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词条 16-cell honeycomb
释义

  1. Alternate names

  2. Coordinates

  3. D4 lattice

  4. Symmetry constructions

  5. Related honeycombs

  6. See also

  7. Notes

  8. References

16-cell honeycomb

Perspective projection: the first layer of adjacent 16-cell facets.
TypeRegular 4-honeycomb
Uniform 4-honeycomb
FamilyAlternated hypercube honeycomb
Schläfli symbol{3,3,4,3}
Coxeter diagramsnode_1|3|node|3|node|4|node|3|node}}
{{CDD|nodes_10ru|split2|node|3|node|4|node}} = {{CDD|node_h1|4|node|3|node|3|node|4|node}}
{{CDD|nodes_10ru|split2|node|split1|nodes}} = {{CDD|node_h1|4|node|3|node|split1|nodes}}
{{CDD|label2|branch_hh|4a4b|nodes|split2|node}}
4-face type{3,3,4}
Cell type{3,3}
Face type{3}
Edge figurecube
Vertex figure
24-cell
Coxeter group = [3,3,4,3]
Dual{3,4,3,3}
Propertiesvertex-transitive, edge-transitive, face-transitive, cell-transitive, 4-face-transitive

In four-dimensional Euclidean geometry, the 16-cell honeycomb is one of the three regular space-filling tessellations (or honeycombs), represented by Schläfli symbol {3,3,4,3}, and constructed by a 4-dimensional packing of 16-cell facets, three around every face.

Its dual is the 24-cell honeycomb. Its vertex figure is a 24-cell. The vertex arrangement is called the B4, D4, or F4 lattice.[1][2]

Alternate names

  • Hexadecachoric tetracomb/honeycomb
  • Demitesseractic tetracomb/honeycomb

Coordinates

Vertices can be placed at all integer coordinates (i,j,k,l), such that the sum of the coordinates is even.

D4 lattice

The vertex arrangement of the 16-cell honeycomb is called the D4 lattice or F4 lattice.[2] The vertices of this lattice are the centers of the 3-spheres in the densest known packing of equal spheres in 4-space;[3] its kissing number is 24, which is also the same as the kissing number in R4, as proved by Oleg Musin in 2003.[4][5]

The D{{sup sub|+|4}} lattice (also called D{{sup sub|2|4}}) can be constructed by the union of two D4 lattices, and is identical to the tesseractic honeycomb:[6]

{{CDD|nodes_10ru|split2|node|split1|nodes}} ∪ {{CDD|nodes_01rd|split2|node|split1|nodes}} = {{CDD|node_1|4|node|3|node|split1|nodes}} = {{CDD|node_1|4|node|3|node|3|node|4|node}}

This packing is only a lattice for even dimensions. The kissing number is 23 = 8, (2n – 1 for n < 8, 240 for n = 8, and 2n(n – 1) for n > 8).[7]

The D{{sup sub|*|4}} lattice (also called D{{sup sub|4|4}} and C{{sup sub|2|4}}) can be constructed by the union of all four D4 lattices, but it is identical to the D4 lattice: It is also the 4-dimensional body centered cubic, the union of two 4-cube honeycombs in dual positions.[8]

{{CDD|nodes_10ru|split2|node|split1|nodes}} ∪ {{CDD|nodes_01rd|split2|node|split1|nodes}} ∪ {{CDD|nodes|split2|node|split1|nodes_10lu}} ∪ {{CDD|nodes|split2|node|split1|nodes_01ld}} = {{CDD|nodes_10ru|split2|node|split1|nodes}} = {{CDD|nodes_10r|4a4b|nodes|split2|node}} ∪ {{CDD|nodes_01r|4a4b|nodes|split2|node}}.

The kissing number of the D{{sup sub|*|4}} lattice (and D4 lattice) is 24[9] and its Voronoi tessellation is a 24-cell honeycomb, {{CDD|node_1|split1|nodes|4a4b|nodes}}, containing all rectified 16-cells (24-cell) Voronoi cells, {{CDD|node|4|node|3|node_1|3|node}} or {{CDD|node_1|3|node|4|node|3|node}}.[10]

Symmetry constructions

There are three different symmetry constructions of this tessellation. Each symmetry can be represented by different arrangements of colored 16-cell facets.

Coxeter groupSchläfli symbolCoxeter diagramVertex figure
Symmetry		Facets/verf
= [3,3,4,3]{3,3,4,3}node_1|3|node|3|node|4|node|3|node}}node_1|3|node|4|node|3|node}}
[3,4,3], order 1152		24: 16-cell
= [31,1,3,4]= h{4,3,3,4}nodes_10ru|split2|node|3|node|4|node}} = {{CDD|node_h1|4|node|3|node|3|node|4|node}}node|3|node_1|3|node|4|node}}
[3,3,4], order 384		16+8: 16-cell
= [31,1,1,1]{3,31,1,1}
= h{4,3,31,1}		nodes_10ru|split2|node|split1|nodes}} = {{CDD|node_h1|4|node|3|node|split1|nodes}}node|3|node_1|split1|nodes}}
[31,1,1], order 192		8+8+8: 16-cell
2×½ = +)">(4,3,3,4,2+)ht0,4{4,3,3,4}label2|branch_hh|4a4b|nodes|split2|node}}8+4+4: 4-demicube
8: 16-cell

Related honeycombs

It is related to the regular hyperbolic 5-space 5-orthoplex honeycomb, {3,3,3,4,3}, with 5-orthoplex facets, the regular 4-polytope 24-cell, {3,4,3} with octahedral (3-orthoplex) cell, and cube {4,3}, with (2-orthoplex) square faces.

It has a 2-dimensional analogue, {3,6}, and as an alternated form (the demitesseractic honeycomb, h{4,3,3,4}) it is related to the alternated cubic honeycomb.

{{D5 honeycombs}}

See also

Regular and uniform honeycombs in 4-space:

  • Tesseractic honeycomb
  • 24-cell honeycomb
  • Truncated 24-cell honeycomb
  • Snub 24-cell honeycomb
  • 5-cell honeycomb
  • Truncated 5-cell honeycomb
  • Omnitruncated 5-cell honeycomb

Notes

1. ^http://www.math.rwth-aachen.de/~Gabriele.Nebe/LATTICES/F4.html
2. ^1 http://www.math.rwth-aachen.de/~Gabriele.Nebe/LATTICES/D4.html
3. ^Conway and Sloane, Sphere packings, lattices, and groups, 1.4 n-dimensional packings, p.9
4. ^Conway and Sloane, Sphere packings, lattices, and groups, 1.5 Sphere packing problem summary of results. , p.12
5. ^{{cite journal |author=O. R. Musin |title=The problem of the twenty-five spheres |year=2003 |journal=Russ. Math. Surv. |volume=58 |pages=794–795 |doi=10.1070/RM2003v058n04ABEH000651|bibcode=2003RuMaS..58..794M }}
6. ^Conway and Sloane, Sphere packings, lattices, and groups, 7.3 The packing D3+, p.119
7. ^Conway and Sloane, Sphere packings, lattices, and groups, p. 119
8. ^Conway and Sloane, Sphere packings, lattices, and groups, 7.4 The dual lattice D3*, p.120
9. ^Conway and Sloane, Sphere packings, lattices, and groups, p. 120
10. ^Conway and Sloane, Sphere packings, lattices, and groups, 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} = {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}}  
    • (Paper 24) H.S.M. Coxeter, Regular and Semi-Regular Polytopes III, [Math. Zeit. 200 (1988) 3-45]
  • George Olshevsky, Uniform Panoploid Tetracombs, Manuscript (2006) (Complete list of 11 convex uniform tilings, 28 convex uniform honeycombs, and 143 convex uniform tetracombs)
  • {{KlitzingPolytopes|flat.htm|4D|Euclidean tesselations}} x3o3o4o3o - hext - O104
  • {{cite book |author=Conway JH, Sloane NJH |year=1998 |title=Sphere Packings, Lattices and Groups |edition=3rd |isbn=0-387-98585-9}}
{{Honeycombs}}{{DEFAULTSORT:Demitesseractic Honeycomb}}

3 : Honeycombs (geometry)|5-polytopes|Regular tessellations
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