2 22 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, {{CDD|node_1|3|node|3|node|split1|nodes|3ab|nodes}}.

Removing a node on the end of one of the 2-node branches leaves the 2₂₁, its only facet type, {{CDD|node_1|3|node|3|node|split1|nodes|3a|nodea}}

The vertex figure is determined by removing the ringed node and ringing the neighboring node. This makes 1₂₂, {{CDD|node_1||3|node|split1|nodes|3ab|nodes}}.

The edge figure is the vertex figure of the vertex figure, here being a birectified 5-simplex, t₂{3⁴}, {{CDD|node_1|split1|nodes|3ab|nodes}}.

The face figure is the vertex figure of the edge figure, here being a triangular duoprism, {3}×{3}, {{CDD|nodes_11|3ab|nodes}}.

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.{{Cite web|url=http://www.math.rwth-aachen.de/~Gabriele.Nebe/LATTICES/E6.html|title = The Lattice E6}}

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

: {{CDD|node|3|node|3|node|split1|nodes|3ab|nodes_10l}} ∪ {{CDD|node|3|node|3|node|split1|nodes|3ab|nodes_01l}}

The E₆^* lattice{{Cite web|url=http://www.math.rwth-aachen.de/~Gabriele.Nebe/LATTICES/Es6.html|title=The Lattice E6*}} (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.[http://home.digital.net/~pervin/publications/vermont.html The Voronoi Cells of the E6* and E7* Lattices] {{Webarchive|url=https://web.archive.org/web/20160130193811/http://home.digital.net/~pervin/publications/vermont.html |date=2016-01-30 }}, Edward Pervin It is constructed by 3 copies of the E₆ lattice vertices, one from each of the three branches of the Coxeter diagram.

: {{CDD|node_1|3|node|3|node|split1|nodes|3ab|nodes}} ∪ {{CDD|node|3|node|3|node|split1|nodes|3ab|nodes_10l}} ∪ {{CDD|node|3|node|3|node|split1|nodes|3ab|nodes_01l}} = dual to {{CDD|node|3|node|3|node_1|split1|nodes|3ab|nodes}}.

The $\{\backslash tilde\{E\}\}\_6$ group is related to the $\{\backslash tilde\{F\}\}\_4$ 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 $\{\backslash tilde\{E\}\}\_6$ 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 {{CDD|node|3|node|3|node_1|split1|nodes|3ab|nodes}}, has rectified 1 22 polytope facets, {{CDD|node|3|node_1|split1|nodes|3ab|nodes}}, and a proprism {3}×{3}×{3} vertex figure.

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

: {{CDD|node_1|3|node|3|node|split1|nodes|3ab|nodes}} ∪ {{CDD|node|3|node|3|node|split1|nodes|3ab|nodes_10l}} ∪ {{CDD|node|3|node|3|node|split1|nodes|3ab|nodes_01l}}

The facet information can be extracted from its Coxeter–Dynkin diagram, {{CDD|node|3|node|3|node_1|split1|nodes|3ab|nodes}}.

The vertex figure is determined by removing the ringed node and ringing the neighboring node. This makes a proprism {3}×{3}×{3}, {{CDD|node|3|node_1|2|nodes_11|3ab|nodes}}.

Removing a node on the end of one of the 3-node branches leaves the rectified 1₂₂, its only facet type, {{CDD|node|3|node_1|split1|nodes|3ab|nodes}}.

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.

{{clear}}

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.

{{k 22 polytopes}}

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

{{2 2k polytopes}}

{{reflist}}

• 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, [https://www.wiley.com/en-us/Kaleidoscopes-p-9780471010036 wiley.com], {{isbn|978-0-471-01003-6}}, [https://books.google.com/books?id=fUm5Mwfx8rAC&dq=Coxeter&pg=PP1 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.
• {{Cite book | first = John H. | last = Conway | authorlink = John Horton Conway | author2 = Sloane, Neil J. A. | year = 1998 | title = Sphere Packings, Lattices and Groups | edition = (3rd ed.) | publisher = Springer-Verlag | location = New York | isbn = 0-387-98585-9 | url-access = registration | url = https://archive.org/details/spherepackingsla0000conw_b8u0 | author2-link = Neil Sloane }} pp. 125–126, 8.3 The 6-dimensional lattices: E6 and E6*
• {{KlitzingPolytopes|flat.htm#6D|6D Hexacombs|x3o3o3o3o *c3o3o - jakoh}}
• {{KlitzingPolytopes|flat.htm#6D|6D Hexacombs|o3o3x3o3o *c3o3o - ramoh}}

{{Honeycombs}}

Category:7-polytopes