6-demicubic honeycomb

6-demicubic honeycomb
(No image)
Type Uniform 6-honeycomb
Family Alternated hypercube honeycomb
Schläfli symbol h{4,3,3,3,3,4}
h{4,3,3,3,31,1}
ht0,6{4,3,3,3,3,4}
Coxeter diagram CDel node h1.pngCDel 4.pngCDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.pngCDel 4.pngCDel node.png = CDel nodes 10ru.pngCDel split2.pngCDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.pngCDel 4.pngCDel node.png
CDel node h1.pngCDel 4.pngCDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.pngCDel split1.pngCDel nodes.png = CDel nodes 10ru.pngCDel split2.pngCDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.pngCDel split1.pngCDel nodes.png
CDel label2.pngCDel branch hh.pngCDel 4a4b.pngCDel nodes.pngCDel 3ab.pngCDel nodes.pngCDel split2.pngCDel node.png
Facets {3,3,3,3,4} 6-cube t5.svg
h{4,3,3,3,3} 6-demicube t0 D6.svg
Vertex figure r{3,3,3,3,4} Rectified hexacross.svg
Coxeter group [4,3,3,3,31,1]
[31,1,3,3,31,1]

The 6-demicubic honeycomb or demihexeractic honeycomb is a uniform space-filling tessellation (or honeycomb) in Euclidean 6-space. It is constructed as an alternation of the regular 6-cube honeycomb.

It is composed of two different types of facets. The 6-cubes become alternated into 6-demicubes h{4,3,3,3,3} and the alternated vertices create 6-orthoplex {3,3,3,3,4} facets.

D6 latticeEdit

The vertex arrangement of the 6-demicubic honeycomb is the D6 lattice.[1] The 60 vertices of the rectified 6-orthoplex vertex figure of the 6-demicubic honeycomb reflect the kissing number 60 of this lattice.[2] The best known is 72, from the E6 lattice and the 222 honeycomb.

The D+
6
lattice (also called D2
6
) can be constructed by the union of two D6 lattices. This packing is only a lattice for even dimensions. The kissing number is 25=32 (2n-1 for n<8, 240 for n=8, and 2n(n-1) for n>8).[3]

                  

The D*
6
lattice (also called D4
6
and C2
6
) can be constructed by the union of all four 6-demicubic lattices:[4] It is also the 6-dimensional body centered cubic, the union of two 6-cube honeycombs in dual positions.

                                     =               .

The kissing number of the D6* lattice is 12 (2n for n≥5).[5] and its Voronoi tessellation is a trirectified 6-cubic honeycomb,        , containing all birectified 6-orthoplex Voronoi cell,            .[6]

Symmetry constructionsEdit

There are three uniform construction symmetries of this tessellation. Each symmetry can be represented by arrangements of different colors on the 64 6-demicube facets around each vertex.

Coxeter group Schläfli symbol Coxeter-Dynkin diagram Vertex figure
Symmetry
Facets/verf
  = [31,1,3,3,3,4]
= [1+,4,3,3,3,3,4]
h{4,3,3,3,3,4}             =                        
[3,3,3,4]
64: 6-demicube
12: 6-orthoplex
  = [31,1,3,31,1]
= [1+,4,3,3,31,1]
h{4,3,3,3,31,1}           =                      
[33,1,1]
32+32: 6-demicube
12: 6-orthoplex
½  = [[(4,3,3,3,4,2+)]] ht0,6{4,3,3,3,3,4}          32+16+16: 6-demicube
12: 6-orthoplex

Related honeycombsEdit

This honeycomb is one of 41 uniform honeycombs constructed by the   Coxeter group, all but 6 repeated in other families by extended symmetry, seen in the graph symmetry of rings in the Coxeter–Dynkin diagrams. The 41 permutations are listed with its highest extended symmetry, and related   and   constructions:

See alsoEdit

NotesEdit

  1. ^ http://www.math.rwth-aachen.de/~Gabriele.Nebe/LATTICES/D6.html
  2. ^ Sphere packings, lattices, and groups, by John Horton Conway, Neil James Alexander Sloane, Eiichi Bannai [1]
  3. ^ Conway (1998), p. 119
  4. ^ http://www.math.rwth-aachen.de/~Gabriele.Nebe/LATTICES/Ds6.html
  5. ^ Conway (1998), p. 120
  6. ^ Conway (1998), p. 466

External linksEdit

  • 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 [2]
    • (Paper 24) H.S.M. Coxeter, Regular and Semi-Regular Polytopes III, [Math. Zeit. 200 (1988) 3-45]
  • Conway JH, Sloane NJH (1998). Sphere Packings, Lattices and Groups (3rd ed.). ISBN 0-387-98585-9.
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