Alternated hypercubic honeycomb

Short description: Family of regular tessellations in geometry

120px An alternated square tiling or checkerboard pattern. or	120px An expanded square tiling.
121px A partially filled alternated cubic honeycomb with tetrahedral and octahedral cells. or	121px 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. 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.

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	{∞}
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}	...

See also

• Hypercubic honeycomb
• Simplicial honeycomb

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
• Kaleidoscopes: Selected Writings of H. S. M. Coxeter, edited 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	{3[3]}	δ3	hδ3	qδ3	Hexagonal
E3	Uniform convex honeycomb	{3[4]}	δ4	hδ4	qδ4
E4	Uniform 4-honeycomb	{3[5]}	δ5	hδ5	qδ5	24-cell honeycomb
E5	Uniform 5-honeycomb	{3[6]}	δ6	hδ6	qδ6
E6	Uniform 6-honeycomb	{3[7]}	δ7	hδ7	qδ7	222
E7	Uniform 7-honeycomb	{3[8]}	δ8	hδ8	qδ8	133 • 331
E8	Uniform 8-honeycomb	{3[9]}	δ9	hδ9	qδ9	152 • 251 • 521
E9	Uniform 9-honeycomb	{3[10]}	δ10	hδ10	qδ10
En-1	Uniform (n-1)-honeycomb	{3[n]}	δn	hδn	qδn	1k2 • 2k1 • k21