HYPERCUBIC HONEYCOMB

A regular ''square tiling''. CDW_ring.png CDW_4.png CDW_dot.png CDW_4.png CDW_dot.png

A partial-filled cubic honeycomb in its regular form. CDW_ring.png CDW_4.png CDW_dot.png CDW_3.png CDW_dot.png CDW_4.png CDW_dot.png

A partially-filled cubic honeycomb in its semiregular form. CD_ring.png CD_4.png CD_3b.png CD_downbranch-00.png CD_3b.png CD_dot.png

In geometry, a 'hypercubic honeycomb' is a family of regular honeycombs (tessellations) in n-dimensions with the Schläfli symbols {4,3...3,4} and containing the symmetry of Coxeter group R_n (or B^~_n-1) for n>=3.
The tessellation is constructed from 4 n-hypercubes per ridge. The vertex figure is a cross-polytope {3...3,4}.
These are also named as - δ_n+1 for an n-dimensional honeycomb.
There's two general forms of the hypercube honeycombs, the ''regular'' form with identical hypercubic facets and one ''semiregular'', with alternating hypercube facets, like a checkerboard.
A more general class of honeycombs are called, 'orthotopic honeycombs', with identical topology, but allow each axial direction to have different edge lengths, for example with rectangle and cuboid facets in 2 and 3 dimensions.
{| class="wikitable"
!rowspan=2| δ_n
!rowspan=2| Name
!rowspan=2| Schläfli
symbol
!colspan=3|Coxeter-Dynkin diagrams
|-
!Orthotopic
!Regular
!Semiregular
|-
| δ₂
| Apeirogon
|{∞}
|
|-
|δ₃
| Square tiling
|{4,4}
|
|
|
|-
| δ₄
| Cubic honeycomb
|{4,3,4}
|
|
|
|-
|δ₅
| ''Tesseractic tetracomb''
|{4,3²,4}
|
|
|
|-
| δ₆
| ''Penteractic pentacomb''
|{4,3³,4}
|
|
|
|-
| δ₇
| ''Hexeractic hexacomb''
|{4,3⁴,4}
|
|
|
|-
| δ₈
| ''Hepteractic heptacomb''
|{4,3⁵,4}
|
|
|
|-
| δ₉
| ''Octeractic octacomb''
|{4,3⁶,4}
|
|
|
|-
| δ₁₀
| ''Enneractic enneacomb''
|{4,3⁷,4}
|colspan=3|...
|}

Contents

Alternated hypercubic honeycombs

References

Alternated hypercubic honeycombs

{| align=right class="wikitable" width=250
| valign=top|
An ''alternated square tiling'' is another ''square tiling'', but having two ''types'' of squares, alternating in a checkerboard pattern.

| valign=top|
A twice alternated square tiling.

|-
| valign=top|
A partially-filled ''alternated cubic honeycomb'' with tetrahedral and octahedral cells.

| valign=top|
A subsymmetry colored alternated cubic honeycomb.

|}
A second infinite family is based on an alternation of the regular family, given a Schläfli symbols h{4,3...3,4} representing the regular form with half the vertices removed and containing the symmetry of Coxeter group S_n (or C^~_n-1) for n>=4. A lower symmetry form Q_n (or B^~_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 cross-polytope facets. The vertex figure for honeycombs of this family are rectified hypercubes.
These are also named as - hδ_n for an (n-1)-dimensional honeycomb.
{| class="wikitable"
!rowspan=2| hδ_n
!rowspan=2| Name
!rowspan=2| Schläfli
symbol
!colspan=3|Coxeter-Dynkin diagrams
|-
!Alternated regular
!Uniform-1
!Uniform-2
|-
| hδ₂
| ''Apeirogon''
|{∞}
|
|
|  
|-
| hδ₃
| ''Alternated square tiling''
(Same as regular square tiling {4,4})
| h{4,4}
|
|
|
|-
| hδ₄
| ''Alternated cubic honeycomb''
| h{4,3,4}
|
|
|
|-
| hδ₅
| ''Alternated tesseractic tetracomb'' or
''demitesseractic tetracomb''
(Same as regular {3,3,4,3})
| h{4,3²,4}
|
|
|
|-
| hδ₆
| ''Demipenteractic pentacomb''
| h{4,3³,4}
|
|
|
|-
| hδ₇
| ''Demihexeractic hexacomb''
| h{4,3⁴,4}
|
|
|
|-
| hδ₈
| ''Demihepteractic heptacomb''
| h{4,3⁵,4}
|
|
|
|-
| δ₉
| ''Demiocteractic octacomb''
|h{4,3⁶,4}
|
|
|
|-
| δ₁₀
| ''Demienneractic enneacomb''
|h{4,3⁷,4}
|colspan=3|...
|}

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 γ_n form the ''cubic honeycombs'', δ_n+1)

★ # pp. 154-156: Partial truncation or alternation, represented by ''h'' prefix: h{4,4}={4,4}; h{4,3,4}={3^1,1,4}, h{4,3,3,4}={3,3,4,3}

★ # p. 296, Table II: Regular honeycombs, δ_n+1