An alternated square tiling or checkerboard pattern. or An expanded square tiling.A partially filled alternated cubic honeycomb with tetrahedral and octahedral cells. or 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 for n ≥ 4. A lower symmetry form can be created by removing another mirror on an order-4 peak.Regular and semi-regular polytopes III, p.318-319
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 hδn for an (n−1)-dimensional honeycomb.
hδnNameSchläflisymbolSymmetry family[4,3n−4,31,1][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 honeycombh{4,36,4}{31,1,35,4}{31,1,34,31,1} hδn+1 n-demicubic honeycombh{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, pp. 122–123, 1973. (The lattice of hypercubes γn form the cubic honeycombs, δn+1) pp. 154–156: Partial truncation or alternation, represented by h prefix: h{4,4}={4,4}; h{4,3,4}={31,1,4}, h{4,3,3,4}={3,3,4,3} 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 24) H.S.M. Coxeter, Regular and Semi-Regular Polytopes III, [Math. Zeit. 200 (1988) 3-45]