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-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 [math]\displaystyle{ {\tilde{F}}_4 }[/math] = [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), 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 related D+4 lattice (also called D24) can be constructed by the union of two D4 lattices, and is identical to the C4 lattice:[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

The kissing number for D+4 is 23 = 8, (2n – 1 for n < 8, 240 for n = 8, and 2n(n – 1) for n > 8).[7]

The related D*4 lattice (also called D44 and C24) 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

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
[math]\displaystyle{ {\tilde{F}}_4 }[/math] = [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
[math]\displaystyle{ {\tilde{B}}_4 }[/math] = [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
[math]\displaystyle{ {\tilde{D}}_4 }[/math] = [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[math]\displaystyle{ {\tilde{C}}_4 }[/math] = (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

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.


See also

Regular and uniform honeycombs in 4-space:

Notes

  1. "The Lattice F4". http://www.math.rwth-aachen.de/~Gabriele.Nebe/LATTICES/F4.html. 
  2. 2.0 2.1 "The Lattice D4". 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 (4): 794–795. doi:10.1070/RM2003v058n04ABEH000651. Bibcode2003RuMaS..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 [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". https://bendwavy.org/klitzing/dimensions/flat.htm.  x3o3o4o3o - hext - O104
  • Conway JH, Sloane NJH (1998). Sphere Packings, Lattices and Groups (3rd ed.). ISBN 0-387-98585-9. https://archive.org/details/spherepackingsla0000conw_b8u0. 
Fundamental convex regular and uniform honeycombs in dimensions 2-9
Space Family [math]\displaystyle{ {\tilde{A}}_{n-1} }[/math] [math]\displaystyle{ {\tilde{C}}_{n-1} }[/math] [math]\displaystyle{ {\tilde{B}}_{n-1} }[/math] [math]\displaystyle{ {\tilde{D}}_{n-1} }[/math] [math]\displaystyle{ {\tilde{G}}_2 }[/math] / [math]\displaystyle{ {\tilde{F}}_4 }[/math] / [math]\displaystyle{ {\tilde{E}}_{n-1} }[/math]
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