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 for n ≥ 5, with = and for quarter n-cubic honeycombs = .Coxeter, Regular and semi-regular honeycoms, 1988, p.318-319
qδnNameSchläflisymbolCoxeter diagramsFacetsVertex figure qδ3 quarter square tiling q{4,4} or
or h{4}={2}{ }×{ }{ }×{ } qδ4 quarter cubic honeycomb q{4,3,4} or or h{4,3} h2{4,3}Elongatedtriangular antiprism qδ5 quarter tesseractic honeycomb q{4,32,4} or or h{4,32} h3{4,32}{3,4}×{} qδ6 quarter 5-cubic honeycomb q{4,33,4}h{4,33} h4{4,33}Rectified 5-cell antiprism qδ7 quarter 6-cubic honeycomb q{4,34,4}h{4,34} h5{4,34}{3,3}×{3,3} qδ8 quarter 7-cubic honeycomb q{4,35,4}h{4,35} h6{4,35}{3,3}×{3,31,1} qδ9 quarter 8-cubic honeycombq{4,36,4}h{4,36} h7{4,36}{3,3}×{3,32,1}{3,31,1}×{3,31,1} qδn quarter n-cubic honeycombq{4,3n−3,4}...h{4,3n−2}hn−2{4,3n−2}...
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, 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, (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