Alternated hypercubic honeycomb

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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 {{\tilde {B}}}_{{n-1}} for n ≥ 4. A lower symmetry form {{\tilde {D}}}_{{n-1}} can be created by removing another mirror on a order-4 peak.

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 hδn for an (n-1)-dimensional honeycomb.

hδn Name Schläfli
symbol		 Symmetry family
{{\tilde {B}}}_{{n-1}}
[4,3n-4,31,1]		 {{\tilde {D}}}_{{n-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 honeycomb h{4,36,4}
{31,1,35,4}
{31,1,34,31,1}

 
hδn n-demicubic honeycomb h{4,3n-3,4}
{31,1,3n-4,4}
{31,1,3n-5,31,1}		 ...

References

  • Coxeter, H.S.M. Regular Polytopes, (3rd edition, 1973), Dover edition, ISBN 0-486-61480-8
    1. pp. 122-123, 1973. (The lattice of hypercubes γn form the cubic honeycombs, δn+1)
    2. 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}
    3. p. 296, Table II: Regular honeycombs, δn+1
Fundamental convex regular and uniform honeycombs in dimensions 2–11
Family {{\tilde {A}}}_{{n-1}} {{\tilde {C}}}_{{n-1}} {{\tilde {B}}}_{{n-1}} {{\tilde {D}}}_{{n-1}} {{\tilde {G}}}_{2} / {{\tilde {F}}}_{4} / {{\tilde {E}}}_{{n-1}}
Uniform tiling {3[3]} δ3 hδ3 qδ3 Hexagonal
Uniform convex honeycomb {3[4]} δ4 hδ4 qδ4
Uniform 5-honeycomb {3[5]} δ5 hδ5 qδ5 24-cell honeycomb
Uniform 6-honeycomb {3[6]} δ6 hδ6 qδ6
Uniform 7-honeycomb {3[7]} δ7 hδ7 qδ7 222
Uniform 8-honeycomb {3[8]} δ8 hδ8 qδ8 133 • 331
Uniform 9-honeycomb {3[9]} δ9 hδ9 qδ9 152 • 251 • 521
Uniform n-honeycomb {3[n]} δn hδn qδn 1k2 • 2k1 • k21
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