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2₂₂ honeycomb
(no image)
Type	Uniform tessellation
Coxeter symbol	2₂₂
Schläfli symbol	{3,3,3^2,2}
Coxeter diagram
6-face type	2₂₁
5-face types	2₁₁ {3⁴}
4-face type	{3³}
Cell type	{3,3}
Face type	{3}
Face figure	{3}×{3} duoprism
Edge figure	{3^2,2}
Vertex figure	1₂₂
Coxeter group	${\tilde {E}}_{6}$ , [[3,3,3^2,2]]
Properties	vertex-transitive, facet-transitive

In geometry, the 2₂₂ honeycomb is a uniform tessellation of the six-dimensional Euclidean space. It can be represented by the Schläfli symbol {3,3,3^2,2}. It is constructed from 2₂₁ facets and has a 1₂₂ vertex figure, with 54 2₂₁ polytopes around every vertex.

Its vertex arrangement is the E₆ lattice, and the root system of the E₆ Lie group so it can also be called the E₆ honeycomb.

Construction

It is created by a Wythoff construction upon a set of 7 hyperplane mirrors in 6-dimensional space.

The facet information can be extracted from its Coxeter–Dynkin diagram, .

Removing a node on the end of one of the 2-node branches leaves the 2₂₁, its only facet type,

The vertex figure is determined by removing the ringed node and ringing the neighboring node. This makes 1₂₂, .

The edge figure is the vertex figure of the vertex figure, here being a birectified 5-simplex, t₂{3⁴}, .

The face figure is the vertex figure of the edge figure, here being a triangular duoprism, {3}×{3}, .

Kissing number

Each vertex of this tessellation is the center of a 5-sphere in the densest known packing in 6 dimensions, with kissing number 72, represented by the vertices of its vertex figure 1₂₂.

E₆ lattice

The 2₂₂ honeycomb's vertex arrangement is called the E₆ lattice.^[1]

The E₆² lattice, with [[3,3,3^2,2]] symmetry, can be constructed by the union of two E₆ lattices:

∪

The E₆^* lattice^[2] (or E₆³) with [[3,3^2,2,2]] symmetry. The Voronoi cell of the E₆^* lattice is the rectified 1₂₂ polytope, and the Voronoi tessellation is a bitruncated 2₂₂ honeycomb.^[3] It is constructed by 3 copies of the E₆ lattice vertices, one from each of the three branches of the Coxeter diagram.

∪

= dual to

.

Geometric folding

The ${\tilde {E}}_{6}$ group is related to the ${\tilde {F}}_{4}$ by a geometric folding, so this honeycomb can be projected into the 4-dimensional 16-cell honeycomb.

${\tilde {E}}_{6}$	${\tilde {F}}_{4}$

{3,3,3^2,2}	{3,3,4,3}

Related honeycombs

The 2₂₂ honeycomb is one of 127 uniform honeycombs (39 unique) with ${\tilde {E}}_{6}$ symmetry. 24 of them have doubled symmetry [[3,3,3^2,2]] with 2 equally ringed branches, and 7 have sextupled (3!) symmetry [[3,3^2,2,2]] with identical rings on all 3 branches. There are no regular honeycombs in the family since its Coxeter diagram a nonlinear graph, but the 2₂₂ and birectified 2₂₂ are isotopic, with only one type of facet: 2₂₁, and rectified 1₂₂ polytopes respectively.

Symmetry	Order	Honeycombs
[3^2,2,2]	Full	8: , , , , , , , .
[[3,3,3^2,2]]	×2	24: , , , , , , , , , , , , , , , , , , , , , , , .
[[3,3^2,2,2]]	×6	7: , , , , , , .

Birectified 2₂₂ honeycomb

Birectified 2₂₂ honeycomb
(no image)
Type	Uniform tessellation
Coxeter symbol	0₂₂₂
Schläfli symbol	{3^2,2,2}
Coxeter diagram
6-face type	0₂₂₁
5-face types	0₂₂ 0₂₁₁
4-face type	0₂₁ 24-cell 0₁₁₁
Cell type	Tetrahedron 0₂₀ Octahedron 0₁₁
Face type	Triangle 0₁₀
Vertex figure	Proprism {3}×{3}×{3}
Coxeter group	6× ${\tilde {E}}_{6}$ , [[3,3^2,2,2]]
Properties	vertex-transitive, facet-transitive

The birectified 2₂₂ honeycomb , has rectified 1 22 polytope facets, , and a proprism {3}×{3}×{3} vertex figure.

Its facets are centered on the vertex arrangement of E₆^* lattice, as:

∪

Construction

The facet information can be extracted from its Coxeter–Dynkin diagram, .

The vertex figure is determined by removing the ringed node and ringing the neighboring node. This makes a proprism {3}×{3}×{3}, .

Removing a node on the end of one of the 3-node branches leaves the rectified 1₂₂, its only facet type, .

Removing a second end node defines 2 types of 5-faces: birectified 5-simplex, 0₂₂ and birectified 5-orthoplex, 0₂₁₁.

Removing a third end node defines 2 types of 4-faces: rectified 5-cell, 0₂₁, and 24-cell, 0₁₁₁.

Removing a fourth end node defines 2 types of cells: octahedron, 0₁₁, and tetrahedron, 0₂₀.

k₂₂ polytopes

The 2₂₂ honeycomb, is fourth in a dimensional series of uniform polytopes, expressed by Coxeter as k₂₂ series. The final is a paracompact hyperbolic honeycomb, 3₂₂. Each progressive uniform polytope is constructed from the previous as its vertex figure.

k₂₂ figures in n dimensions
Space	Finite			Euclidean	Hyperbolic
n	4	5	6	7	8
Coxeter group	A₂A₂	E₆	${\tilde {E}}_{6}$ =E₆⁺	${\bar {T}}_{7}$ =E₆⁺⁺
Coxeter diagram
Symmetry	[[3^2,2,-1]]	[[3^2,2,0]]	[[3^2,2,1]]	[[3^2,2,2]]	[[3^2,2,3]]
Order	72	1440	103,680	∞
Graph				∞	∞
Name	−1₂₂	0₂₂	1₂₂	2₂₂	3₂₂

The 2₂₂ honeycomb is third in another dimensional series 2_2k.

2_2k figures of n dimensions
Space	Finite			Euclidean	Hyperbolic
n	4	5	6	7	8
Coxeter group	A₂A₂	A₅	E₆	${\tilde {E}}_{6}$ =E₆⁺	E₆⁺⁺
Coxeter diagram
Graph				∞	∞
Name	2_2,-1	2₂₀	2₂₁	2₂₂	2₂₃

Notes

^ "The Lattice E6".
^ "The Lattice E6".
^ The Voronoi Cells of the E6* and E7* Lattices Archived 2016-01-30 at the Wayback Machine, Edward Pervin

References

Coxeter The Beauty of Geometry: Twelve Essays, Dover Publications, 1999, ISBN 978-0-486-40919-1 (Chapter 3: Wythoff's Construction for Uniform Polytopes)
Coxeter Regular Polytopes (1963), Macmillan Company
- Regular Polytopes, Third edition, (1973), Dover edition, ISBN 0-486-61480-8 (Chapter 5: The Kaleidoscope)
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] GoogleBook
- (Paper 24) H.S.M. Coxeter, Regular and Semi-Regular Polytopes III, [Math. Zeit. 200 (1988) 3–45]
R. T. Worley, The Voronoi Region of E6*. J. Austral. Math. Soc. Ser. A, 43 (1987), 268–278.
Conway, John H.; Sloane, Neil J. A. (1998). Sphere Packings, Lattices and Groups ((3rd ed.) ed.). New York: Springer-Verlag. ISBN 0-387-98585-9. p125-126, 8.3 The 6-dimensional lattices: E6 and E6*
Klitzing, Richard. "6D Hexacombs x3o3o3o3o *c3o3o - jakoh".
Klitzing, Richard. "6D Hexacombs o3o3x3o3o *c3o3o - ramoh".

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