E6 Honeycomb

In geometry, the 2₂₂ honeycomb is a uniform tessellation of the six-dimensional Euclidean space. It can be represented by the Schläfli symbol . 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, ''t''₂, .

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

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.

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

The E₆^* lattice (or E₆³) with [3^2,2,2 symmetry. The Voronoi cell">[3<sup>2,2,2<_sup>.html" ;"title="[3^2,2,2"> rectified 1₂₂ polytope, and the Voronoi tessellation">^2,2,2 symmetry. The Voronoi cell of the E₆^* lattice is the Rectified 1 22 polytope">rectified 1₂₂ polytope, and the bitruncated 2₂₂ honeycomb.The Voronoi Cells of the E6* and E7* Lattices
, Edward Pervin 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 $_6$ group is related to the $_4$ by a geometric folding, so this honeycomb can be projected into the 4-dimensional Coxeter-Dynkin diagram#Geometric folding">folding, so this honeycomb can be projected into the 4-dimensional 16-cell honeycomb.

Related honeycombs

The 2₂₂ honeycomb is one of 127 uniform honeycombs (39 unique) with $_6$ symmetry. 24 of them have doubled symmetry 3,3,3^2,2 with 2 equally ringed branches, and 7 have sextupled (3 !) symmetry [3^2,2,2 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">Face-tra.html" ;"title="#Bitruncated 2 22 honeycomb">birectified 2₂₂ are Face-transitive#Related terms">isotopic, with only one type of facet: 2₂₁, and Rectified 1 22 polytope">rectified 1₂₂ polytopes respectively.

Birectified 2₂₂ honeycomb

The birectified 2₂₂ honeycomb , has rectified 1 22 polytope facets, , and a proprism ×× 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 ××, .

Removing a node on the end of one of the 3-node branches leaves the 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.

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

Notes

References

* Coxeter ''The Beauty of Geometry: Twelve Essays'', Dover Publications, 1999, (Chapter 3: Wythoff's Construction for Uniform Polytopes)
* Coxeter ''Regular Polytopes'' (1963), Macmillan Company
** ''Regular Polytopes'', Third edition, (1973), Dover edition, (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,
GoogleBook
** (Paper 24) H.S.M. Coxeter, ''Regular and Semi-Regular Polytopes III'', ath. Zeit. 200 (1988) 3–45* R. T. Worley, ''The Voronoi Region of E6*''. J. Austral. Math. Soc. Ser. A, 43 (1987), 268-278.
* p125-126, 8.3 The 6-dimensional lattices: E6 and E6*
*
*

{{Honeycombs

7-polytopes