# Alternated hypercubic 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 ${\displaystyle {\tilde {B}}_{n-1}}$ for n ≥ 4. A lower symmetry form ${\displaystyle {\tilde {D}}_{n-1}}$ can be created by removing another mirror on an order-4 peak.[1]

 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.

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
${\displaystyle {\tilde {B}}_{n-1}}$
[4,3n-4,31,1]
${\displaystyle {\tilde {D}}_{n-1}}$
[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

1. Regular and semi-regular polytopes III, p.318-319
• 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
• Kaleidoscopes: Selected Writings of H. S. M. Coxeter, editied by F. Arthur Sherk, Peter McMullen, Anthony C. Thompson, Asia Ivic Weiss, Wiley-Interscience Publication, 1995, ISBN 978-0-471-01003-6
• (Paper 24) H.S.M. Coxeter, Regular and Semi-Regular Polytopes III, [Math. Zeit. 200 (1988) 3-45]
Fundamental convex regular and uniform honeycombs in dimensions 2-9
Space Family ${\displaystyle {\tilde {A}}_{n-1}}$ ${\displaystyle {\tilde {C}}_{n-1}}$ ${\displaystyle {\tilde {B}}_{n-1}}$ ${\displaystyle {\tilde {D}}_{n-1}}$ ${\displaystyle {\tilde {G}}_{2}}$ / ${\displaystyle {\tilde {F}}_{4}}$ / ${\displaystyle {\tilde {E}}_{n-1}}$
E2 Uniform tiling {3[3]} δ3 hδ3 qδ3 Hexagonal
E3 Uniform convex honeycomb {3[4]} δ4 hδ4 qδ4
E4 Uniform 4-honeycomb {3[5]} δ5 hδ5 qδ5 24-cell honeycomb
E5 Uniform 5-honeycomb {3[6]} δ6 hδ6 qδ6
E6 Uniform 6-honeycomb {3[7]} δ7 hδ7 qδ7 222
E7 Uniform 7-honeycomb {3[8]} δ8 hδ8 qδ8 133331
E8 Uniform 8-honeycomb {3[9]} δ9 hδ9 qδ9 152251521
E9 Uniform 9-honeycomb {3[10]} δ10 hδ10 qδ10
En-1 Uniform (n-1)-honeycomb {3[n]} δn hδn qδn 1k22k1k21