“8-demicubic honeycomb”的意思、由来-开放百科全书

8-demicubic honeycomb

(No image)

Type

Uniform 8-honeycomb

Family

Alternated hypercube honeycomb

Schläfli symbol

h{4,3,3,3,3,3,3,4}

Coxeter diagrams

nodes_10ru|split2|node|3|node|3|node|3|node|3|node|3|node|4|node}} = {{CDD|node_h1|4|node|3|node|3|node|3|node|3|node|3|node|3|node|4|node}}
{{CDD|nodes_10ru|split2|node|3|node|3|node|3|node|3|node|split1|nodes}} = {{CDD|node_h1|4|node|3|node|3|node|3|node|3|node|3|node|split1|nodes}}
{{CDD|label2|branch_hh|4a4b|nodes|3ab|nodes|3ab|nodes|split2|node}}

Facets

{3,3,3,3,3,3,4}
h{4,3,3,3,3,3,3}

Vertex figure

Rectified 8-orthoplex

Coxeter group

[4,3,3,3,3,3,3^1,1]

[3^1,1,3,3,3,3,3^1,1]

The 8-demicubic honeycomb, or demiocteractic honeycomb is a uniform space-filling tessellation (or honeycomb) in Euclidean 8-space. It is constructed as an alternation of the regular 8-cubic honeycomb.

It is composed of two different types of facets. The 8-cubes become alternated into 8-demicubes h{4,3,3,3,3,3,3} and the alternated vertices create 8-orthoplex {3,3,3,3,3,3,4} facets .

D8 lattice

The vertex arrangement of the 8-demicubic honeycomb is the D₈ lattice.^[1] The 112 vertices of the rectified 8-orthoplex vertex figure of the 8-demicubic honeycomb reflect the kissing number 112 of this lattice.^[2] The best known is 240, from the E₈ lattice and the 5₂₁ honeycomb.

contains

as a subgroup of index 270.^[3] Both

and

can be seen as affine extensions of

from different nodes:

The D{{sup sub|+|8}} lattice (also called D{{sup sub|2|8}}) can be constructed by the union of two D8 lattices.^[4] This packing is only a lattice for even dimensions. The kissing number is 240. (2^n-1 for n<8, 240 for n=8, and 2n(n-1) for n>8).^[5] It is identical to the E8 lattice. At 8-dimensions, the 240 contacts contain both the 2⁷=128 from lower dimension contact progression (2^n-1), and 16*7=112 from higher dimensions (2n(n-1)).

The D{{sup sub|*|8}} lattice (also called D{{sup sub|4|8}} and C{{sup sub|2|8}}) can be constructed by the union of all four D8 lattices:^[6] It is also the 7-dimensional body centered cubic, the union of two 7-cube honeycombs in dual positions.

The kissing number of the D{{sup sub|*|8}} lattice is 16 (2n for n≥5).^[7] and its Voronoi tessellation is a quadrirectified 8-cubic honeycomb, {{CDD|node_1|split1|nodes|3ab|nodes|3ab|nodes|4a4b|nodes}}, containing all trirectified 8-orthoplex Voronoi cell, {{CDD|node|4|node|3|node|3|node|3|node_1|3|node|3|node|3|node}}.^[8]

Symmetry constructions

There are three uniform construction symmetries of this tessellation. Each symmetry can be represented by arrangements of different colors on the 256 8-demicube facets around each vertex.

Coxeter group

Schläfli symbol

Coxeter-Dynkin diagram

Vertex figure
Symmetry

Facets/verf

= [3^1,1,3,3,3,3,3,4]
= [1⁺,4,3,3,3,3,3,3,4]

h{4,3,3,3,3,3,3,4}

nodes_10ru|split2|node|3|node|3|node|3|node|3|node|3|node|4|node}} = {{CDD|node_h1|4|node|3|node|3|node|3|node|3|node|3|node|3|node|4|node}}

node|3|node_1|3|node|3|node|3|node|3|node|3|node|4|node}}
[3,3,3,3,3,3,4]

256: 8-demicube
16: 8-orthoplex

= [3^1,1,3,3,3,3^1,1]
= [1⁺,4,3,3,3,3,3^1,1]

h{4,3,3,3,3,3,3^1,1}

nodes_10ru|split2|node|3|node|3|node|3|node|3|node|split1|nodes}} = {{CDD|node_h1|4|node|3|node|3|node|3|node|3|node|3|node|split1|nodes}}

node|3|node_1|3|node|3|node|3|node|3|node|split1|nodes}}
[3^6,1,1]

128+128: 8-demicube
16: 8-orthoplex

2×½

= +)">(4,3,3,3,3,3,4,2⁺)

ht_0,8{4,3,3,3,3,3,3,4}

label2|branch_hh|4a4b|nodes|3ab|nodes|3ab|nodes|split2|node}}

128+64+64: 8-demicube
16: 8-orthoplex

See also

Notes

1. ^http://www.math.rwth-aachen.de/~Gabriele.Nebe/LATTICES/D8.html
2. ^Sphere packings, lattices, and groups, by John Horton Conway, Neil James Alexander Sloane, Eiichi Bannai[https://books.google.com/books?id=upYwZ6cQumoC&lpg=PP1&dq=Sphere%20Packings%2C%20Lattices%20and%20Groups&pg=PR19#v=onepage&q=&f=false]
3. ^Johnson (2015) p.177
4. ^Kaleidoscopes: Selected Writings of H. S. M. Coxeter, Paper 18, "Extreme forms" (1950)
5. ^Conway (1998), p. 119
6. ^http://www.math.rwth-aachen.de/~Gabriele.Nebe/LATTICES/Ds8.html
7. ^Conway (1998), p. 120
8. ^Conway (1998), p. 466

References

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}={3^1,1,4}, 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}}

(Paper 24) H.S.M. Coxeter, Regular and Semi-Regular Polytopes III, [Math. Zeit. 200 (1988) 3-45]

N.W. Johnson: Geometries and Transformations, (2018)

{{cite book |author=Conway JH, Sloane NJH |year=1998 |title=Sphere Packings, Lattices and Groups |edition=3rd |isbn=0-387-98585-9}}

D8 lattice

Symmetry constructions

See also

Notes

References

External links