Simplectic honeycomb

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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 (octahedron)
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 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 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[edit]

n Tessellation Vertex figure Facets per vertex figure Vertices per vertex figure Edge figure
1 Regular apeirogon.png
Apeirogon
CDel node 1.pngCDel infin.pngCDel node.png
CDel node 1.png 1 2 -
2 Uniform tiling 333-t1.png
Triangular tiling
2-simplex honeycomb
CDel node 1.pngCDel split1.pngCDel branch.png
Truncated triangle.png
Hexagon
(Truncated triangle)
CDel node 1.pngCDel 3.pngCDel node 1.png
3+3 triangles 6 Line segment
CDel node 1.png
3 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
CDel node 1.pngCDel 2.pngCDel node 1.png
4 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
CDel node h.pngCDel 3.pngCDel node h.pngCDel 2x.pngCDel node h.png
5 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
CDel node.pngCDel 3.pngCDel node.pngCDel 4.pngCDel node h.pngCDel 2x.pngCDel node h.png
6 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 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 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 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 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 11-simplex honeycomb ... ... ... ...

Projection by folding[edit]

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:

CDel node 1.pngCDel split1.pngCDel branch.png CDel node 1.pngCDel split1.pngCDel nodes.pngCDel 3ab.pngCDel branch.png CDel node 1.pngCDel split1.pngCDel nodes.pngCDel 3ab.pngCDel nodes.pngCDel 3ab.pngCDel branch.png CDel node 1.pngCDel split1.pngCDel nodes.pngCDel 3ab.pngCDel nodes.pngCDel 3ab.pngCDel nodes.pngCDel 3ab.pngCDel branch.png CDel node 1.pngCDel split1.pngCDel nodes.pngCDel 3ab.pngCDel nodes.pngCDel 3ab.pngCDel nodes.pngCDel 3ab.pngCDel nodes.pngCDel 3ab.pngCDel branch.png ...
CDel nodes 10r.pngCDel splitcross.pngCDel nodes.png CDel node 1.pngCDel split1.pngCDel nodes.pngCDel split2.pngCDel node.png CDel node 1.pngCDel split1.pngCDel nodes.pngCDel 3ab.pngCDel nodes.pngCDel split2.pngCDel node.png CDel node 1.pngCDel split1.pngCDel nodes.pngCDel 3ab.pngCDel nodes.pngCDel 3ab.pngCDel nodes.pngCDel split2.pngCDel node.png CDel node 1.pngCDel split1.pngCDel nodes.pngCDel 3ab.pngCDel nodes.pngCDel 3ab.pngCDel nodes.pngCDel 3ab.pngCDel nodes.pngCDel split2.pngCDel node.png ...
CDel node 1.pngCDel infin.pngCDel node.png CDel node 1.pngCDel 4.pngCDel node.pngCDel 4.pngCDel node.png CDel node 1.pngCDel 4.pngCDel node.pngCDel 3.pngCDel node.pngCDel 4.pngCDel node.png CDel node 1.pngCDel 4.pngCDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.pngCDel 4.pngCDel node.png 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[edit]

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 an 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[edit]

References[edit]

  • 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 / /
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