8-simplex honeycomb
(No image)
Type Uniform 8-honeycomb
Family Simplectic honeycomb
Schläfli symbol {3[9]} = 0[9]
Coxeter diagram
6-face types {37} , t1{37}
t2{37} , t3{37}
6-face types {36} , t1{36}
t2{36} , t3{36}
6-face types {35} , t1{35}
t2{35}
5-face types {34} , t1{34}
t2{34}
4-face types {33} , t1{33}
Cell types {3,3} , t1{3,3}
Face types {3}
Vertex figure t0,7{37}
Symmetry ×2, [[3[9]]]
Properties vertex-transitive

In eighth-dimensional Euclidean geometry, the 8-simplex honeycomb is a space-filling tessellation (or honeycomb). The tessellation fills space by 8-simplex, rectified 8-simplex, birectified 8-simplex, and trirectified 8-simplex facets. These facet types occur in proportions of 1:1:1:1 respectively in the whole honeycomb.

A8 lattice

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This vertex arrangement is called the A8 lattice or 8-simplex lattice. The 72 vertices of the expanded 8-simplex vertex figure represent the 72 roots of the   Coxeter group.[1] It is the 8-dimensional case of a simplectic honeycomb. Around each vertex figure are 510 facets: 9+9 8-simplex, 36+36 rectified 8-simplex, 84+84 birectified 8-simplex, 126+126 trirectified 8-simplex, with the count distribution from the 10th row of Pascal's triangle.

  contains   as a subgroup of index 5760.[2] Both   and   can be seen as affine extensions of   from different nodes:  

The A3
8
lattice is the union of three A8 lattices, and also identical to the E8 lattice.[3]

                            =                .

The A*
8
lattice (also called A9
8
) is the union of nine A8 lattices, and has the vertex arrangement of the dual honeycomb to the omnitruncated 8-simplex honeycomb, and therefore the Voronoi cell of this lattice is an omnitruncated 8-simplex

                                                                                  = dual of          .

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This honeycomb is one of 45 unique uniform honeycombs[4] constructed by the   Coxeter group. The symmetry can be multiplied by the ring symmetry of the Coxeter diagrams:

A8 honeycombs
Enneagon
symmetry
Symmetry Extended
diagram
Extended
group
Honeycombs
a1 [3[9]]            

                                                                     

                                                                     

i2 [[3[9]]]            ×2

         1                               2

                                       

                                                           

                                                           

                             

                                       

         

i6 [3[3[9]]]            ×6                    
r18 [9[3[9]]]            ×18           3

Projection by folding

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The 8-simplex honeycomb can be projected into the 4-dimensional tesseractic honeycomb by a geometric folding operation that maps two pairs of mirrors into each other, sharing the same vertex arrangement:

           
           

See also

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Notes

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  1. ^ "The Lattice A8".
  2. ^ N.W. Johnson: Geometries and Transformations, (2018) Chapter 12: Euclidean symmetry groups, p.294
  3. ^ Kaleidoscopes: Selected Writings of H. S. M. Coxeter, Paper 18, "Extreme forms" (1950)
  4. ^ * Weisstein, Eric W. "Necklace". MathWorld., OEIS sequence A000029 46-1 cases, skipping one with zero marks

References

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  • Norman Johnson Uniform Polytopes, Manuscript (1991)
  • 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 22) H.S.M. Coxeter, Regular and Semi Regular Polytopes I, [Math. Zeit. 46 (1940) 380–407, MR 2,10] (1.9 Uniform space-fillings)
    • (Paper 24) H.S.M. Coxeter, Regular and Semi-Regular Polytopes III, [Math. Zeit. 200 (1988) 3–45]
Space Family           /   /  
E2 Uniform tiling 0[3] δ3 3 3 Hexagonal
E3 Uniform convex honeycomb 0[4] δ4 4 4
E4 Uniform 4-honeycomb 0[5] δ5 5 5 24-cell honeycomb
E5 Uniform 5-honeycomb 0[6] δ6 6 6
E6 Uniform 6-honeycomb 0[7] δ7 7 7 222
E7 Uniform 7-honeycomb 0[8] δ8 8 8 133331
E8 Uniform 8-honeycomb 0[9] δ9 9 9 152251521
E9 Uniform 9-honeycomb 0[10] δ10 10 10
E10 Uniform 10-honeycomb 0[11] δ11 11 11
En-1 Uniform (n-1)-honeycomb 0[n] δn n n 1k22k1k21