Oseen equations

222 honeycomb
(no image)
Type	Uniform tessellation
Coxeter symbol	222
Schläfli symbol	{3,3,32,2}
Coxeter–Dynkin diagram	Template:CDD
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	t2{34}
Vertex figure	122
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 Schlafli 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, Template:CDD.

Removing a node on the end of one of the 2-node branches leaves the 2₂₁, its only facet type, Template:CDD

The vertex figure is determined by removing the ringed node and ringing the neighboring node. This makes 1₂₂, Template:CDD.

The edge figure is the vertex figure of the vertex figure, here being a birectified 5-simplex, t₂{3⁴}, Template:CDD.

The face figure is the vertex figure of the edge figure, here being a triangular duoprism, {3}×{3}, Template:CDD.

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 E₆^{2 lattice}, with [[3,3,3^2,2]] symmetry, can be constructed by the union of two E₆ lattices:

Template:CDD + Template:CDD

The E₆^* lattice (or E₆^{3) 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. It is constructed by 3 copies of the E₆ lattice vertices, one from each of the three branches of the Coxeter diagram.}

Template:CDD + Template:CDD + Template:CDD = dual to Template:CDD.

Related honeycombs

The 2₂₂ honeycomb is one of 127 uniform honeycombs (39 unique) with ${\tilde {E}}_{6}$ symmetry. 24 of them have doubled symmetry [[3,3,3^2,2]] with 2 equally ringed branches and, 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.

Symmetry	Order	Honeycombs
[32,2,2]	Full	8: Template:CDD, Template:CDD, Template:CDD, Template:CDD, Template:CDD, Template:CDD, Template:CDD, Template:CDD.
[[3,3,32,2]]	×2	24: Template:CDD, Template:CDD, Template:CDD, Template:CDD, Template:CDD, Template:CDD, Template:CDD, Template:CDD, Template:CDD, Template:CDD, Template:CDD, Template:CDD, Template:CDD, Template:CDD, Template:CDD, Template:CDD, Template:CDD, Template:CDD, Template:CDD, Template:CDD, Template:CDD, Template:CDD, Template:CDD, Template:CDD.
[3[32,2,2]]	×6	7: Template:CDD, Template:CDD, Template:CDD, Template:CDD, Template:CDD, Template:CDD, Template:CDD.

Bitruncated 2 22 honeycomb

The bitruncated 2₂₂ honeycomb Template:CDD, has within its symmetry construction 3 copies of Template:CDD facets. Its vertex arrangement can also be constructed as an E₆^* lattice, as:

Template:CDD + Template:CDD + Template:CDD

Geometric folding

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.

${\tilde {E}}_{6}$	${\tilde {F}}_{4}$
Template:CDD	Template:CDD
{3,3,32,2}	{3,3,4,3}

k₂₂ polytopes

The 2₂₂ honeycomb, is fourth in a dimensional series of uniform polytopes, expressed by Coxeter as k₂₂ series. The final is a noncompact hyperbolic honeycomb, 3<sub>22. Each progressive uniform polytope is constructed from the previous as its vertex figure. TҺe wonderful achieve wіth our phone sex assist is uѕually that yoս wߋuld poѕsibly uѕe іt anyplace inside of earth - so inside the event you're an ex-pat іn lookup of horny phone sex ƴou may get in contact ѡith us. That you will Ƅe capable of't uѕе excellent tοp quality pace sex chat traces fгom external the UK but yߋu'll have the option tо usе our phone sex assist credited tοwards tɦe truth уoս shell out to yoսr bank card.

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Notes

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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), Macmillian 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.
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  My blog: http://www.primaboinca.com/view_profile.php?userid=5889534 p125-126, 8.3 The 6-dimensional lattices: E6 and E6*

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