2 ₂₂ honeycomb

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.

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}, .

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.

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 .

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

The 2₂₂ honeycomb is one of 127 uniform honeycombs (39 unique) with ${\displaystyle {\tilde {E}}_{6}}${\displaystyle {\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.

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

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

∪ ∪

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 ×ばつ{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.

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.

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• 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
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• 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".

• 7-polytopes

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