5 21 honeycomb

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521 honeycomb
Type Uniform honeycomb
Family k21 polytope
Schläfli symbol {3,3,3,3,3,32,1}
Coxeter symbol 521
Coxeter-Dynkin diagram CDel nodea.pngCDel 3a.pngCDel nodea.pngCDel 3a.pngCDel branch.pngCDel 3a.pngCDel nodea.pngCDel 3a.pngCDel nodea.pngCDel 3a.pngCDel nodea.pngCDel 3a.pngCDel nodea.pngCDel 3a.pngCDel nodea 1.png
8-faces 511 Cross graph 8 Nodes highlighted.svg
{37} 8-simplex t0.svg
7-faces {36} 7-simplex t0.svg
6-faces {35} 6-simplex t0.svg
5-faces {34} 5-simplex t0.svg
4-faces {33} 4-simplex t0.svg
Cells {32} 3-simplex t0.svg
Faces {3} 2-simplex t0.svg
Cell figure 121 5-demicube.svg
Face figure 221 E6 graph.svg
Edge figure 321 E7 graph.svg
Vertex figure 421 E8 graph.svg
Symmetry group {\tilde{E}}_8, [35,2,1]

In geometry, the 521 honeycomb is a uniform tessellation of 8-dimensional Euclidean space. The symbol 521 is from Coxeter, named for the length of the 3 branches of its Coxeter-Dynkin diagram.[1]

This honeycomb was first studied by Gosset who called it a 9-ic semi-regular figure[2] (Gosset regarded honeycombs in n dimensions as degenerate n+1 polytopes).

Each vertex of the 521 honeycomb is surrounded by 2160 8-orthoplexes and 17280 8-simplices.

The vertex figure of Gosset's honeycomb is the semiregular 421 polytope. It is the final figure in the k21 family.

This honeycomb is highly regular in the sense that its symmetry group (the affine {\tilde{E}}_8 Weyl group) acts transitively on the k-faces for k ≤ 6. All of the k-faces for k ≤ 7 are simplices.

Construction[edit]

It is created by a Wythoff construction upon a set of 9 hyperplane mirrors in 8-dimensional space.

The facet information can be extracted from its Coxeter-Dynkin diagram.

CDel nodea.pngCDel 3a.pngCDel nodea.pngCDel 3a.pngCDel branch.pngCDel 3a.pngCDel nodea.pngCDel 3a.pngCDel nodea.pngCDel 3a.pngCDel nodea.pngCDel 3a.pngCDel nodea.pngCDel 3a.pngCDel nodea 1.png

Removing the node on the end of the 2-length branch leaves the 8-orthoplex, 611.

CDel nodea.pngCDel 3a.pngCDel branch.pngCDel 3a.pngCDel nodea.pngCDel 3a.pngCDel nodea.pngCDel 3a.pngCDel nodea.pngCDel 3a.pngCDel nodea.pngCDel 3a.pngCDel nodea 1.png

Removing the node on the end of the 1-length branch leaves the 8-simplex.

CDel nodea.pngCDel 3a.pngCDel nodea.pngCDel 3a.pngCDel nodea.pngCDel 3a.pngCDel nodea.pngCDel 3a.pngCDel nodea.pngCDel 3a.pngCDel nodea.pngCDel 3a.pngCDel nodea 1.png

The vertex figure is determined by removing the ringed node and ringing the neighboring node. This makes the 421 polytope.

CDel nodea.pngCDel 3a.pngCDel nodea.pngCDel 3a.pngCDel branch.pngCDel 3a.pngCDel nodea.pngCDel 3a.pngCDel nodea.pngCDel 3a.pngCDel nodea.pngCDel 3a.pngCDel nodea 1.png

The edge figure is determined from the vertex figure by removing the ringed node and ringing the neighboring node. This makes the 321 polytope.

CDel nodea.pngCDel 3a.pngCDel nodea.pngCDel 3a.pngCDel branch.pngCDel 3a.pngCDel nodea.pngCDel 3a.pngCDel nodea.pngCDel 3a.pngCDel nodea 1.png

The face figure is determined from the edge figure by removing the ringed node and ringing the neighboring node. This makes the 221 polytope.

CDel nodea.pngCDel 3a.pngCDel nodea.pngCDel 3a.pngCDel branch.pngCDel 3a.pngCDel nodea.pngCDel 3a.pngCDel nodea 1.png

The cell figure is determined from the face figure by removing the ringed node and ringing the neighboring node. This makes the 121 polytope.

CDel nodea.pngCDel 3a.pngCDel nodea.pngCDel 3a.pngCDel branch.pngCDel 3a.pngCDel nodea 1.png

Kissing number[edit]

Each vertex of this tessellation is the center of a 7-sphere in the densest known packing in 8 dimensions; its kissing number is 240, represented by the vertices of its vertex figure 421.

E8 lattice[edit]

{\tilde{E}}_8 contains {\tilde{A}}_8 as a subgroup of index 5760.[3] Both {\tilde{E}}_8 and {\tilde{A}}_8 can be seen as affine extensions of A_8 from different nodes: Affine A8 E8 relations.png

{\tilde{E}}_8 contains {\tilde{D}}_8 as a subgroup of index 270.[4] Both {\tilde{E}}_8 and {\tilde{D}}_8 can be seen as affine extensions of D_8 from different nodes: Affine D8 E8 relations.png

The vertex arrangement of 521 is called the E8 lattice.[5]

The E8 lattice can also be constructed as a union of the vertices of two 8-demicube honeycombs (called a D82 or D8+ lattice), as well as the union of the vertices of three 8-simplex honeycombs (called an A83 lattice):

CDel nodea 1.pngCDel 3a.pngCDel nodea.pngCDel 3a.pngCDel nodea.pngCDel 3a.pngCDel nodea.pngCDel 3a.pngCDel nodea.pngCDel 3a.pngCDel branch.pngCDel 3a.pngCDel nodea.pngCDel 3a.pngCDel nodea.png = CDel nodes 10ru.pngCDel split2.pngCDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.pngCDel split1.pngCDel nodes.pngCDel nodes.pngCDel split2.pngCDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.pngCDel split1.pngCDel nodes 10lu.png = CDel node 1.pngCDel split1.pngCDel nodes.pngCDel 3ab.pngCDel nodes.pngCDel 3ab.pngCDel nodes.pngCDel 3ab.pngCDel branch.pngCDel node.pngCDel split1.pngCDel nodes.pngCDel 3ab.pngCDel nodes.pngCDel 3ab.pngCDel nodes 10lr.pngCDel 3ab.pngCDel branch.pngCDel node.pngCDel split1.pngCDel nodes.pngCDel 3ab.pngCDel nodes.pngCDel 3ab.pngCDel nodes 01lr.pngCDel 3ab.pngCDel branch.png

Related polytopes and honeycombs[edit]

The 521 is seventh in a dimensional series of semiregular polytopes, identified in 1900 by Thorold Gosset. Each member of the sequence has the previous member as its vertex figure. All facets of these polytopes are regular polytopes, namely simplexes and orthoplexes.

k21 figures in n dimensional
Space Finite Euclidean Hyperbolic
En 3 4 5 6 7 8 9 10
Coxeter
group
E3=A2A1 E4=A4 E5=D5 E6 E7 E8 E9 = {\tilde{E}}_{8} = E8+ E10 = {\bar{T}}_8 = E8++
Coxeter
diagram
CDel node.pngCDel 3.pngCDel node 1.pngCDel 2.pngCDel node 1.png CDel nodea.pngCDel 3a.pngCDel nodea.pngCDel 3a.pngCDel branch 10.png CDel nodea.pngCDel 3a.pngCDel nodea.pngCDel 3a.pngCDel branch.pngCDel 3a.pngCDel nodea 1.png CDel nodea.pngCDel 3a.pngCDel nodea.pngCDel 3a.pngCDel branch.pngCDel 3a.pngCDel nodea.pngCDel 3a.pngCDel nodea 1.png CDel nodea.pngCDel 3a.pngCDel nodea.pngCDel 3a.pngCDel branch.pngCDel 3a.pngCDel nodea.pngCDel 3a.pngCDel nodea.pngCDel 3a.pngCDel nodea 1.png CDel nodea.pngCDel 3a.pngCDel nodea.pngCDel 3a.pngCDel branch.pngCDel 3a.pngCDel nodea.pngCDel 3a.pngCDel nodea.pngCDel 3a.pngCDel nodea.pngCDel 3a.pngCDel nodea 1.png CDel nodea.pngCDel 3a.pngCDel nodea.pngCDel 3a.pngCDel branch.pngCDel 3a.pngCDel nodea.pngCDel 3a.pngCDel nodea.pngCDel 3a.pngCDel nodea.pngCDel 3a.pngCDel nodea.pngCDel 3a.pngCDel nodea 1.png CDel nodea.pngCDel 3a.pngCDel nodea.pngCDel 3a.pngCDel branch.pngCDel 3a.pngCDel nodea.pngCDel 3a.pngCDel nodea.pngCDel 3a.pngCDel nodea.pngCDel 3a.pngCDel nodea.pngCDel 3a.pngCDel nodea.pngCDel 3a.pngCDel nodea 1.png
Symmetry [3−1,2,1] [30,2,1] [31,2,1] [32,2,1] [33,2,1] [34,2,1] [35,2,1] [36,2,1]
Order 12 120 192 51,840 2,903,040 696,729,600
Graph Triangular prism.png 4-simplex t1.svg Demipenteract graph ortho.svg E6 graph.svg E7 graph.svg E8 graph.svg - -
Name −121 021 121 221 321 421 521 621

See also[edit]

Notes[edit]

  1. ^ Coxeter, 1973, Chapter 5: The Kaleidoscope
  2. ^ Gosset, Thorold (1900). "On the regular and semi-regular figures in space of n dimensions". Messenger of Mathematics 29: 43–48. 
  3. ^ N.W. Johnson: Geometries and Transformations, Manuscript, (2011) Chapter 12: Euclidean symmetry groups, p 177
  4. ^ Johnson (2011) p.177
  5. ^ http://www.math.rwth-aachen.de/~Gabriele.Nebe/LATTICES/E8.html

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, H. S. M. (1973). Regular Polytopes ((3rd ed.) ed.). New York: Dover Publications. ISBN 0-486-61480-8. 
  • 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]
    • (Paper 24) H.S.M. Coxeter, Regular and Semi-Regular Polytopes III, [Math. Zeit. 200 (1988) 3-45]
  • N.W. Johnson: Geometries and Transformations, (2015)