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 B ~ n − 1 D ~ n − 1 peak .[ 1] 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
Symmetry family
B ~ n − 1 n-4 ,31,1 ]
D ~ n − 1 1,1 ,3n-5 ,31,1 ]
Coxeter-Dynkin diagrams by family
hδ2
Apeirogon {∞}
File:CDel node h1.png File:CDel infin.png File:CDel node.png File:CDel node 1.png File:CDel infin.png File:CDel node 1.png
hδ3
Alternated square tiling h{4,4}=t1 {4,4}0,2 {4,4}
File:CDel node h1.png File:CDel 4.png File:CDel node.png File:CDel 4.png File:CDel node.png File:CDel nodes hh.png File:CDel split2-44.png File:CDel node.png File:CDel nodes.png File:CDel split2-44.png File:CDel node 1.png File:CDel nodes 11.png File:CDel split2-44.png File:CDel node.png
hδ4
Alternated cubic honeycomb h{4,3,4}1,1 ,4}
File:CDel node h1.png File:CDel 4.png File:CDel node.png File:CDel 3.png File:CDel node.png File:CDel 4.png File:CDel node.png File:CDel nodes hh.png File:CDel 4a4b.png File:CDel branch.png File:CDel nodes 10ru.png File:CDel split2.png File:CDel node.png File:CDel 4.png File:CDel node.png File:CDel node 1.png File:CDel split1.png File:CDel nodes.png File:CDel split2.png File:CDel node.png
hδ5
16-cell tetracomb h{4,32 ,4}1,1 ,3,4}1,1,1,1 }
File:CDel node h1.png File:CDel 4.png File:CDel node.png File:CDel 3.png File:CDel node.png File:CDel 3.png File:CDel node.png File:CDel 4.png File:CDel node.png File:CDel nodes hh.png File:CDel 4a4b.png File:CDel nodes.png File:CDel split2.png File:CDel node.png File:CDel nodes 10ru.png File:CDel split2.png File:CDel node.png File:CDel 3.png File:CDel node.png File:CDel 4.png File:CDel node.png File:CDel nodes 10ru.png File:CDel split2.png File:CDel node.png File:CDel split1.png File:CDel nodes.png
hδ6
5-demicube honeycomb h{4,33 ,4}1,1 ,32 ,4}1,1 ,3,31,1 }
File:CDel node h1.png File:CDel 4.png File:CDel node.png File:CDel 3.png File:CDel node.png File:CDel 3.png File:CDel node.png File:CDel 3.png File:CDel node.png File:CDel 4.png File:CDel node.png File:CDel nodes hh.png File:CDel 4a4b.png File:CDel nodes.png File:CDel 3ab.png File:CDel branch.png File:CDel nodes 10ru.png File:CDel split2.png File:CDel node.png File:CDel 3.png File:CDel node.png File:CDel 3.png File:CDel node.png File:CDel 4.png File:CDel node.png File:CDel nodes 10ru.png File:CDel split2.png File:CDel node.png File:CDel 3.png File:CDel node.png File:CDel split1.png File:CDel nodes.png
hδ7
6-demicube honeycomb h{4,34 ,4}1,1 ,33 ,4}1,1 ,32 ,31,1 }
File:CDel node h1.png File:CDel 4.png File:CDel node.png File:CDel 3.png File:CDel node.png File:CDel 3.png File:CDel node.png File:CDel 3.png File:CDel node.png File:CDel 3.png File:CDel node.png File:CDel 4.png File:CDel node.png File:CDel nodes hh.png File:CDel 4a4b.png File:CDel nodes.png File:CDel 3ab.png File:CDel nodes.png File:CDel split2.png File:CDel node.png File:CDel nodes 10ru.png File:CDel split2.png File:CDel node.png File:CDel 3.png File:CDel node.png File:CDel 3.png File:CDel node.png File:CDel 3.png File:CDel node.png File:CDel 4.png File:CDel node.png File:CDel nodes 10ru.png File:CDel split2.png File:CDel node.png File:CDel 3.png File:CDel node.png File:CDel 3.png File:CDel node.png File:CDel split1.png File:CDel nodes.png
hδ8
7-demicube honeycomb h{4,35 ,4}1,1 ,34 ,4}1,1 ,33 ,31,1 }
File:CDel node h1.png File:CDel 4.png File:CDel node.png File:CDel 3.png File:CDel node.png File:CDel 3.png File:CDel node.png File:CDel 3.png File:CDel node.png File:CDel 3.png File:CDel node.png File:CDel 3.png File:CDel node.png File:CDel 4.png File:CDel node.png File:CDel nodes hh.png File:CDel 4a4b.png File:CDel nodes.png File:CDel 3ab.png File:CDel nodes.png File:CDel 3ab.png File:CDel branch.png File:CDel nodes 10ru.png File:CDel split2.png File:CDel node.png File:CDel 3.png File:CDel node.png File:CDel 3.png File:CDel node.png File:CDel 3.png File:CDel node.png File:CDel 3.png File:CDel node.png File:CDel 4.png File:CDel node.png File:CDel nodes 10ru.png File:CDel split2.png File:CDel node.png File:CDel 3.png File:CDel node.png File:CDel 3.png File:CDel node.png File:CDel 3.png File:CDel node.png File:CDel split1.png File:CDel nodes.png
hδ9
8-demicube honeycomb h{4,36 ,4}1,1 ,35 ,4}1,1 ,34 ,31,1 }
File:CDel node h1.png File:CDel 4.png File:CDel node.png File:CDel 3.png File:CDel node.png File:CDel 3.png File:CDel node.png File:CDel 3.png File:CDel node.png File:CDel 3.png File:CDel node.png File:CDel 3.png File:CDel node.png File:CDel 3.png File:CDel node.png File:CDel 4.png File:CDel node.png File:CDel nodes hh.png File:CDel 4a4b.png File:CDel nodes.png File:CDel 3ab.png File:CDel nodes.png File:CDel 3ab.png File:CDel nodes.png File:CDel split2.png File:CDel node.png File:CDel nodes 10ru.png File:CDel split2.png File:CDel node.png File:CDel 3.png File:CDel node.png File:CDel 3.png File:CDel node.png File:CDel 3.png File:CDel node.png File:CDel 3.png File:CDel node.png File:CDel 3.png File:CDel node.png File:CDel 4.png File:CDel node.png File:CDel nodes 10ru.png File:CDel split2.png File:CDel node.png File:CDel 3.png File:CDel node.png File:CDel 3.png File:CDel node.png File:CDel 3.png File:CDel node.png File:CDel 3.png File:CDel node.png File:CDel split1.png File:CDel nodes.png
hδn
n-demicubic honeycomb
h{4,3n-3 ,4}1,1 ,3n-4 ,4}1,1 ,3n-5 ,31,1 }
...
References
↑ Regular and semi-regular polytopes III, p.318-319
Coxeter, H.S.M. Regular Polytopes ISBN 0-486-61480-8
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 , editied by F. Arthur Sherk, Peter McMullen, Anthony C. Thompson, Asia Ivic Weiss, Wiley-Interscience Publication, 1995, ISBN 978-0-471-01003-6 [1]
(Paper 24) H.S.M. Coxeter, Regular and Semi-Regular Polytopes III , [Math. Zeit. 200 (1988) 3-45]
