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 hδ_n for an (n-1)-dimensional honeycomb.

hδn	Name	Schläfli symbol	Coxeter-Dynkin diagrams
			${\tilde{B}}_{n-1}$	${\tilde{D}}_{n-1}$
hδ2	Apeirogon	{∞}
hδ3	Alternated square tiling (Same as regular square tiling {4,4})	h{4,4}=t1{4,4} t0,2{4,4}
hδ4	Alternated cubic honeycomb	h{4,3,4} {31,1,4}
hδ5	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}
hδ6	Demipenteractic honeycomb	h{4,33,4} {31,1,32,4} {31,1,3,31,1}
hδ7	Demihexeractic honeycomb	h{4,34,4} {31,1,33,4} {31,1,32,31,1}
hδ8	Demihepteractic honeycomb	h{4,35,4} {31,1,34,4} {31,1,33,31,1}
hδ9	Demiocteractic honeycomb	h{4,36,4} {31,1,35,4} {31,1,34,31,1}
hδ10	Demienneractic honeycomb	h{4,37,4} {31,1,36,4} {31,1,35,31,1}
hδ11	Demidekeractic honeycomb	h{4,38,4} {31,1,37,4} {31,1,36,31,1}
hδn	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 γ_n form the cubic honeycombs, δ_n+1)
  2. pp. 154-156: Partial truncation or alternation, represented by h prefix: h{4,4}={4,4}; h{4,3,4}={3^1,1,4}, h{4,3,3,4}={3,3,4,3}
  3. p. 296, Table II: Regular honeycombs, δ_n+1