In geometry, the quarter hypercubic honeycomb (or quarter n-cubic honeycomb) is a dimensional infinite series of honeycombs, based on the hypercube honeycomb. It is given a Schläfli symbol q{4,3...3,4} or Coxeter symbol qδ4 representing the regular form with three quarters of the vertices removed and containing the symmetry of Coxeter group D ~ n − 1 {\displaystyle {\tilde {D}}_{n-1}} for n ≥ 5, with D ~ 4 {\displaystyle {\tilde {D}}_{4}} = A ~ 4 {\displaystyle {\tilde {A}}_{4}} and for quarter n-cubic honeycombs D ~ 5 {\displaystyle {\tilde {D}}_{5}} = B ~ 5 {\displaystyle {\tilde {B}}_{5}} .
See also Hypercubic honeycomb Alternated hypercubic honeycomb Simplectic honeycomb Truncated simplectic honeycomb Omnitruncated simplectic honeycomb
References
Coxeter, H.S.M. Regular Polytopes, (3rd edition, 1973), Dover edition, ISBN 0-486-61480-8 pp. 122–123, 1973. (The lattice of hypercubes γn form the cubic honeycombs, δn+1) pp. 154–156: Partial truncation or alternation, represented by q prefix p. 296, Table II: Regular honeycombs, δn+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] Archived 2016-07-11 at the Wayback Machine (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 [2] Klitzing, Richard. "1D-8D Euclidean tesselations".
