16-cell honeycomb

From Wikipedia, the free encyclopedia
Jump to: navigation, search
16-cell honeycomb
Demitesseractic tetra hc.png
Perspective projection: the first layer of adjacent 16-cell facets.
Type Regular 4-honeycomb
Uniform 4-honeycomb
Family Alternated hypercube honeycomb
Schläfli symbol {3,3,4,3}
Coxeter diagrams CDel node 1.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.pngCDel 4.pngCDel node.pngCDel 3.pngCDel node.png
CDel nodes 10ru.pngCDel split2.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 4.pngCDel node.png
CDel nodes 10ru.pngCDel split2.pngCDel node.pngCDel split1.pngCDel nodes.png = CDel node h1.pngCDel 4.pngCDel node.pngCDel 3.pngCDel node.pngCDel split1.pngCDel nodes.png
CDel label2.pngCDel branch hh.pngCDel 4a4b.pngCDel nodes.pngCDel split2.pngCDel node.png
4-face type {3,3,4} Schlegel wireframe 16-cell.png
Cell type {3,3} Tetrahedron.png
Face type {3}
Edge figure cube
Vertex figure 24-cell t0 F4.svg
24-cell
Coxeter group = [3,3,4,3]
Dual {3,4,3,3}
Properties vertex-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) in Euclidean 4-space. The other two are its dual the 24-cell honeycomb, and the tesseractic honeycomb. This honeycomb is constructed from 16-cell facets, three around every face. It has a 24-cell vertex figure.

This vertex arrangement or lattice is called the B4, D4, or F4 lattice.[1][2]

Alternate names[edit]

  • Hexadecachoric tetracomb/honeycomb
  • Demitesseractic tetracomb/honeycomb

Coordinates[edit]

As a regular honeycomb, {3,3,4,3}, 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.

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

D4 lattice[edit]

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+
4
lattice (also called D2
4
) can be constructed by the union of two D4 lattices, and is identical to the tesseractic honeycomb:[6]

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

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*
4
lattice (also called D4
4
and C2
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]

CDel nodes 10ru.pngCDel split2.pngCDel node.pngCDel split1.pngCDel nodes.pngCDel nodes 01rd.pngCDel split2.pngCDel node.pngCDel split1.pngCDel nodes.pngCDel nodes.pngCDel split2.pngCDel node.pngCDel split1.pngCDel nodes 10lu.pngCDel nodes.pngCDel split2.pngCDel node.pngCDel split1.pngCDel nodes 01ld.png = CDel nodes 10ru.pngCDel split2.pngCDel node.pngCDel split1.pngCDel nodes.png = CDel nodes 10r.pngCDel 4a4b.pngCDel nodes.pngCDel split2.pngCDel node.pngCDel nodes 01r.pngCDel 4a4b.pngCDel nodes.pngCDel split2.pngCDel node.png.

The kissing number of the D*
4
lattice (and D4 lattice) is 24[9] and its Voronoi tessellation is a 24-cell honeycomb, CDel node 1.pngCDel split1.pngCDel nodes.pngCDel 4a4b.pngCDel nodes.png, containing all rectified 16-cells (24-cell) Voronoi cells, CDel node.pngCDel 4.pngCDel node.pngCDel 3.pngCDel node 1.pngCDel 3.pngCDel node.png or CDel node 1.pngCDel 3.pngCDel node.pngCDel 4.pngCDel node.pngCDel 3.pngCDel node.png.[10]

Symmetry constructions[edit]

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

Coxeter group Schläfli symbol Coxeter diagram Vertex figure
Symmetry
Facets/verf
= [3,3,4,3] {3,3,4,3} CDel node 1.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.pngCDel 4.pngCDel node.pngCDel 3.pngCDel node.png CDel node 1.pngCDel 3.pngCDel node.pngCDel 4.pngCDel node.pngCDel 3.pngCDel node.png
[3,4,3], order 1152
24: 16-cell
= [31,1,3,4] = h{4,3,3,4} CDel nodes 10ru.pngCDel split2.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 4.pngCDel node.png CDel node.pngCDel 3.pngCDel node 1.pngCDel 3.pngCDel node.pngCDel 4.pngCDel node.png
[3,3,4], order 384
16+8: 16-cell
= [31,1,1,1] {3,31,1,1}
= h{4,3,31,1}
CDel nodes 10ru.pngCDel split2.pngCDel node.pngCDel split1.pngCDel nodes.png = CDel node h1.pngCDel 4.pngCDel node.pngCDel 3.pngCDel node.pngCDel split1.pngCDel nodes.png CDel node.pngCDel 3.pngCDel node 1.pngCDel split1.pngCDel nodes.png
[31,1,1], order 192
8+8+8: 16-cell
2×½ = [[(4,3,3,4,2+)]] ht0,4{4,3,3,4} CDel label2.pngCDel branch hh.pngCDel 4a4b.pngCDel nodes.pngCDel split2.pngCDel node.png 8+4+4: 4-demicube
8: 16-cell

Related honeycombs[edit]

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.

This honeycomb is one of 20 uniform honeycombs constructed by the 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:

See also[edit]

Regular and uniform honeycombs in 4-space:

Notes[edit]

  1. ^ http://www.math.rwth-aachen.de/~Gabriele.Nebe/LATTICES/F4.html
  2. ^ a b 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. ^ O. R. Musin (2003). "The problem of the twenty-five spheres". Russ. Math. Surv. 58: 794–795. doi:10.1070/RM2003v058n04ABEH000651. 
  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[edit]

  • 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 [1]
    • (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)
  • Klitzing, Richard. "4D Euclidean tesselations".  x3o3o4o3o - hext - O104
  • Conway JH, Sloane NJH (1998). Sphere Packings, Lattices and Groups (3rd ed.). ISBN 0-387-98585-9. 
Fundamental convex regular and uniform honeycombs in dimensions 2-9
Space Family / /
E2 Uniform tiling {3[3]} δ3 3 3 Hexagonal
E3 Uniform convex honeycomb {3[4]} δ4 4 4
E4 Uniform 4-honeycomb {3[5]} δ5 5 5 24-cell honeycomb
E5 Uniform 5-honeycomb {3[6]} δ6 6 6
E6 Uniform 6-honeycomb {3[7]} δ7 7 7 222
E7 Uniform 7-honeycomb {3[8]} δ8 8 8 133331
E8 Uniform 8-honeycomb {3[9]} δ9 9 9 152251521
E9 Uniform 9-honeycomb {3[10]} δ10 10 10
En-1 Uniform (n-1)-honeycomb {3[n]} δn n n 1k22k1k21