Simplectic honeycomb

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Short description: Tiling of n-dimensional space
[math]\displaystyle{ {\tilde{A}}_2 }[/math] [math]\displaystyle{ {\tilde{A}}_3 }[/math]
Triangular tiling Tetrahedral-octahedral honeycomb
Uniform tiling 333-t1.png
With red and yellow equilateral triangles
Tetrahedral-octahedral honeycomb2.png
With cyan and yellow tetrahedra, and red rectified tetrahedra (octahedra)
CDel node 1.pngCDel split1.pngCDel branch.png CDel node 1.pngCDel split1.pngCDel nodes.pngCDel split2.pngCDel node.png

In geometry, the simplectic honeycomb (or n-simplex honeycomb) is a dimensional infinite series of honeycombs, based on the [math]\displaystyle{ {\tilde{A}}_n }[/math] affine Coxeter group symmetry. It is represented by a Coxeter-Dynkin diagram as a cyclic graph of n + 1 nodes with one node ringed. It is composed of n-simplex facets, along with all rectified n-simplices. It can be thought of as an n-dimensional hypercubic honeycomb that has been subdivided along all hyperplanes [math]\displaystyle{ x+y+\cdots\in\mathbb{Z} }[/math], then stretched along its main diagonal until the simplices on the ends of the hypercubes become regular. The vertex figure of an n-simplex honeycomb is an expanded n-simplex.

In 2 dimensions, the honeycomb represents the triangular tiling, with Coxeter graph CDel node 1.pngCDel split1.pngCDel branch.png filling the plane with alternately colored triangles. In 3 dimensions it represents the tetrahedral-octahedral honeycomb, with Coxeter graph CDel node 1.pngCDel split1.pngCDel nodes.pngCDel split2.pngCDel node.png filling space with alternately tetrahedral and octahedral cells. In 4 dimensions it is called the 5-cell honeycomb, with Coxeter graph CDel node 1.pngCDel split1.pngCDel nodes.pngCDel 3ab.pngCDel branch.png, with 5-cell and rectified 5-cell facets. In 5 dimensions it is called the 5-simplex honeycomb, with Coxeter graph CDel node 1.pngCDel split1.pngCDel nodes.pngCDel 3ab.pngCDel nodes.pngCDel split2.pngCDel node.png, filling space by 5-simplex, rectified 5-simplex, and birectified 5-simplex facets. In 6 dimensions it is called the 6-simplex honeycomb, with Coxeter graph CDel node 1.pngCDel split1.pngCDel nodes.pngCDel 3ab.pngCDel nodes.pngCDel 3ab.pngCDel branch.png, filling space by 6-simplex, rectified 6-simplex, and birectified 6-simplex facets.

By dimension

n [math]\displaystyle{ {\tilde{A}}_{2+} }[/math] Tessellation Vertex figure Facets per vertex figure Vertices per vertex figure Edge figure
1 [math]\displaystyle{ {\tilde{A}}_1 }[/math] Regular apeirogon.svg
Apeirogon
CDel node 1.pngCDel infin.pngCDel node.png
Line segment
CDel node 1.png
2 2 Point
2 [math]\displaystyle{ {\tilde{A}}_2 }[/math] Uniform tiling 333-t1.png
Triangular tiling
2-simplex honeycomb
CDel node 1.pngCDel split1.pngCDel branch.png
Truncated triangle.svg
Hexagon
(Truncated triangle)
CDel node 1.pngCDel 3.pngCDel node 1.png
3+3 triangles 6 Line segment
3 [math]\displaystyle{ {\tilde{A}}_3 }[/math] Tetrahedral-octahedral honeycomb2.png
Tetrahedral-octahedral honeycomb
3-simplex honeycomb
CDel node 1.pngCDel split1.pngCDel nodes.pngCDel split2.pngCDel node.png
Uniform t0 3333 honeycomb verf2.png
Cuboctahedron
(Cantellated tetrahedron)
CDel node 1.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node 1.png
4+4 tetrahedron
6 rectified tetrahedra
12 Cuboctahedron vertfig.png
Rectangle
4 [math]\displaystyle{ {\tilde{A}}_4 }[/math] 4-simplex honeycomb
CDel node 1.pngCDel split1.pngCDel nodes.pngCDel 3ab.pngCDel branch.png
4-simplex honeycomb verf.png
Runcinated 5-cell
CDel node 1.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node 1.png
5+5 5-cells
10+10 rectified 5-cells
20 Runcinated 5-cell verf.png
Triangular antiprism
5 [math]\displaystyle{ {\tilde{A}}_5 }[/math] 5-simplex honeycomb
CDel node 1.pngCDel split1.pngCDel nodes.pngCDel 3ab.pngCDel nodes.pngCDel split2.pngCDel node.png
5-simplex t04 A4.svg
Stericated 5-simplex
CDel node 1.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node 1.png
6+6 5-simplex
15+15 rectified 5-simplex
20 birectified 5-simplex
30 Stericated hexateron verf.png
Tetrahedral antiprism
6 [math]\displaystyle{ {\tilde{A}}_6 }[/math] 6-simplex honeycomb
CDel node 1.pngCDel split1.pngCDel nodes.pngCDel 3ab.pngCDel nodes.pngCDel 3ab.pngCDel branch.png
6-simplex t05.svg
Pentellated 6-simplex
CDel node 1.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node 1.png
7+7 6-simplex
21+21 rectified 6-simplex
35+35 birectified 6-simplex
42 4-simplex antiprism
7 [math]\displaystyle{ {\tilde{A}}_7 }[/math] 7-simplex honeycomb
CDel node 1.pngCDel split1.pngCDel nodes.pngCDel 3ab.pngCDel nodes.pngCDel 3ab.pngCDel nodes.pngCDel split2.pngCDel node.png
7-simplex t06 A6.svg
Hexicated 7-simplex
CDel node 1.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node 1.png
8+8 7-simplex
28+28 rectified 7-simplex
56+56 birectified 7-simplex
70 trirectified 7-simplex
56 5-simplex antiprism
8 [math]\displaystyle{ {\tilde{A}}_8 }[/math] 8-simplex honeycomb
CDel node 1.pngCDel split1.pngCDel nodes.pngCDel 3ab.pngCDel nodes.pngCDel 3ab.pngCDel nodes.pngCDel 3ab.pngCDel branch.png
8-simplex t07.svg
Heptellated 8-simplex
CDel node 1.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node 1.png
9+9 8-simplex
36+36 rectified 8-simplex
84+84 birectified 8-simplex
126+126 trirectified 8-simplex
72 6-simplex antiprism
9 [math]\displaystyle{ {\tilde{A}}_9 }[/math] 9-simplex honeycomb
CDel node 1.pngCDel split1.pngCDel nodes.pngCDel 3ab.pngCDel nodes.pngCDel 3ab.pngCDel nodes.pngCDel 3ab.pngCDel nodes.pngCDel split2.pngCDel node.png
9-simplex t08.svg
Octellated 9-simplex
CDel node 1.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node 1.png
10+10 9-simplex
45+45 rectified 9-simplex
120+120 birectified 9-simplex
210+210 trirectified 9-simplex
252 quadrirectified 9-simplex
90 7-simplex antiprism
10 [math]\displaystyle{ {\tilde{A}}_{10} }[/math] 10-simplex honeycomb
CDel node 1.pngCDel split1.pngCDel nodes.pngCDel 3ab.pngCDel nodes.pngCDel 3ab.pngCDel nodes.pngCDel 3ab.pngCDel nodes.pngCDel 3ab.pngCDel branch.png
10-simplex t09.svg
Ennecated 10-simplex
CDel node 1.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node 1.png
11+11 10-simplex
55+55 rectified 10-simplex
165+165 birectified 10-simplex
330+330 trirectified 10-simplex
462+462 quadrirectified 10-simplex
110 8-simplex antiprism
11 [math]\displaystyle{ {\tilde{A}}_{11} }[/math] 11-simplex honeycomb ... ... ... ...

Projection by folding

The (2n-1)-simplex honeycombs and 2n-simplex honeycombs can be projected into the n-dimensional hypercubic honeycomb by a geometric folding operation that maps two pairs of mirrors into each other, sharing the same vertex arrangement:

[math]\displaystyle{ {\tilde{A}}_2 }[/math] CDel node 1.pngCDel split1.pngCDel branch.png [math]\displaystyle{ {\tilde{A}}_4 }[/math] CDel node 1.pngCDel split1.pngCDel nodes.pngCDel 3ab.pngCDel branch.png [math]\displaystyle{ {\tilde{A}}_6 }[/math] CDel node 1.pngCDel split1.pngCDel nodes.pngCDel 3ab.pngCDel nodes.pngCDel 3ab.pngCDel branch.png [math]\displaystyle{ {\tilde{A}}_8 }[/math] CDel node 1.pngCDel split1.pngCDel nodes.pngCDel 3ab.pngCDel nodes.pngCDel 3ab.pngCDel nodes.pngCDel 3ab.pngCDel branch.png [math]\displaystyle{ {\tilde{A}}_{10} }[/math] CDel node 1.pngCDel split1.pngCDel nodes.pngCDel 3ab.pngCDel nodes.pngCDel 3ab.pngCDel nodes.pngCDel 3ab.pngCDel nodes.pngCDel 3ab.pngCDel branch.png ...
[math]\displaystyle{ {\tilde{A}}_3 }[/math] CDel nodes 10r.pngCDel splitcross.pngCDel nodes.png [math]\displaystyle{ {\tilde{A}}_3 }[/math] CDel node 1.pngCDel split1.pngCDel nodes.pngCDel split2.pngCDel node.png [math]\displaystyle{ {\tilde{A}}_5 }[/math] CDel node 1.pngCDel split1.pngCDel nodes.pngCDel 3ab.pngCDel nodes.pngCDel split2.pngCDel node.png [math]\displaystyle{ {\tilde{A}}_7 }[/math] CDel node 1.pngCDel split1.pngCDel nodes.pngCDel 3ab.pngCDel nodes.pngCDel 3ab.pngCDel nodes.pngCDel split2.pngCDel node.png [math]\displaystyle{ {\tilde{A}}_9 }[/math] CDel node 1.pngCDel split1.pngCDel nodes.pngCDel 3ab.pngCDel nodes.pngCDel 3ab.pngCDel nodes.pngCDel 3ab.pngCDel nodes.pngCDel split2.pngCDel node.png ...
[math]\displaystyle{ {\tilde{C}}_1 }[/math] CDel node 1.pngCDel infin.pngCDel node.png [math]\displaystyle{ {\tilde{C}}_2 }[/math] CDel node 1.pngCDel 4.pngCDel node.pngCDel 4.pngCDel node.png [math]\displaystyle{ {\tilde{C}}_3 }[/math] CDel node 1.pngCDel 4.pngCDel node.pngCDel 3.pngCDel node.pngCDel 4.pngCDel node.png [math]\displaystyle{ {\tilde{C}}_4 }[/math] CDel node 1.pngCDel 4.pngCDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.pngCDel 4.pngCDel node.png [math]\displaystyle{ {\tilde{C}}_5 }[/math] CDel node 1.pngCDel 4.pngCDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.pngCDel 4.pngCDel node.png ...

Kissing number

These honeycombs, seen as tangent n-spheres located at the center of each honeycomb vertex have a fixed number of contacting spheres and correspond to the number of vertices in the vertex figure. This represents the highest kissing number for 2 and 3 dimensions, but falls short on higher dimensions. In 2-dimensions, the triangular tiling defines a circle packing of 6 tangent spheres arranged in a regular hexagon, and for 3 dimensions there are 12 tangent spheres arranged in a cuboctahedral configuration. For 4 to 8 dimensions, the kissing numbers are 20, 30, 42, 56, and 72 spheres, while the greatest solutions are 24, 40, 72, 126, and 240 spheres respectively.

See also

References

  • George Olshevsky, Uniform Panoploid Tetracombs, Manuscript (2006) (Complete list of 11 convex uniform tilings, 28 convex uniform honeycombs, and 143 convex uniform tetracombs)
  • Branko Grünbaum, Uniform tilings of 3-space. Geombinatorics 4(1994), 49 - 56.
  • Norman Johnson Uniform Polytopes, Manuscript (1991)
  • Coxeter, H.S.M. Regular Polytopes, (3rd edition, 1973), Dover edition, ISBN 0-486-61480-8
  • 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 22) H.S.M. Coxeter, Regular and Semi Regular Polytopes I, [Math. Zeit. 46 (1940) 380-407, MR 2,10] (1.9 Uniform space-fillings)
    • (Paper 24) H.S.M. Coxeter, Regular and Semi-Regular Polytopes III, [Math. Zeit. 200 (1988) 3-45]
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