Crossing number (knot theory)

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

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	Symmetry family
			${\tilde {B}}_{n-1}$ [4,3n-4,31,1]	${\tilde {D}}_{n-1}$ [31,1,3n-5,31,1]
			Coxeter-Dynkin diagrams by family
hδ2	Apeirogon	{∞}	Template:CDD Template:CDD
hδ3	Alternated square tiling (Same as {4,4})	h{4,4}=t1{4,4} t0,2{4,4}	Template:CDD Template:CDD Template:CDD	Template:CDD
hδ4	Alternated cubic honeycomb	h{4,3,4} {31,1,4}	Template:CDD Template:CDD Template:CDD	Template:CDD
hδ5	16-cell tetracomb (Same as {3,3,4,3})	h{4,32,4} {31,1,3,4} {31,1,1,1}	Template:CDD Template:CDD Template:CDD	Template:CDD
hδ6	5-demicube honeycomb	h{4,33,4} {31,1,32,4} {31,1,3,31,1}	Template:CDD Template:CDD Template:CDD	Template:CDD
hδ7	6-demicube honeycomb	h{4,34,4} {31,1,33,4} {31,1,32,31,1}	Template:CDD Template:CDD Template:CDD	Template:CDD
hδ8	7-demicube honeycomb	h{4,35,4} {31,1,34,4} {31,1,33,31,1}	Template:CDD Template:CDD Template:CDD	Template:CDD
hδ9	8-demicube honeycomb	h{4,36,4} {31,1,35,4} {31,1,34,31,1}	Template:CDD Template:CDD Template:CDD	Template:CDD
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

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