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16-cell honeycomb

16-cell honeycomb
[[Image:Demitesseractic tetra hc.png280px]]
Perspective projection: the first layer of adjacent 16-cell facets.
Type
Family
Schläfli symbol
Coxeter diagrams
4-face type
Cell type
Face type
Edge figure
Vertex figure
Coxeter group
Dual
Properties

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 (triangular) 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.

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. The vertices of this lattice are the centers of the 3-spheres in the densest known packing of equal spheres in 4-space; its kissing number is 24, which is also the same as the kissing number in R4, as proved by Oleg Musin in 2003.

The related D lattice (also called D) can be constructed by the union of two D4 lattices, and is identical to the C4 lattice: : ∪ = =

The kissing number for D is 23 = 8, (2n − 1 for n 8).

The related D lattice (also called D and C) 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. : ∪ ∪ ∪ = = ∪ .

The kissing number of the D lattice (and D4 lattice) is 24 and its Voronoi tessellation is a 24-cell honeycomb, , containing all rectified 16-cells (24-cell) Voronoi cells, or .

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 figureSymmetryFacets/verf
[3,3,4,3]]]{3,3,4,3}[3,4,3], order 115224: [16-cell](16-cell)
[31,1,3,4]]]= h{4,3,3,4}=[3,3,4], order 38416+8: [16-cell](16-cell)
[31,1,1,1]]]{3,31,1,1} = h{4,3,31,1}=[31,1,1], order 1928+8+8: [16-cell](16-cell)
2×½{\tilde{C}}_4 = [(4,3,3,4,2+)](4-3-3-4-2-sup-sup)ht0,4{4,3,3,4}8+4+4: [4-demicube](4-demicube)8: [16-cell](16-cell)

Notes

References

  • Coxeter, H.S.M. Regular Polytopes, (3rd edition, 1973), Dover edition,
    • 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, http://www.wiley.com/WileyCDA/WileyTitle/productCd-0471010030.html
    • (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)
  • x3o3o4o3o - hext - O104

References

  1. "The Lattice F4".
  2. "The Lattice D4".
  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. O. R. Musin. (2003). "The problem of the twenty-five spheres". Russ. Math. Surv..
  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
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honeycombs-(geometry) 5-polytopes regular-tessellations
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