Alternated hypercubic honeycomb

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 ${\tilde {B}}_{n-1}$ for n ≥ 4. A lower symmetry form ${\tilde {D}}_{n-1}$ 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 hδでるた_n for an (n-1)-dimensional honeycomb.

hδでるたn	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
hδでるた2	Apeirogon	{∞}
hδでるた3	Alternated square tiling (Same as {4,4})	h{4,4}=t1{4,4} t0,2{4,4}
hδでるた4	Alternated cubic honeycomb	h{4,3,4} {31,1,4}
hδでるた5	16-cell tetracomb (Same as {3,3,4,3})	h{4,32,4} {31,1,3,4} {31,1,1,1}
hδでるた6	5-demicube honeycomb	h{4,33,4} {31,1,32,4} {31,1,3,31,1}
hδでるた7	6-demicube honeycomb	h{4,34,4} {31,1,33,4} {31,1,32,31,1}
hδでるた8	7-demicube honeycomb	h{4,35,4} {31,1,34,4} {31,1,33,31,1}
hδでるた9	8-demicube honeycomb	h{4,36,4} {31,1,35,4} {31,1,34,31,1}
hδでるた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 γがんま_n form the cubic honeycombs, δでるた_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でるた_n+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]

Fundamental convex regular and uniform honeycombs in dimensions 2–9
Space	Family	${\tilde {A}}_{n-1}$	${\tilde {C}}_{n-1}$	${\tilde {B}}_{n-1}$	${\tilde {D}}_{n-1}$	${\tilde {G}}_{2}$ / ${\tilde {F}}_{4}$ / ${\tilde {E}}_{n-1}$
E2	Uniform tiling	0[3]	δでるた3	hδでるた3	qδでるた3	Hexagonal
E3	Uniform convex honeycomb	0[4]	δでるた4	hδでるた4	qδでるた4
E4	Uniform 4-honeycomb	0[5]	δでるた5	hδでるた5	qδでるた5	24-cell honeycomb
E5	Uniform 5-honeycomb	0[6]	δでるた6	hδでるた6	qδでるた6
E6	Uniform 6-honeycomb	0[7]	δでるた7	hδでるた7	qδでるた7	222
E7	Uniform 7-honeycomb	0[8]	δでるた8	hδでるた8	qδでるた8	133 • 331
E8	Uniform 8-honeycomb	0[9]	δでるた9	hδでるた9	qδでるた9	152 • 251 • 521
E9	Uniform 9-honeycomb	0[10]	δでるた10	hδでるた10	qδでるた10
E10	Uniform 10-honeycomb	0[11]	δでるた11	hδでるた11	qδでるた11
En-1	Uniform (n-1)-honeycomb	0[n]	δでるたn	hδでるたn	qδでるたn	1k2 • 2k1 • k21