y+...\in \mathbb {Z}$ , then stretched along its main diagonal until the simplices on the ends of the hypercubes become regular. The vertex figure of an n-simplex honeycomb is an expanded n-simplex.

In 2 dimensions, the honeycomb represents the triangular tiling, with Coxeter graph filling the plane with alternately colored triangles. In 3 dimensions it represents the tetrahedral-octahedral honeycomb, with Coxeter graph filling space with alternately tetrahedral and octahedral cells. In 4 dimensions it is called the 5-cell honeycomb, with Coxeter graph , with 5-cell and rectified 5-cell facets. In 5 dimensions it is called the 5-simplex honeycomb, with Coxeter graph , 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 , filling space by 6-simplex, rectified 6-simplex, and birectified 6-simplex facets.

By dimension

n	${{\tilde {A}}}_{{2+}}$	Tessellation	Vertex figure	Facets per vertex figure	Vertices per vertex figure	Edge figure
1	${\tilde {A}}_{1}$	Apeirogon		1	2	-
2	${\tilde {A}}_{2}$	Triangular tiling 2-simplex honeycomb	Hexagon (Truncated triangle)	3+3 triangles	6	Line segment
3	${\tilde {A}}_{3}$	Tetrahedral-octahedral honeycomb 3-simplex honeycomb	Cuboctahedron (Cantellated tetrahedron)	4+4 tetrahedron 6 rectified tetrahedra	12	Rectangle
4	${{\tilde {A}}}_{4}$	4-simplex honeycomb	Runcinated 5-cell	5+5 5-cells 10+10 rectified 5-cells	20	Triangular antiprism
5	${\tilde {A}}_{5}$	5-simplex honeycomb	Stericated 5-simplex	6+6 5-simplex 15+15 rectified 5-simplex 20 birectified 5-simplex	30	Tetrahedral antiprism
6	${\tilde {A}}_{6}$	6-simplex honeycomb	Pentellated 6-simplex	7+7 6-simplex 21+21 rectified 6-simplex 35+35 birectified 6-simplex	42	4-simplex antiprism
7	${\tilde {A}}_{7}$	7-simplex honeycomb	Hexicated 7-simplex	8+8 7-simplex 28+28 rectified 7-simplex 56+56 birectified 7-simplex 70 trirectified 7-simplex	56	5-simplex antiprism
8	${{\tilde {A}}}_{8}$	8-simplex honeycomb	Heptellated 8-simplex	9+9 8-simplex 36+36 rectified 8-simplex 84+84 birectified 8-simplex 126+126 trirectified 8-simplex	72	6-simplex antiprism
9	${{\tilde {A}}}_{9}$	9-simplex honeycomb	Octellated 9-simplex	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
10	${{\tilde {A}}}_{{10}}$	10-simplex honeycomb	Ennecated 10-simplex	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
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:

${\tilde {A}}_{2}$	${{\tilde {A}}}_{4}$	${\tilde {A}}_{6}$	${{\tilde {A}}}_{8}$	${{\tilde {A}}}_{{10}}$	...
${\tilde {A}}_{3}$	${\tilde {A}}_{3}$	${\tilde {A}}_{5}$	${\tilde {A}}_{7}$	${{\tilde {A}}}_{9}$	...
${\tilde {C}}_{1}$	${\tilde {C}}_{2}$	${\tilde {C}}_{3}$	${{\tilde {C}}}_{4}$	${{\tilde {C}}}_{5}$	...

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. 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 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
- (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]

Fundamental convex regular and uniform honeycombs in dimensions 3–10 (or 2-9)
Family	${\tilde {A}}_{n-1}$	${\tilde {C}}_{n-1}$	${\tilde {B}}_{n-1}$	${\tilde {D}}_{n-1}$	${\tilde {G}}_{2}$ / ${\tilde {F}}_{4}$ / ${\tilde {E}}_{n-1}$
Uniform tiling	{3^[3]}	δ₃	hδ₃	qδ₃	Hexagonal
Uniform convex honeycomb	{3^[4]}	δ₄	hδ₄	qδ₄
Uniform 5-honeycomb	{3^[5]}	δ₅	hδ₅	qδ₅	24-cell honeycomb
Uniform 6-honeycomb	{3^[6]}	δ₆	hδ₆	qδ₆
Uniform 7-honeycomb	{3^[7]}	δ₇	hδ₇	qδ₇	2₂₂
Uniform 8-honeycomb	{3^[8]}	δ₈	hδ₈	qδ₈	1₃₃ • 3₃₁
Uniform 9-honeycomb	{3^[9]}	δ₉	hδ₉	qδ₉	1₅₂ • 2₅₁ • 5₂₁
Uniform 10-honeycomb	{3^[10]}	δ₁₀	hδ₁₀	qδ₁₀
Uniform n-honeycomb	{3^[n]}	δ_n	hδ_n	qδ_n	1_k2 • 2_k1 • k₂₁

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Simplectic honeycomb

By dimension

Projection by folding

Kissing number

See also

References