# Alternated hypercubic honeycomb

 An alternated square tiling or checkerboard pattern. or An expanded square tiling. A partially filled alternated cubic honeycomb with tetrahedral and octahedral cells. or A subsymmetry colored alternated cubic honeycomb.

In geometry, the alternated hypercube honeycomb (or demicubic honeycomb) is a dimensional infinite series of honeycombs, based on the hypercube honeycomb with an alternation operation. It is given a Schläfli symbol h{4,3...3,4} representing the regular form with half the vertices removed and containing the symmetry of Coxeter group ${\displaystyle {\tilde {B}}_{n-1}}$ for n ≥ 4. A lower symmetry form ${\displaystyle {\tilde {D}}_{n-1}}$ can be created by removing another mirror on an order-4 peak.[1]

The alternated hypercube facets become demihypercubes, and the deleted vertices create new orthoplex facets. The vertex figure for honeycombs of this family are rectified orthoplexes.

These are also named as hδn for an (n-1)-dimensional honeycomb.

n Name Schläfli
symbol
Symmetry family
${\displaystyle {\tilde {B}}_{n-1}}$
[4,3n-4,31,1]
${\displaystyle {\tilde {D}}_{n-1}}$
[31,1,3n-5,31,1]
Coxeter-Dynkin diagrams by family
2 Apeirogon {∞}
3 Alternated square tiling
(Same as {4,4})
h{4,4}=t1{4,4}
t0,2{4,4}

4 Alternated cubic honeycomb h{4,3,4}
{31,1,4}

5 16-cell tetracomb
(Same as {3,3,4,3})
h{4,32,4}
{31,1,3,4}
{31,1,1,1}

6 5-demicube honeycomb h{4,33,4}
{31,1,32,4}
{31,1,3,31,1}

7 6-demicube honeycomb h{4,34,4}
{31,1,33,4}
{31,1,32,31,1}

8 7-demicube honeycomb h{4,35,4}
{31,1,34,4}
{31,1,33,31,1}

9 8-demicube honeycomb h{4,36,4}
{31,1,35,4}
{31,1,34,31,1}

n n-demicubic honeycomb h{4,3n-3,4}
{31,1,3n-4,4}
{31,1,3n-5,31,1}
...

## References

1. ^ Regular and semi-regular polytopes III, p.318-319
• Coxeter, H.S.M. Regular Polytopes, (3rd edition, 1973), Dover edition, ISBN 0-486-61480-8
1. pp. 122–123, 1973. (The lattice of hypercubes γn form the cubic honeycombs, δn+1)
2. pp. 154–156: Partial truncation or alternation, represented by h prefix: h{4,4}={4,4}; h{4,3,4}={31,1,4}, h{4,3,3,4}={3,3,4,3}
3. p. 296, Table II: Regular honeycombs, δn+1
• Kaleidoscopes: Selected Writings of H. S. M. Coxeter, editied 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
Space Family ${\displaystyle {\tilde {A}}_{n-1}}$ ${\displaystyle {\tilde {C}}_{n-1}}$ ${\displaystyle {\tilde {B}}_{n-1}}$ ${\displaystyle {\tilde {D}}_{n-1}}$ ${\displaystyle {\tilde {G}}_{2}}$ / ${\displaystyle {\tilde {F}}_{4}}$ / ${\displaystyle {\tilde {E}}_{n-1}}$
E2 Uniform tiling {3[3]} δ3 3 3 Hexagonal
E3 Uniform convex honeycomb {3[4]} δ4 4 4
E4 Uniform 4-honeycomb {3[5]} δ5 5 5 24-cell honeycomb
E5 Uniform 5-honeycomb {3[6]} δ6 6 6
E6 Uniform 6-honeycomb {3[7]} δ7 7 7 222
E7 Uniform 7-honeycomb {3[8]} δ8 8 8 133331
E8 Uniform 8-honeycomb {3[9]} δ9 9 9 152251521
E9 Uniform 9-honeycomb {3[10]} δ10 10 10
En-1 Uniform (n-1)-honeycomb {3[n]} δn n n 1k22k1k21