6-cubic honeycomb

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6-cubic honeycomb
(no image)
Type Regular 6-honeycomb
Uniform 6-honeycomb
Family Hypercube honeycomb
Schläfli symbol {4,34,4}
{4,33,31,1}
Coxeter-Dynkin diagrams

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6-face type {4,34}
5-face type {4,33}
4-face type {4,3,3}
Cell type {4,3}
Face type {4}
Face figure {4,3}
(octahedron)
Edge figure 8 {4,3,3}
(16-cell)
Vertex figure 64 {4,34}
(6-orthoplex)
Coxeter group , [4,34,4]
, [4,33,31,1]
Dual self-dual
Properties vertex-transitive, edge-transitive, face-transitive, cell-transitive

The 6-cubic honeycomb or hexeractic honeycomb is the only regular space-filling tessellation (or honeycomb) in Euclidean 6-space.

It is analogous to the square tiling of the plane and to the cubic honeycomb of 3-space.

ConstructionsEdit

There are many different Wythoff constructions of this honeycomb. The most symmetric form is regular, with Schläfli symbol {4,34,4}. Another form has two alternating 6-cube facets (like a checkerboard) with Schläfli symbol {4,33,31,1}. The lowest symmetry Wythoff construction has 64 types of facets around each vertex and a prismatic product Schläfli symbol {∞}6.

Related honeycombsEdit

The [4,34,4],              , Coxeter group generates 127 permutations of uniform tessellations, 71 with unique symmetry and 70 with unique geometry. The expanded 6-cubic honeycomb is geometrically identical to the 6-cubic honeycomb.

The 6-cubic honeycomb can be alternated into the 6-demicubic honeycomb, replacing the 6-cubes with 6-demicubes, and the alternated gaps are filled by 6-orthoplex facets.

Trirectified 6-cubic honeycombEdit

A trirectified 6-cubic honeycomb,        , contains all birectified 6-orthoplex facets and is the Voronoi tessellation of the D6* lattice. Facets can be identically colored from a doubled  ×2, [[4,34,4]] symmetry, alternately colored from  , [4,34,4] symmetry, three colors from  , [4,33,31,1] symmetry, and 4 colors from  , [31,1,3,3,31,1] symmetry.

See alsoEdit

ReferencesEdit

  • Coxeter, H.S.M. Regular Polytopes, (3rd edition, 1973), Dover edition, ISBN 0-486-61480-8 p. 296, Table II: Regular honeycombs
  • 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]
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