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词条 Simplectic honeycomb
释义

  1. By dimension

  2. Projection by folding

  3. Kissing number

  4. See also

  5. References

Triangular tilingTetrahedral-octahedral honeycomb

With red and yellow equilateral triangles
With cyan and yellow tetrahedra, and red rectified tetrahedra (octahedra)
node_1|split1|branchnode_1|split1|nodes|split2|node

In geometry, the simplectic honeycomb (or n-simplex honeycomb) is a dimensional infinite series of honeycombs, based on the affine Coxeter group symmetry. It is given a Schläfli symbol {3[n+1]}, and 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 , 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 {{CDD|node_1|split1|branch}} filling the plane with alternately colored triangles. In 3 dimensions it represents the tetrahedral-octahedral honeycomb, with Coxeter graph {{CDD|node_1|split1|nodes|split2|node}} filling space with alternately tetrahedral and octahedral cells. In 4 dimensions it is called the 5-cell honeycomb, with Coxeter graph {{CDD|node_1|split1|nodes|3ab|branch}}, with 5-cell and rectified 5-cell facets. In 5 dimensions it is called the 5-simplex honeycomb, with Coxeter graph {{CDD|node_1|split1|nodes|3ab|nodes|split2|node}}, 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 {{CDD|node_1|split1|nodes|3ab|nodes|3ab|branch}}, filling space by 6-simplex, rectified 6-simplex, and birectified 6-simplex facets.

By dimension

nTessellationVertex figureFacets per vertex figureVertices per vertex figureEdge figure
1Apeirogon
{{CDD>node_1|infin|node}}		node_1}}12 -
2Triangular tiling
2-simplex honeycomb
{{CDD>node_1|split1|branch}}		Hexagon
(Truncated triangle)
{{CDD>node_1|3|node_1}}		3+3 triangles6{{CDD>node_1}}
3Tetrahedral-octahedral honeycomb
3-simplex honeycomb
{{CDD>node_1|split1|nodes|split2|node}}		Cuboctahedron
(Cantellated tetrahedron)
{{CDD>node_1|3|node|3|node_1}}		4+4 tetrahedron
6 rectified tetrahedra		12Rectangle
{{CDD>node_1|2|node_1}}
4{{CDD>node_1|split1|nodes|3ab|branch}}Runcinated 5-cell
{{CDD>node_1|3|node|3|node|3|node_1}}		5+5 5-cells
10+10 rectified 5-cells		20Triangular antiprism
{{CDD>node_h|3|node_h|2x|node_h}}
5{{CDD>node_1|split1|nodes|3ab|nodes|split2|node}}Stericated 5-simplex
{{CDD>node_1|3|node|3|node|3|node|3|node_1}}		6+6 5-simplex
15+15 rectified 5-simplex
20 birectified 5-simplex		30Tetrahedral antiprism
{{CDD>node|3|node|4|node_h|2x|node_h}}
6{{CDD>node_1|split1|nodes|3ab|nodes|3ab|branch}}Pentellated 6-simplex
{{CDD>node_1|3|node|3|node|3|node|3|node|3|node_1}}		7+7 6-simplex
21+21 rectified 6-simplex
35+35 birectified 6-simplex		424-simplex antiprism
7{{CDD>node_1|split1|nodes|3ab|nodes|3ab|nodes|split2|node}}Hexicated 7-simplex
{{CDD>node_1|3|node|3|node|3|node|3|node|3|node|3|node_1}}		8+8 7-simplex
28+28 rectified 7-simplex
56+56 birectified 7-simplex
70 trirectified 7-simplex		565-simplex antiprism
8{{CDD>node_1|split1|nodes|3ab|nodes|3ab|nodes|3ab|branch}}Heptellated 8-simplex
{{CDD>node_1|3|node|3|node|3|node|3|node|3|node|3|node|3|node_1}}		9+9 8-simplex
36+36 rectified 8-simplex
84+84 birectified 8-simplex
126+126 trirectified 8-simplex		726-simplex antiprism
9{{CDD>node_1|split1|nodes|3ab|nodes|3ab|nodes|3ab|nodes|split2|node}}Octellated 9-simplex
{{CDD>node_1|3|node|3|node|3|node|3|node|3|node|3|node|3|node|3|node_1}}		10+10 9-simplex
45+45 rectified 9-simplex
120+120 birectified 9-simplex
210+210 trirectified 9-simplex
252 quadrirectified 9-simplex		907-simplex antiprism
10{{CDD>node_1|split1|nodes|3ab|nodes|3ab|nodes|3ab|nodes|3ab|branch}}Ennecated 10-simplex
{{CDD>node_1|3|node|3|node|3|node|3|node|3|node|3|node|3|node|3|node|3|node_1}}		11+11 10-simplex
55+55 rectified 10-simplex
165+165 birectified 10-simplex
330+330 trirectified 10-simplex
462+462 quadrirectified 10-simplex		1108-simplex antiprism
1111-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:

node_1|split1|branch}}node_1|split1|nodes|3ab|branch}}node_1|split1|nodes|3ab|nodes|3ab|branch}}node_1|split1|nodes|3ab|nodes|3ab|nodes|3ab|branch}}node_1|split1|nodes|3ab|nodes|3ab|nodes|3ab|nodes|3ab|branch}}...
nodes_10r|splitcross|nodes}}node_1|split1|nodes|split2|node}}node_1|split1|nodes|3ab|nodes|split2|node}}node_1|split1|nodes|3ab|nodes|3ab|nodes|split2|node}}node_1|split1|nodes|3ab|nodes|3ab|nodes|3ab|nodes|split2|node}}...
node_1|infin|node}}node_1|4|node|4|node}}node_1|4|node|3|node|4|node}}node_1|4|node|3|node|3|node|4|node}}node_1|4|node|3|node|3|node|3|node|4|node}}...

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. For 2 and 3 dimensions, this represents the highest kissing number for 2 and 3 dimensions, but fall 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

  • Hypercubic honeycomb
  • Alternated hypercubic honeycomb
  • Quarter hypercubic honeycomb
  • Truncated simplectic honeycomb
  • Omnitruncated simplectic honeycomb

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}}  
    • (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]
{{Honeycombs}}

2 : Honeycombs (geometry)|Polytopes
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