5-demicubic honeycomb

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Demipenteractic honeycomb
(No image)
Type uniform honeycomb
Family Alternated hypercubic honeycomb
Schläfli symbol h{4,3,3,3,4}
Coxeter diagram

CDel nodes 10ru.pngCDel split2.pngCDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.pngCDel 4.pngCDel node.png or 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 10ru.pngCDel split2.pngCDel node.pngCDel 3.pngCDel node.pngCDel split1.pngCDel nodes.png or CDel node h1.pngCDel 4.pngCDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.pngCDel split1.pngCDel nodes.png
CDel node h.pngCDel 4.pngCDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.pngCDel 4.pngCDel node h.png
CDel node h.pngCDel 4.pngCDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.pngCDel 4.pngCDel node.pngCDel 2.pngCDel node h.pngCDel infin.pngCDel node.png
CDel node h.pngCDel 4.pngCDel node.pngCDel 3.pngCDel node.pngCDel split1.pngCDel nodes.pngCDel 2.pngCDel node h.pngCDel infin.pngCDel node.png
CDel node h.pngCDel 4.pngCDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.pngCDel 4.pngCDel node h.pngCDel 2.pngCDel node h.pngCDel infin.pngCDel node.png
CDel node h.pngCDel 4.pngCDel node.pngCDel 3.pngCDel node.pngCDel 4.pngCDel node.pngCDel 2.pngCDel node h.pngCDel infin.pngCDel node.pngCDel 2.pngCDel node h.pngCDel infin.pngCDel node.png
CDel node h.pngCDel 4.pngCDel node.pngCDel split1.pngCDel nodes.pngCDel 2.pngCDel node h.pngCDel infin.pngCDel node.pngCDel 2.pngCDel node h.pngCDel infin.pngCDel node.png
CDel node h.pngCDel 4.pngCDel node.pngCDel 3.pngCDel node.pngCDel 4.pngCDel node h.pngCDel 2.pngCDel node h.pngCDel infin.pngCDel node.pngCDel 2.pngCDel node h.pngCDel infin.pngCDel node.png
CDel node h.pngCDel 4.pngCDel node.pngCDel 4.pngCDel node.pngCDel 2.pngCDel node h.pngCDel infin.pngCDel node.pngCDel 2.pngCDel node h.pngCDel infin.pngCDel node.pngCDel 2.pngCDel node h.pngCDel infin.pngCDel node.png
CDel node h.pngCDel 4.pngCDel node.pngCDel 4.pngCDel node h.pngCDel 2.pngCDel node h.pngCDel infin.pngCDel node.pngCDel 2.pngCDel node h.pngCDel infin.pngCDel node.pngCDel 2.pngCDel node h.pngCDel infin.pngCDel node.png
CDel node h.pngCDel infin.pngCDel node.pngCDel 2.pngCDel node h.pngCDel infin.pngCDel node.pngCDel 2.pngCDel node h.pngCDel infin.pngCDel node.pngCDel 2.pngCDel node h.pngCDel infin.pngCDel node.pngCDel 2.pngCDel node h.pngCDel infin.pngCDel node.png

Facets {3,3,3,4} 5-cube t4.svg
h{4,3,3,3} 5-demicube t0 D5.svg
Vertex figure t1{3,3,3,4} Rectified pentacross.svg
Coxeter group {\tilde{B}}_5 [4,3,3,31,1]
{\tilde{D}}_5 [31,1,3,31,1]

The 5-demicube honeycomb, or demipenteractic honeycomb is a uniform space-filling tessellation (or honeycomb) in Euclidean 5-space. It is constructed as an alternation of the regular 5-cube honeycomb.

It is the first tessellation in the demihypercube honeycomb family which, with all the next ones, is not regular, being composed of two different types of uniform facets. The 5-cubes become alternated into 5-demicubes h{4,3,3,3} and the alternated vertices create 5-orthoplex {3,3,3,4} facets.

D5 lattice[edit]

The vertex arrangement of the 5-demicubic honeycomb is the D5 lattice which is the densest known sphere packing in 5 dimensions.[1] The 40 vertices of the rectified 5-orthoplex vertex figure of the 5-demicubic honeycomb reflect the kissing number 40 of this lattice.[2]

The D+
5
packing (also called D2
5
) can be constructed by the union of two D5 lattices. The analogous packings form lattices only in even dimensions. The kissing number is 24=16 (2n-1 for n<8, 240 for n=8, and 2n(n-1) for n>8).[3]

CDel nodes 10ru.pngCDel split2.pngCDel node.pngCDel 3.pngCDel node.pngCDel split1.pngCDel nodes.pngCDel nodes.pngCDel split2.pngCDel node.pngCDel 3.pngCDel node.pngCDel split1.pngCDel nodes 10lu.png

The D*
5
[4] lattice (also called D4
5
and C2
5
) can be constructed by the union of all four 5-demicubic lattices:[5] It is also the 5-dimensional body centered cubic, the union of two 5-cube honeycombs in dual positions.

CDel nodes 10ru.pngCDel split2.pngCDel node.pngCDel 3.pngCDel node.pngCDel split1.pngCDel nodes.pngCDel nodes 01rd.pngCDel split2.pngCDel node.pngCDel 3.pngCDel node.pngCDel split1.pngCDel nodes.pngCDel nodes.pngCDel split2.pngCDel node.pngCDel 3.pngCDel node.pngCDel split1.pngCDel nodes 10lu.pngCDel nodes.pngCDel split2.pngCDel node.pngCDel 3.pngCDel node.pngCDel split1.pngCDel nodes 01ld.png = CDel node 1.pngCDel 4.pngCDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.pngCDel 4.pngCDel node.pngCDel node.pngCDel 4.pngCDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.pngCDel 4.pngCDel node 1.png

The kissing number of the D*
5
lattice is 10 (2n for n≥5) and it Voronoi tessellation is a tritruncated 5-cubic honeycomb, CDel branch 11.pngCDel 3ab.pngCDel nodes.pngCDel 4a4b.pngCDel nodes.png, containing all with bitruncated 5-orthoplex, CDel node.pngCDel 4.pngCDel node.pngCDel 3.pngCDel node 1.pngCDel 3.pngCDel node 1.pngCDel 3.pngCDel node.png Voronoi cells.[6]

Symmetry constructions[edit]

There are three uniform construction symmetries of this tessellation. Each symmetry can be represented by arrangements of differened colors on the 32 5-demicube facets around each vertex.

Coxeter group Schläfli symbol Coxeter-Dynkin diagram Vertex figure
Symmetry
Facets/verf
{\tilde{B}}_5 = [31,1,3,3,4]
= [1+,4,3,3,4]
= h{4,3,3,3,4} CDel nodes 10ru.pngCDel split2.pngCDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.pngCDel 4.pngCDel node.png = 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 node.pngCDel 3.pngCDel node 1.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.pngCDel 4.pngCDel node.png
[3,3,3,4]
32: 5-demicube
10: 5-orthoplex
{\tilde{D}}_5 = [31,1,3,31,1]
= [1+,4,3,31,1]
= h{4,3,3,31,1} CDel nodes 10ru.pngCDel split2.pngCDel node.pngCDel 3.pngCDel node.pngCDel split1.pngCDel nodes.png = CDel node h1.pngCDel 4.pngCDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.pngCDel split1.pngCDel nodes.png CDel node.pngCDel 3.pngCDel node 1.pngCDel 3.pngCDel node.pngCDel split1.pngCDel nodes.png
[32,1,1]
16+16: 5-demicube
10: 5-orthoplex
{\tilde{C}}_5 = [[(4,3,3,3,4,2+)]] ht0,5{4,3,3,3,4} CDel node h.pngCDel 4.pngCDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.pngCDel 4.pngCDel node h.png 16+8+8: 5-demicube
10: 5-orthoplex

Related honeycombs[edit]

This honeycomb is one of 20 uniform honycombs constructed by the {\tilde{D}}_5 Coxeter group, all but 3 repeated in other families by extended symmetry, seen in the graph symmetry of rings in the Coxeter–Dynkin diagrams. The 20 permutations are listed with its highest extended symmetry relation:

Extended
symmetry
Extended
diagram
Order Honeycombs
[31,1,3,31,1] CDel nodes.pngCDel split2.pngCDel node.pngCDel 3.pngCDel node.pngCDel split1.pngCDel nodes.png ×1 CDel nodes 10ru.pngCDel split2.pngCDel node 1.pngCDel 3.pngCDel node.pngCDel split1.pngCDel nodes 10lu.png
[[31,1,3,31,1]] CDel nodeab c1-2.pngCDel split2.pngCDel node c3.pngCDel 3.pngCDel node c3.pngCDel split1.pngCDel nodeab c1-2.png ×2 CDel nodes 10ru.pngCDel split2.pngCDel node.pngCDel 3.pngCDel node.pngCDel split1.pngCDel nodes 10lu.png, CDel nodes 10ru.pngCDel split2.pngCDel node 1.pngCDel 3.pngCDel node 1.pngCDel split1.pngCDel nodes 10lu.png
<[31,1,3,31,1]>
= [31,1,3,3,4]
CDel nodeab c1-2.pngCDel split2.pngCDel node c3.pngCDel 3.pngCDel node c4.pngCDel split1.pngCDel nodeab c5.png
= CDel nodeab c1-2.pngCDel split2.pngCDel node c3.pngCDel 3.pngCDel node c4.pngCDel 3.pngCDel node c5.pngCDel 4.pngCDel node.png
×2 CDel nodes 10ru.pngCDel split2.pngCDel node.pngCDel 3.pngCDel node.pngCDel split1.pngCDel nodes.png, CDel nodes 10ru.pngCDel split2.pngCDel node 1.pngCDel 3.pngCDel node.pngCDel split1.pngCDel nodes.png, CDel nodes 10ru.pngCDel split2.pngCDel node.pngCDel 3.pngCDel node 1.pngCDel split1.pngCDel nodes.png, CDel nodes 10ru.pngCDel split2.pngCDel node 1.pngCDel 3.pngCDel node 1.pngCDel split1.pngCDel nodes.png

CDel nodes 10ru.pngCDel split2.pngCDel node.pngCDel 3.pngCDel node.pngCDel split1.pngCDel nodes 11.png, CDel nodes 10ru.pngCDel split2.pngCDel node 1.pngCDel 3.pngCDel node.pngCDel split1.pngCDel nodes 11.png, CDel nodes 10ru.pngCDel split2.pngCDel node.pngCDel 3.pngCDel node 1.pngCDel split1.pngCDel nodes 11.png, CDel nodes 10ru.pngCDel split2.pngCDel node 1.pngCDel 3.pngCDel node 1.pngCDel split1.pngCDel nodes 11.png

<<[31,1,3,31,1]>>
= [4,3,3,3,4]
CDel nodeab c1.pngCDel split2.pngCDel node c3.pngCDel 3.pngCDel node c2.pngCDel split1.pngCDel nodeab c4.png
= CDel node.pngCDel 4.pngCDel node c1.pngCDel 3.pngCDel node c3.pngCDel 3.pngCDel node c2.pngCDel 3.pngCDel node c4.pngCDel 4.pngCDel node.png
×4 CDel nodes 11.pngCDel split2.pngCDel node.pngCDel 3.pngCDel node.pngCDel split1.pngCDel nodes.png, CDel nodes.pngCDel split2.pngCDel node 1.pngCDel 3.pngCDel node.pngCDel split1.pngCDel nodes.png, CDel nodes 11.pngCDel split2.pngCDel node 1.pngCDel 3.pngCDel node.pngCDel split1.pngCDel nodes.png, CDel nodes 11.pngCDel split2.pngCDel node.pngCDel 3.pngCDel node 1.pngCDel split1.pngCDel nodes.png, CDel nodes 11.pngCDel split2.pngCDel node 1.pngCDel 3.pngCDel node 1.pngCDel split1.pngCDel nodes.png, CDel nodes 11.pngCDel split2.pngCDel node 1.pngCDel 3.pngCDel node.pngCDel split1.pngCDel nodes 11.png
[<<[31,1,3,1,1]>>]
= [[4,3,3,3,4]]
CDel nodeab c1.pngCDel split2.pngCDel node c2.pngCDel 3.pngCDel node c2.pngCDel split1.pngCDel nodeab c1.png
= CDel node.pngCDel 4.pngCDel node c1.pngCDel 3.pngCDel node c2.pngCDel 3.pngCDel node c2.pngCDel 3.pngCDel node c1.pngCDel 4.pngCDel node.png
×8 CDel nodes.pngCDel split2.pngCDel node 1.pngCDel 3.pngCDel node 1.pngCDel split1.pngCDel nodes.png, CDel nodes 11.pngCDel split2.pngCDel node.pngCDel 3.pngCDel node.pngCDel split1.pngCDel nodes 11.png, CDel nodes 11.pngCDel split2.pngCDel node 1.pngCDel 3.pngCDel node 1.pngCDel split1.pngCDel nodes 11.png

See also[edit]

Regular and uniform honeycombs in 5-space:

References[edit]

  1. ^ http://www.math.rwth-aachen.de/~Gabriele.Nebe/LATTICES/D5.html
  2. ^ Sphere packings, lattices, and groups, by John Horton Conway, Neil James Alexander Sloane, Eiichi Bannai [1]
  3. ^ Conway (1998), p. 119
  4. ^ http://www.math.rwth-aachen.de/~Gabriele.Nebe/LATTICES/Ds5.html
  5. ^ Conway (1998), p. 120
  6. ^ Conway (1998), p. 466
  • Coxeter, H.S.M. Regular Polytopes, (3rd edition, 1973), Dover edition, ISBN 0-486-61480-8
    • 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}, ...
  • Kaleidoscopes: Selected Writings of H.S.M. Coxeter, edited by F. Arthur Sherk, Peter McMullen, Anthony C. Thompson, Asia Ivic Weiss, Wiley-Interscience Publication, 1995, ISBN 978-0-471-01003-6 [2]
    • (Paper 24) H.S.M. Coxeter, Regular and Semi-Regular Polytopes III, [Math. Zeit. 200 (1988) 3-45]
  • Conway JH, Sloane NJH (1998). Sphere Packings, Lattices and Groups (3rd ed.). ISBN 0-387-98585-9. 

External links[edit]