Simplectic honeycomb

${\tilde{A}}_2$	${\tilde{A}}_3$
Triangular tiling	Tetrahedral-octahedral honeycomb
With red and yellow equilateral triangles	160px 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	${\tilde{A}}_3$	80px Tetrahedral-octahedral honeycomb 3-simplex honeycomb	80px Cuboctahedron (Cantellated tetrahedron)	4+4 tetrahedron 6 rectified tetrahedra	12	Rectangle
4	${\tilde{A}}_4$	4-simplex honeycomb	80px Runcinated 5-cell	5+5 5-cells 10+10 rectified 5-cells	20	60px Triangular antiprism
5	${\tilde{A}}_5$	5-simplex honeycomb	80px 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	80px 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	80px 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	80px 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.

See also

• Hypercubic honeycomb
• Alternated hypercubic honeycomb
• Quarter hypercubic honeycomb
• Truncated simplectic honeycomb
• Omnitruncated simplectic honeycomb

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 [1]
  • (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]