16-cell honeycomb

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16-cell honeycomb
Demitesseractic tetra hc.png
Perspective projection: the first layer of adjacent 16-cell facets.
Type Regular 4-space honeycomb
Uniform 4-honeycomb
Family Alternated hypercube honeycomb
Schläfli symbol {3,3,4,3}
Coxeter-Dynkin diagram CDel node 1.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.pngCDel 4.pngCDel node.pngCDel 3.pngCDel node.png
CDel node h.pngCDel 4.pngCDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.pngCDel 4.pngCDel node.png
CDel node h.pngCDel 4.pngCDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.pngCDel 4.pngCDel node h.png
CDel nodes 10ru.pngCDel split2.pngCDel node.pngCDel 3.pngCDel node.pngCDel 4.pngCDel node.png
CDel nodes 10ru.pngCDel split2.pngCDel node.pngCDel split1.pngCDel nodes.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 {\tilde{F}}_4 = [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 the one of three regular space-filling tessellation (or honeycomb) 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]

Its vertex arrangement is called the D4 lattice or F4 lattice.[2] The vertices of this lattice are the centers of the 3-spheres in the densest possible packing of equal spheres in 4-space; its kissing number is 24, which is also the highest possible in 4-space.[3]

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

The D+
4
lattice (also called D2
4
) can be constructed by the union of two 4-demicubic lattices, and is identical to the tesseractic honeycomb:

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).[4]

The D*
4
lattice (also called D4
4
and C2
4
) can be constructed by the union of all four 5-demicubic honeycombs, 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.

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 node 1.pngCDel 4.pngCDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.pngCDel 4.pngCDel node.pngCDel node.pngCDel 4.pngCDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.pngCDel 4.pngCDel node 1.png.

The kissing number of the D*
4
lattice (and D4 lattice) is 24[5] 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.[6]

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.

Name Coxeter group Schläfli symbol Coxeter diagram Vertex figure
Symmetry
Facets/verf
16-cell honeycomb {\tilde{F}}_4 = [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
4-demicube honeycomb {\tilde{B}}_4 = [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
{\tilde{D}}_4 = [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

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.

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. ^ O. R. Musin (2003). "The problem of the twenty-five spheres". Russ. Math. Surv. 58: 794–795. doi:10.1070/RM2003v058n04ABEH000651. 
  4. ^ Conway (1998), p. 119
  5. ^ Conway (1998), p. 120
  6. ^ Conway (1998), 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)
  • Richard Klitzing, 4D, Euclidean tesselations x3o3o4o3o - hext - O104
  • Conway JH, Sloane NJH (1998). Sphere Packings, Lattices and Groups (3rd ed.). ISBN 0-387-98585-9.