Alternated hypercubic 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]

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.

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. In the four-dimensional case, the demihypercube and the orthoplex are the same (both are the 16-cell), and the rectified orthoplex is also regular (the 24-cell), so the 4-demicubic honeycomb is regular (the 16-cell honeycomb).

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

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hδn	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
hδ2	Apeirogon	{∞}
hδ3	Alternated square tiling (Same as {4,4})	h{4,4}=t1{4,4} t0,2{4,4}
hδ4	Alternated cubic honeycomb	h{4,3,4} {31,1,4}
hδ5	16-cell tetracomb (Same as {3,3,4,3})	h{4,32,4} {31,1,3,4} {31,1,1,1}
hδ6	5-demicube honeycomb	h{4,33,4} {31,1,32,4} {31,1,3,31,1}
hδ7	6-demicube honeycomb	h{4,34,4} {31,1,33,4} {31,1,32,31,1}
hδ8	7-demicube honeycomb	h{4,35,4} {31,1,34,4} {31,1,33,31,1}
hδ9	8-demicube honeycomb	h{4,36,4} {31,1,35,4} {31,1,34,31,1}
hδn+1	n-demicubic honeycomb	h{4,3n-2,4} {31,1,3n-3,4} {31,1,3n-4,31,1}	...

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