Alternated hypercubic honeycomb explained

{\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.^[1]

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} [4,3<sup>n-4</sup>,3<sup>1,1</sup>]		{\tilde{D}} _{n-1} [3<sup>1,1</sup>,3<sup>n-5</sup>,3<sup>1,1</sup>]
	Coxeter-Dynkin diagrams by family
hδ2	Apeirogon
hδ3	Alternated square tiling (Same as)	h=t1 t0,2
hδ4	Alternated cubic honeycomb	h
hδ5	16-cell tetracomb (Same as)	h
hδ6	5-demicube honeycomb	h
hδ7	6-demicube honeycomb	h
hδ8	7-demicube honeycomb	h
hδ9	8-demicube honeycomb	h
hδn	n-demicubic honeycomb	h	...

References

• Coxeter, H.S.M. Regular Polytopes, (3rd edition, 1973), Dover edition,
  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=; h=, h=
  3. p. 296, Table II: Regular honeycombs, δ_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, http://www.wiley.com/WileyCDA/WileyTitle/productCd-0471010030.html
  • (Paper 24) H.S.M. Coxeter, Regular and Semi-Regular Polytopes III, [Math. Zeit. 200 (1988) 3-45]

Notes and References

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