Quarter hypercubic honeycomb explained

{\tilde{D}}_n-1

for n ≥ 5, with

{\tilde{D}}₄

{\tilde{A}}₄

and for quarter n-cubic honeycombs

{\tilde{D}}₅

{\tilde{B}}₅

.^[1]

qδ_n	Name	Schläfli symbol	Coxeter diagrams	Facets			Vertex figure
qδ₃	quarter square tiling	q	or or	h=	×		×
qδ₄	quarter cubic honeycomb	q	or or		h₂		Elongated triangular antiprism
qδ₅	quarter tesseractic honeycomb	q	or or		h₃		×
qδ₆	quarter 5-cubic honeycomb	q			h₄		Rectified 5-cell antiprism
qδ₇	quarter 6-cubic honeycomb	q			h₅	×
qδ₈	quarter 7-cubic honeycomb	q			h₆	×
qδ₉	quarter 8-cubic honeycomb	q			h₇	× ×

qδ_n	quarter n-cubic honeycomb	q	...	h	h_n-2	...

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 q prefix
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, http://www.wiley.com/WileyCDA/WileyTitle/productCd-0471010030.html
- (Paper 22) H.S.M. Coxeter, Regular and Semi Regular Polytopes I, [Math. Zeit. 46 (1940) 380-407, MR 2,10] (1.9 Uniform space-fillings)
- (Paper 24) H.S.M. Coxeter, Regular and Semi-Regular Polytopes III, [Math. Zeit. 200 (1988) 3-45] See p318 https://books.google.com/books?id=fUm5Mwfx8rAC&dq=%22quarter+cubic+honeycomb%22+q%7B4%2C3%2C4%7D&pg=PA318