Alternated hypercubic honeycomb

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Uniform tiling 44-t1.png
An alternated square tiling or checkerboard pattern.
CDel node h1.pngCDel 4.pngCDel node.pngCDel 4.pngCDel node.png or CDel nodes.pngCDel split2-44.pngCDel node 1.png
Uniform tiling 44-t02.png
An expanded square tiling.
CDel nodes 11.pngCDel split2-44.pngCDel node.png
Tetrahedral-octahedral honeycomb.png
A partially filled alternated cubic honeycomb with tetrahedral and octahedral cells.
CDel node h1.pngCDel 4.pngCDel node.pngCDel 3.pngCDel node.pngCDel 4.pngCDel node.png or CDel nodes 10ru.pngCDel split2.pngCDel node.pngCDel 4.pngCDel node.png
Tetrahedral-octahedral honeycomb2.png
A subsymmetry colored alternated cubic honeycomb.
CDel node 1.pngCDel split1.pngCDel nodes.pngCDel split2.pngCDel node.png

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.

n 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
2 Apeirogon {∞} CDel node h1.pngCDel infin.pngCDel node.png
CDel node 1.pngCDel infin.pngCDel node 1.png
3 Alternated square tiling
(Same as {4,4})
h{4,4}=t1{4,4}
t0,2{4,4}
CDel node h1.pngCDel 4.pngCDel node.pngCDel 4.pngCDel node.png
CDel nodes hh.pngCDel split2-44.pngCDel node.png
CDel nodes.pngCDel split2-44.pngCDel node 1.png
CDel nodes 11.pngCDel split2-44.pngCDel node.png
4 Alternated cubic honeycomb h{4,3,4}
{31,1,4}
CDel node h1.pngCDel 4.pngCDel node.pngCDel 3.pngCDel node.pngCDel 4.pngCDel node.png
CDel nodes hh.pngCDel 4a4b.pngCDel branch.png
CDel nodes 10ru.pngCDel split2.pngCDel node.pngCDel 4.pngCDel node.png
CDel node 1.pngCDel split1.pngCDel nodes.pngCDel split2.pngCDel node.png
5 16-cell tetracomb
(Same as {3,3,4,3})
h{4,32,4}
{31,1,3,4}
{31,1,1,1}
CDel node h1.pngCDel 4.pngCDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.pngCDel 4.pngCDel node.png
CDel nodes hh.pngCDel 4a4b.pngCDel nodes.pngCDel split2.pngCDel node.png
CDel nodes 10ru.pngCDel split2.pngCDel node.pngCDel 3.pngCDel node.pngCDel 4.pngCDel node.png
CDel nodes 10ru.pngCDel split2.pngCDel node.pngCDel split1.pngCDel nodes.png
6 5-demicube honeycomb h{4,33,4}
{31,1,32,4}
{31,1,3,31,1}
CDel node h1.pngCDel 4.pngCDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.pngCDel 4.pngCDel node.png
CDel nodes hh.pngCDel 4a4b.pngCDel nodes.pngCDel 3ab.pngCDel branch.png
CDel nodes 10ru.pngCDel split2.pngCDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.pngCDel 4.pngCDel node.png
CDel nodes 10ru.pngCDel split2.pngCDel node.pngCDel 3.pngCDel node.pngCDel split1.pngCDel nodes.png
7 6-demicube honeycomb h{4,34,4}
{31,1,33,4}
{31,1,32,31,1}
CDel node h1.pngCDel 4.pngCDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.pngCDel 4.pngCDel node.png
CDel nodes hh.pngCDel 4a4b.pngCDel nodes.pngCDel 3ab.pngCDel nodes.pngCDel split2.pngCDel node.png
CDel nodes 10ru.pngCDel split2.pngCDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.pngCDel 4.pngCDel node.png
CDel nodes 10ru.pngCDel split2.pngCDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.pngCDel split1.pngCDel nodes.png
8 7-demicube honeycomb h{4,35,4}
{31,1,34,4}
{31,1,33,31,1}
CDel node h1.pngCDel 4.pngCDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.pngCDel 4.pngCDel node.png
CDel nodes hh.pngCDel 4a4b.pngCDel nodes.pngCDel 3ab.pngCDel nodes.pngCDel 3ab.pngCDel branch.png
CDel nodes 10ru.pngCDel split2.pngCDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.pngCDel 4.pngCDel node.png
CDel nodes 10ru.pngCDel split2.pngCDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.pngCDel split1.pngCDel nodes.png
9 8-demicube honeycomb h{4,36,4}
{31,1,35,4}
{31,1,34,31,1}
CDel node h1.pngCDel 4.pngCDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.pngCDel 4.pngCDel node.png
CDel nodes hh.pngCDel 4a4b.pngCDel nodes.pngCDel 3ab.pngCDel nodes.pngCDel 3ab.pngCDel nodes.pngCDel split2.pngCDel node.png
CDel nodes 10ru.pngCDel split2.pngCDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.pngCDel 4.pngCDel node.png
CDel nodes 10ru.pngCDel split2.pngCDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.pngCDel split1.pngCDel nodes.png
 
n n-demicubic honeycomb h{4,3n-3,4}
{31,1,3n-4,4}
{31,1,3n-5,31,1}
...

References[edit]

  • 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