6-simplex honeycomb

6-simplex honeycomb
(No image)
Type Uniform 6-honeycomb
Family Simplectic honeycomb
Schläfli symbol {3[7]}
Coxeter diagram CDel node 1.pngCDel split1.pngCDel nodes.pngCDel 3ab.pngCDel nodes.pngCDel 3ab.pngCDel branch.png
6-face types {35} 6-simplex t0.svg, t1{35} 6-simplex t1.svg
t2{35} 6-simplex t2.svg
5-face types {34} 5-simplex t0.svg, t1{34} 5-simplex t1.svg
t2{34} 5-simplex t2.svg
4-face types {33} 4-simplex t0.svg, t1{33} 4-simplex t1.svg
Cell types {3,3} 3-simplex t0.svg, t1{3,3} 3-simplex t1.svg
Face types {3} 2-simplex t0.svg
Vertex figure t0,5{35} 6-simplex t05.svg
Symmetry ×2, [[3[7]]]
Properties vertex-transitive

In six-dimensional Euclidean geometry, the 6-simplex honeycomb is a space-filling tessellation (or honeycomb). The tessellation fills space by 6-simplex, rectified 6-simplex, and birectified 6-simplex facets. These facet types occur in proportions of 1:1:1 respectively in the whole honeycomb.

A6 latticeEdit

This vertex arrangement is called the A6 lattice or 6-simplex lattice. The 42 vertices of the expanded 6-simplex vertex figure represent the 42 roots of the   Coxeter group.[1] It is the 6-dimensional case of a simplectic honeycomb. Around each vertex figure are 126 facets: 7+7 6-simplex, 21+21 rectified 6-simplex, 35+35 birectified 6-simplex, with the count distribution from the 8th row of Pascal's triangle.

The A*
6
lattice (also called A7
6
) is the union of seven A6 lattices, and has the vertex arrangement of the dual to the omnitruncated 6-simplex honeycomb, and therefore the Voronoi cell of this lattice is the omnitruncated 6-simplex.

                                                  = dual of        

Related polytopes and honeycombsEdit

This honeycomb is one of 17 unique uniform honeycombs[2] constructed by the   Coxeter group, grouped by their extended symmetry of the Coxeter–Dynkin diagrams:

Projection by foldingEdit

The 6-simplex honeycomb can be projected into the 3-dimensional cubic honeycomb by a geometric folding operation that maps two pairs of mirrors into each other, sharing the same vertex arrangement:

         
         

See alsoEdit

Regular and uniform honeycombs in 6-space:

NotesEdit

  1. ^ http://www2.research.att.com/~njas/lattices/A6.html
  2. ^ * Weisstein, Eric W. "Necklace". MathWorld., OEIS sequence A000029 18-1 cases, skipping one with zero marks

ReferencesEdit

  • 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]
Fundamental convex regular and uniform honeycombs in dimensions 2-9
          /   /  
{3[3]} δ3 3 3 Hexagonal
{3[4]} δ4 4 4
{3[5]} δ5 5 5 24-cell honeycomb
{3[6]} δ6 6 6
{3[7]} δ7 7 7 222
{3[8]} δ8 8 8 133331
{3[9]} δ9 9 9 152251521
{3[10]} δ10 10 10
{3[n]} δn n n 1k22k1k21