# Alternated hypercubic honeycomb

Alternated hypercubic honeycomb
 An alternated square tiling is another square tiling, but having two types of squares, alternating in a checkerboard pattern. A twice alternated square tiling. A partially-filled alternated cubic honeycomb with tetrahedral and octahedral cells. 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 ${\tilde{B}}_{n-1}$ for n ≥ 4. A lower symmetry form ${\tilde{D}}_{n-1}$ can be created by removing another mirror on a order-4 peak.

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
Coxeter-Dynkin diagrams
${\tilde{B}}_{n-1}$ ${\tilde{D}}_{n-1}$
2 Apeirogon {∞}

3 Alternated square tiling
(Same as regular square tiling {4,4})
h{4,4}=t1{4,4}
t0,2{4,4}

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

5 Alternated tesseractic honeycomb or
demitesseractic tetracomb
(Same as regular {3,3,4,3})
h{4,32,4}
{31,1,3,4}
{31,1,1,1}

6 Demipenteractic honeycomb h{4,33,4}
{31,1,32,4}
{31,1,3,31,1}

7 Demihexeractic honeycomb h{4,34,4}
{31,1,33,4}
{31,1,32,31,1}

8 Demihepteractic honeycomb h{4,35,4}
{31,1,34,4}
{31,1,33,31,1}

9 Demiocteractic honeycomb h{4,36,4}
{31,1,35,4}
{31,1,34,31,1}

10 Demienneractic honeycomb h{4,37,4}
{31,1,36,4}
{31,1,35,31,1}

11 Demidekeractic honeycomb h{4,38,4}
{31,1,37,4}
{31,1,36,31,1}

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

## References

• 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

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