1. Construction

2. Kissing number

3. E₆ lattice

4. Geometric folding

5. Related honeycombs
     Birectified 2₂₂ honeycomb   Construction   k₂₂ polytopes  

6. Notes

7. References

222 honeycomb
(no image)
Type	Uniform tessellation
Coxeter symbol	222
Schläfli symbol	{3,3,32,2}
Coxeter diagram	node_1|3|node|3|node|split1|nodes|3ab|nodes}}
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	{32,2}
Vertex figure	122
Coxeter group	, 2,2">3,3,32,2
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, {{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}}.

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 2,2">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^[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.

{{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}}.

Geometric folding

The group is related to the by a geometric folding, so this honeycomb can be projected into the 4-dimensional 16-cell honeycomb.

node_1|3|node|3|node|split1|nodes|3ab|nodes}}	node_1|3|node|3|node|4|node|3|node}}
{3,3,32,2}	{3,3,4,3}

Related honeycombs

The 2₂₂ honeycomb is one of 127 uniform honeycombs (39 unique) with symmetry. 24 of them have doubled symmetry 2,2">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.

Birectified 2₂₂ honeycomb

Birectified 222 honeycomb
(no image)
Type	Uniform tessellation
Coxeter symbol	0222
Schläfli symbol	{32,2,2}
Coxeter diagram	node|3|node|3|node_1|split1|nodes|3ab|nodes}}
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×, [3[32,2,2]]
Properties	vertex-transitive, facet-transitive

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}}

Construction

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 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.

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, {{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}}   [https://books.google.com/books?id=fUm5Mwfx8rAC&lpg=PP1&dq=Coxeter&pg=PP1#v=onepage&q&f=false 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. (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}} p125-126, 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}}

• Slavery in Greece
• Slavery in Haiti
• Slavery in Hispaniola
• Slavery in Iceland
• Slavery in Illinois
• Slavery in International Law
• Slavery in international law
• Slavery in Ireland
• Slavery in Italy
• Slavery in Jubaland
• Slavery in Kansas
• Slavery in La Española
• Slavery in Latin America
• Slavery in Lithuania
• Slavery in Louisiana
• Slavery in Madras Presidency
• Slavery in Mali
• Slavery in Maryland
• Slavery in Missouri
• Slavery in modern Africa
• Slavery in Nebraska
• Slavery in Niger
• Slavery in North America
• Slavery in north america
• Slavery in North Korea