# 6-demicubic honeycomb

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.

6-demicubic honeycomb
(No image)
TypeUniform 6-honeycomb
FamilyAlternated hypercube honeycomb
Schläfli symbolh{4,3,3,3,3,4}
h{4,3,3,3,31,1}
ht0,6{4,3,3,3,3,4}
Coxeter diagram =
=
Facets{3,3,3,3,4}
h{4,3,3,3,3}
Vertex figurer{3,3,3,3,4}
Coxeter group${\displaystyle {\tilde {B}}_{6}}$ [4,3,3,3,31,1]
${\displaystyle {\tilde {D}}_{6}}$ [31,1,3,3,31,1]

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 lattice

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 constructions

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
${\displaystyle {\tilde {B}}_{6}}$ = [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
${\displaystyle {\tilde {D}}_{6}}$ = [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${\displaystyle {\tilde {C}}_{6}}$ = [[(4,3,3,3,4,2<sup>+</sup>)]]ht0,6{4,3,3,3,3,4} 32+16+16: 6-demicube
12: 6-orthoplex

This honeycomb is one of 41 uniform honeycombs constructed by the ${\displaystyle {\tilde {D}}_{6}}$ 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 ${\displaystyle {\tilde {B}}_{6}}$ and ${\displaystyle {\tilde {C}}_{6}}$ constructions: