Simplectic honeycomb

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 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 dimensionEdit

n   Tessellation Vertex figure Facets per vertex figure Vertices per vertex figure Edge figure
1    
Apeirogon
   
  1 2 -
2    
Triangular tiling
2-simplex honeycomb
   
 
Hexagon
(Truncated triangle)
   
3+3 triangles 6 Line segment
 
3    
Tetrahedral-octahedral honeycomb
3-simplex honeycomb
     
 
Cuboctahedron
(Cantellated tetrahedron)
     
4+4 tetrahedron
6 rectified tetrahedra
12  
Rectangle
   
4   4-simplex honeycomb
     
 
Runcinated 5-cell
       
5+5 5-cells
10+10 rectified 5-cells
20  
Triangular antiprism
     
5   5-simplex honeycomb
       
 
Stericated 5-simplex
         
6+6 5-simplex
15+15 rectified 5-simplex
20 birectified 5-simplex
30  
Tetrahedral antiprism
       
6   6-simplex honeycomb
       
 
Pentellated 6-simplex
           
7+7 6-simplex
21+21 rectified 6-simplex
35+35 birectified 6-simplex
42 4-simplex antiprism
7   7-simplex honeycomb
         
 
Hexicated 7-simplex
             
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
         
 
Heptellated 8-simplex
               
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
           
 
Octellated 9-simplex
                 
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
           
 
Ennecated 10-simplex
                   
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 foldingEdit

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:

                                                  ...
                                                  ...
                                                  ...

Kissing numberEdit

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 alsoEdit

ReferencesEdit

  • 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
          /   /  
{3[3]} δ3 3 3 Hexagonal
{3[4]} δ4 4 4
{3[5]} δ5 5 5 24-cell honeycomb
{3[6]} δ6 6 6
{3[7]} δ7 7 7 222
{3[8]} δ8 8 8 133331
{3[9]} δ9 9 9 152251521
{3[10]} δ10 10 10
{3[n]} δn n n 1k22k1k21