2 22 honeycomb

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222 honeycomb
(no image)
Type Uniform tessellation
Coxeter symbol 222
Schläfli symbol {3,3,32,2}
Coxeter diagram CDel node 1.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.pngCDel split1.pngCDel nodes.pngCDel 3ab.pngCDel nodes.png
6-face type 221 E6 graph.svg
5-face types 2115-orthoplex.svg
{34}5-simplex t0.svg
4-face type {33}4-simplex t0.svg
Cell type {3,3}3-simplex t0.svg
Face type {3}2-simplex t0.svg
Face figure {3}×{3} duoprism
Edge figure t2{34} 5-simplex t2.svg
Vertex figure 122 Gosset 1 22 polytope.svg
Coxeter group {\tilde{E}}_6, [[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 {3,3,32,2}. 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[edit]

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, CDel node 1.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.pngCDel split1.pngCDel nodes.pngCDel 3ab.pngCDel nodes.png.

Removing a node on the end of one of the 2-node branches leaves the 221, its only facet type, CDel node 1.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.pngCDel split1.pngCDel nodes.pngCDel 3a.pngCDel nodea.png

The vertex figure is determined by removing the ringed node and ringing the neighboring node. This makes 122, CDel node 1.pngCDel 3.pngCDel node.pngCDel split1.pngCDel nodes.pngCDel 3ab.pngCDel nodes.png.

The edge figure is the vertex figure of the vertex figure, here being a birectified 5-simplex, t2{34}, CDel node 1.pngCDel split1.pngCDel nodes.pngCDel 3ab.pngCDel nodes.png.

The face figure is the vertex figure of the edge figure, here being a triangular duoprism, {3}×{3}, CDel nodes 11.pngCDel 3ab.pngCDel nodes.png.

Kissing number[edit]

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[edit]

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

The E62 lattice, with [[3,3,32,2]] symmetry, can be constructed by the union of two E6 lattices:

CDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.pngCDel split1.pngCDel nodes.pngCDel 3ab.pngCDel nodes 10l.pngCDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.pngCDel split1.pngCDel nodes.pngCDel 3ab.pngCDel nodes 01l.png

The E6* lattice[2] (or E63) with [3[32,2,2]] 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.

CDel node 1.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.pngCDel split1.pngCDel nodes.pngCDel 3ab.pngCDel nodes.pngCDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.pngCDel split1.pngCDel nodes.pngCDel 3ab.pngCDel nodes 10l.pngCDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.pngCDel split1.pngCDel nodes.pngCDel 3ab.pngCDel nodes 01l.png = dual to CDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node 1.pngCDel split1.pngCDel nodes.pngCDel 3ab.pngCDel nodes.png.

Related honeycombs[edit]

The 222 honeycomb is one of 127 uniform honeycombs (39 unique) with {\tilde{E}}_6 symmetry. 24 of them have doubled symmetry [[3,3,32,2]] with 2 equally ringed branches and, and 7 have sextupled (3!) symmetry [3[32,2,2]] 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
[32,2,2] Full

8: CDel node 1.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.pngCDel split1.pngCDel nodes 10lur.pngCDel 3ab.pngCDel nodes.png, CDel node 1.pngCDel 3.pngCDel node 1.pngCDel 3.pngCDel node.pngCDel split1.pngCDel nodes.pngCDel 3ab.pngCDel nodes 10l.png, CDel node 1.pngCDel 3.pngCDel node 1.pngCDel 3.pngCDel node.pngCDel split1.pngCDel nodes 10lur.pngCDel 3ab.pngCDel nodes.png, CDel node 1.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node 1.pngCDel split1.pngCDel nodes 10lur.pngCDel 3ab.pngCDel nodes.png, CDel node 1.pngCDel 3.pngCDel node 1.pngCDel 3.pngCDel node.pngCDel split1.pngCDel nodes 10lur.pngCDel 3ab.pngCDel nodes 01l.png, CDel node 1.pngCDel 3.pngCDel node 1.pngCDel 3.pngCDel node 1.pngCDel split1.pngCDel nodes.pngCDel 3ab.pngCDel nodes 10l.png, CDel node 1.pngCDel 3.pngCDel node 1.pngCDel 3.pngCDel node 1.pngCDel split1.pngCDel nodes 10lur.pngCDel 3ab.pngCDel nodes.png, CDel node 1.pngCDel 3.pngCDel node 1.pngCDel 3.pngCDel node 1.pngCDel split1.pngCDel nodes 10lur.pngCDel 3ab.pngCDel nodes 01l.png.

[[3,3,32,2]] ×2

24: CDel node 1.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.pngCDel split1.pngCDel nodes.pngCDel 3ab.pngCDel nodes.png, CDel node.pngCDel 3.pngCDel node 1.pngCDel 3.pngCDel node.pngCDel split1.pngCDel nodes.pngCDel 3ab.pngCDel nodes.png, CDel node 1.pngCDel 3.pngCDel node 1.pngCDel 3.pngCDel node.pngCDel split1.pngCDel nodes.pngCDel 3ab.pngCDel nodes.png, CDel node 1.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node 1.pngCDel split1.pngCDel nodes.pngCDel 3ab.pngCDel nodes.png, CDel node.pngCDel 3.pngCDel node 1.pngCDel 3.pngCDel node 1.pngCDel split1.pngCDel nodes.pngCDel 3ab.pngCDel nodes.png, CDel node 1.pngCDel 3.pngCDel node 1.pngCDel 3.pngCDel node 1.pngCDel split1.pngCDel nodes.pngCDel 3ab.pngCDel nodes.png,

CDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.pngCDel split1.pngCDel nodes 11.pngCDel 3ab.pngCDel nodes.png, CDel node 1.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.pngCDel split1.pngCDel nodes 11.pngCDel 3ab.pngCDel nodes.png, CDel node 1.pngCDel 3.pngCDel node 1.pngCDel 3.pngCDel node.pngCDel split1.pngCDel nodes 11.pngCDel 3ab.pngCDel nodes.png, CDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node 1.pngCDel split1.pngCDel nodes 11.pngCDel 3ab.pngCDel nodes.png, CDel node 1.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node 1.pngCDel split1.pngCDel nodes 11.pngCDel 3ab.pngCDel nodes.png, CDel node 1.pngCDel 3.pngCDel node 1.pngCDel 3.pngCDel node 1.pngCDel split1.pngCDel nodes 11.pngCDel 3ab.pngCDel nodes.png,

CDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.pngCDel split1.pngCDel nodes.pngCDel 3ab.pngCDel nodes 11.png, CDel node.pngCDel 3.pngCDel node 1.pngCDel 3.pngCDel node.pngCDel split1.pngCDel nodes.pngCDel 3ab.pngCDel nodes 11.png, CDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node 1.pngCDel split1.pngCDel nodes.pngCDel 3ab.pngCDel nodes 11.png, CDel node 1.pngCDel 3.pngCDel node 1.pngCDel 3.pngCDel node.pngCDel split1.pngCDel nodes.pngCDel 3ab.pngCDel nodes 11.png, CDel node.pngCDel 3.pngCDel node 1.pngCDel 3.pngCDel node 1.pngCDel split1.pngCDel nodes.pngCDel 3ab.pngCDel nodes 11.png, CDel node 1.pngCDel 3.pngCDel node 1.pngCDel 3.pngCDel node 1.pngCDel split1.pngCDel nodes.pngCDel 3ab.pngCDel nodes 11.png,

CDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.pngCDel split1.pngCDel nodes 11.pngCDel 3ab.pngCDel nodes 11.png, CDel node 1.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.pngCDel split1.pngCDel nodes 11.pngCDel 3ab.pngCDel nodes 11.png, CDel node.pngCDel 3.pngCDel node 1.pngCDel 3.pngCDel node.pngCDel split1.pngCDel nodes 11.pngCDel 3ab.pngCDel nodes 11.png, CDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node 1.pngCDel split1.pngCDel nodes 11.pngCDel 3ab.pngCDel nodes 11.png, CDel node 1.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node 1.pngCDel split1.pngCDel nodes 11.pngCDel 3ab.pngCDel nodes 11.png, CDel node.pngCDel 3.pngCDel node 1.pngCDel 3.pngCDel node 1.pngCDel split1.pngCDel nodes 11.pngCDel 3ab.pngCDel nodes 11.png.

[3[32,2,2]] ×6

7: CDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node 1.pngCDel split1.pngCDel nodes.pngCDel 3ab.pngCDel nodes.png, CDel node.pngCDel 3.pngCDel node 1.pngCDel 3.pngCDel node.pngCDel split1.pngCDel nodes 11.pngCDel 3ab.pngCDel nodes.png, CDel node.pngCDel 3.pngCDel node 1.pngCDel 3.pngCDel node 1.pngCDel split1.pngCDel nodes 11.pngCDel 3ab.pngCDel nodes.png, CDel node 1.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.pngCDel split1.pngCDel nodes.pngCDel 3ab.pngCDel nodes 11.png, CDel node 1.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node 1.pngCDel split1.pngCDel nodes.pngCDel 3ab.pngCDel nodes 11.png, CDel node 1.pngCDel 3.pngCDel node 1.pngCDel 3.pngCDel node.pngCDel split1.pngCDel nodes 11.pngCDel 3ab.pngCDel nodes 11.png, CDel node 1.pngCDel 3.pngCDel node 1.pngCDel 3.pngCDel node 1.pngCDel split1.pngCDel nodes 11.pngCDel 3ab.pngCDel nodes 11.png.

Bitruncated 2 22 honeycomb[edit]

The bitruncated 222 honeycomb CDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node 1.pngCDel split1.pngCDel nodes.pngCDel 3ab.pngCDel nodes.png, has within its symmetry construction 3 copies of CDel node.pngCDel 3.pngCDel node 1.pngCDel split1.pngCDel nodes.pngCDel 3ab.pngCDel nodes.png facets. Its facets are centered on the vertex arrangement of E6* lattice, as:

CDel node 1.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.pngCDel split1.pngCDel nodes.pngCDel 3ab.pngCDel nodes.png + CDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.pngCDel split1.pngCDel nodes.pngCDel 3ab.pngCDel nodes 10l.png + CDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.pngCDel split1.pngCDel nodes.pngCDel 3ab.pngCDel nodes 01l.png

Geometric folding[edit]

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
CDel node 1.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.pngCDel split1.pngCDel nodes.pngCDel 3ab.pngCDel nodes.png CDel node 1.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.pngCDel 4.pngCDel node.pngCDel 3.pngCDel node.png
{3,3,32,2} {3,3,4,3}

k22 polytopes[edit]

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
2A2 A5 E6 {\tilde{E}}_{6}=E6+ {\bar{T}}_7=E6++
Coxeter
diagram
CDel nodes.pngCDel 3ab.pngCDel nodes 11.png CDel nodes.pngCDel 3ab.pngCDel nodes.pngCDel split2.pngCDel node 1.png CDel nodes.pngCDel 3ab.pngCDel nodes.pngCDel split2.pngCDel node.pngCDel 3.pngCDel node 1.png CDel nodes.pngCDel 3ab.pngCDel nodes.pngCDel split2.pngCDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node 1.png CDel nodes.pngCDel 3ab.pngCDel nodes.pngCDel split2.pngCDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node 1.png
Symmetry [[32,2,-1]] [[32,2,0]] [[32,2,1]] [[32,2,2]] [[32,2,3]]
Order 72 1440 103,680
Graph 3-3 duoprism ortho-skew.png 5-simplex t2.svg Up 1 22 t0 E6.svg
Name −122 022 122 222 322

Notes[edit]

References[edit]

  • 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. (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*