Alternated hypercubic honeycomb

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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 B ~ n − 1 {\displaystyle {\tilde {B}}_{n-1}} for n ≥ 4. A lower symmetry form D ~ n − 1 {\displaystyle {\tilde {D}}_{n-1}} can be created by removing another mirror on an 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.

hδn	Name	Schläflisymbol	Symmetry family
			B ~ n − 1 {\displaystyle {\tilde {B}}_{n-1}} [4,3n-4,31,1]	D ~ n − 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	n-demicubic honeycomb	h{4,3n-3,4}{31,1,3n-4,4}{31,1,3n-5,31,1}	...

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]

v t e Fundamental convex regular and uniform honeycombs in dimensions 2–9
Space	Family	A ~ n − 1 {\displaystyle {\tilde {A}}_{n-1}}	C ~ n − 1 {\displaystyle {\tilde {C}}_{n-1}}	B ~ n − 1 {\displaystyle {\tilde {B}}_{n-1}}	D ~ n − 1 {\displaystyle {\tilde {D}}_{n-1}}	G ~ 2 {\displaystyle {\tilde {G}}_{2}} / F ~ 4 {\displaystyle {\tilde {F}}_{4}} / E ~ n − 1 {\displaystyle {\tilde {E}}_{n-1}}
E2	Uniform tiling	0[3]	δ3	hδ3	qδ3	Hexagonal
E3	Uniform convex honeycomb	0[4]	δ4	hδ4	qδ4
E4	Uniform 4-honeycomb	0[5]	δ5	hδ5	qδ5	24-cell honeycomb
E5	Uniform 5-honeycomb	0[6]	δ6	hδ6	qδ6
E6	Uniform 6-honeycomb	0[7]	δ7	hδ7	qδ7	222
E7	Uniform 7-honeycomb	0[8]	δ8	hδ8	qδ8	133 • 331
E8	Uniform 8-honeycomb	0[9]	δ9	hδ9	qδ9	152 • 251 • 521
E9	Uniform 9-honeycomb	0[10]	δ10	hδ10	qδ10
E10	Uniform 10-honeycomb	0[11]	δ11	hδ11	qδ11
En−1	Uniform (n−1)-honeycomb	0[n]	δn	hδn	qδn	1k2 • 2k1 • k21

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References

Regular and semi-regular polytopes III, p.318-319 ↩