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16-cell honeycomb

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!bgcolor=#e7dcc3 colspan=2|16-cell honeycomb

bgcolor=#ffffff align=center colspan=2|280px
Perspective projection: the first layer of adjacent 16-cell facets.
bgcolor=#e7dcc3|TypeRegular 4-honeycomb
Uniform 4-honeycomb
bgcolor=#e7dcc3|FamilyAlternated hypercube honeycomb
bgcolor=#e7dcc3|Schläfli symbol{3,3,4,3}
bgcolor=#e7dcc3|Coxeter diagrams{{CDD|node_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}}
bgcolor=#e7dcc3|4-face type{3,3,4} 40px
bgcolor=#e7dcc3|Cell type{3,3} 20px
bgcolor=#e7dcc3|Face type{3}
bgcolor=#e7dcc3|Edge figurecube
bgcolor=#e7dcc3|Vertex figure80px
24-cell
bgcolor=#e7dcc3|Coxeter group{\tilde{F}}_4 = [3,3,4,3]
bgcolor=#e7dcc3|Dual{3,4,3,3}
bgcolor=#e7dcc3|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.{{Cite web|url=http://www.math.rwth-aachen.de/~Gabriele.Nebe/LATTICES/F4.html|title = The Lattice F4}}{{Cite web|url=http://www.math.rwth-aachen.de/~Gabriele.Nebe/LATTICES/D4.html|title = The Lattice D4}}

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.

D<sub>4</sub> 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;Conway and Sloane, Sphere packings, lattices, and groups, 1.4 n-dimensional packings, p.9 its kissing number is 24, which is also the same as the kissing number in R4, as proved by Oleg Musin in 2003.Conway and Sloane, Sphere packings, lattices, and groups, 1.5 Sphere packing problem summary of results, p. 12{{cite journal |author=O. R. Musin |title=The problem of the twenty-five spheres |year=2003 |journal=Russ. Math. Surv. |volume=58 |issue=4 |pages=794–795 |doi=10.1070/RM2003v058n04ABEH000651|bibcode=2003RuMaS..58..794M }}

The related 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 C4 lattice:Conway and Sloane, Sphere packings, lattices, and groups, 7.3 The packing D3+, p.119

:{{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}}

The kissing number for D{{sup sub|+|4}} is 23 = 8, (2n − 1 for n < 8, 240 for n = 8, and 2n(n − 1) for n > 8).Conway and Sloane, Sphere packings, lattices, and groups, p. 119

The related 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.Conway and Sloane, Sphere packings, lattices, and groups, 7.4 The dual lattice D3*, p.120

:{{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 24Conway and Sloane, Sphere packings, lattices, and groups, p. 120 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}}.Conway and Sloane, Sphere packings, lattices, and groups, p. 466

Symmetry constructions

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

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!Coxeter group

!Schläfli symbol

!Coxeter diagram

!Vertex figure
Symmetry

!Facets/verf

{\tilde{F}}_4 = [3,3,4,3]{3,3,4,3}{{CDD|node_1|3|node|3|node|4|node|3|node}}{{CDD|node_1|3|node|4|node|3|node}}
[3,4,3], order 1152		24: 16-cell
{\tilde{B}}_4 = [31,1,3,4]= h{4,3,3,4}{{CDD|nodes_10ru|split2|node|3|node|4|node}} = {{CDD|node_h1|4|node|3|node|3|node|4|node}}{{CDD|node|3|node_1|3|node|4|node}}
[3,3,4], order 384		16+8: 16-cell
{\tilde{D}}_4 = [31,1,1,1]{3,31,1,1}
= h{4,3,31,1}		{{CDD|nodes_10ru|split2|node|split1|nodes}} = {{CDD|node_h1|4|node|3|node|split1|nodes}}{{CDD|node|3|node_1|split1|nodes}}
[31,1,1], order 192		8+8+8: 16-cell
2×½{\tilde{C}}_4 = (4,3,3,4,2+)ht0,4{4,3,3,4}{{CDD|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:

Notes

{{reflist}}

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}} [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)
  • {{KlitzingPolytopes|flat.htm|4D|Euclidean tesselations}} x3o3o4o3o - hext - O104
  • {{cite book |author=Conway JH, Sloane NJH |year=1998 |title=Sphere Packings, Lattices and Groups |publisher=Springer |edition=3rd |isbn=0-387-98585-9 |url-access=registration |url=https://archive.org/details/spherepackingsla0000conw_b8u0 }}

{{Honeycombs}}

Category:Honeycombs (geometry)

Category:5-polytopes

Category:Regular tessellations