Simplicial honeycomb

Last updated November 26, 2024

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

In geometry, the simplicial honeycomb (or $n$ -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 $n + 1$ 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 $x+y+\cdots \in \mathbb {Z}$ , then stretched along its main diagonal until the simplices on the ends of the hypercubes become regular. The vertex figure of an $n$ -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	Line segment	2	2	Point
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. 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.

Related Research Articles

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<span class="mw-page-title-main">Birectified 16-cell honeycomb</span>

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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]

v t e Fundamental convex regular and uniform honeycombs in dimensions 2–9
Space	Family	${\tilde {A}}_{n-1}$	${\tilde {C}}_{n-1}$	${\tilde {B}}_{n-1}$	${\tilde {D}}_{n-1}$	${\tilde {G}}_{2}$ / ${\tilde {F}}_{4}$ / ${\tilde {E}}_{n-1}$
E²	Uniform tiling	0_[3]	δ₃	hδ₃	qδ₃	Hexagonal
E³	Uniform convex honeycomb	0_[4]	δ₄	hδ₄	qδ₄
E⁴	Uniform 4-honeycomb	0_[5]	δ₅	hδ₅	qδ₅	24-cell honeycomb
E⁵	Uniform 5-honeycomb	0_[6]	δ₆	hδ₆	qδ₆
E⁶	Uniform 6-honeycomb	0_[7]	δ₇	hδ₇	qδ₇	2₂₂
E⁷	Uniform 7-honeycomb	0_[8]	δ₈	hδ₈	qδ₈	1₃₃ • 3₃₁
E⁸	Uniform 8-honeycomb	0_[9]	δ₉	hδ₉	qδ₉	1₅₂ • 2₅₁ • 5₂₁
E⁹	Uniform 9-honeycomb	0_[10]	δ₁₀	hδ₁₀	qδ₁₀
E¹⁰	Uniform 10-honeycomb	0_[11]	δ₁₁	hδ₁₁	qδ₁₁
E^n-1	Uniform (n-1)-honeycomb	0_[n]	δ_n	hδ_n	qδ_n	1_k2 • 2_k1 • k₂₁

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