Birectified 16-cell honeycomb

Birectified 16-cell honeycomb
(No image)
Type Uniform honeycomb
Schläfli symbol t2{3,3,4,3}
Coxeter-Dynkin diagram
=
4-face type Rectified tesseract
Rectified 24-cell
Cell type Cube
Cuboctahedron
Tetrahedron
Face type {3}, {4}
Vertex figure
{3}×{3} duoprism
Coxeter group = [3,3,4,3]
= [4,3,31,1]
= [31,1,1,1]
Dual ?
Properties vertex-transitive

In four-dimensional Euclidean geometry, the birectified 16-cell honeycomb (or runcic tesseractic honeycomb) is a uniform space-filling tessellation (or honeycomb) in Euclidean 4-space.

Symmetry constructions

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There are 3 different symmetry constructions, all with 3-3 duoprism vertex figures. The   symmetry doubles on   in three possible ways, while   contains the highest symmetry.

Affine Coxeter group  
[3,3,4,3]
 
[4,3,31,1]
 
[31,1,1,1]
Coxeter diagram                         
Vertex figure      
Vertex figure
symmetry
[3,2,3]
(order 36)
[3,2]
(order 12)
[3]
(order 6)
4-faces        
       
     
       
       
    
     
Cells      
     
     
     
     
     
     
     
     
     
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The [3,4,3,3],          , Coxeter group generates 31 permutations of uniform tessellations, 28 are unique in this family and ten are shared in the [4,3,3,4] and [4,3,31,1] families. The alternation (13) is also repeated in other families.

F4 honeycombs
Extended
symmetry
Extended
diagram
Order Honeycombs
[3,3,4,3]           ×1

          1,           3,           5,           6,           8,
          9,           10,           11,           12

[3,4,3,3]           ×1

          2,           4,           7,           13,
          14,           15,           16,           17,
          18,           19,           20,           21,
          22           23,           24,           25,
          26,           27,           28,           29

[(3,3)[3,3,4,3*]]
=[(3,3)[31,1,1,1]]
=[3,4,3,3]
       
=     
=          
×4

          (2),           (4),           (7),           (13)

The [4,3,31,1],        , Coxeter group generates 31 permutations of uniform tessellations, 23 with distinct symmetry and 4 with distinct geometry. There are two alternated forms: the alternations (19) and (24) have the same geometry as the 16-cell honeycomb and snub 24-cell honeycomb respectively.

B4 honeycombs
Extended
symmetry
Extended
diagram
Order Honeycombs
[4,3,31,1]:         ×1

        5,         6,         7,         8

<[4,3,31,1]>:
↔[4,3,3,4]
       
         
×2

        9,         10,         11,         12,         13,         14,

        (10),         15,         16,         (13),         17,         18,         19

[3[1+,4,3,31,1]]
↔ [3[3,31,1,1]]
↔ [3,3,4,3]
       
      
         
×3

        1,         2,         3,         4

[(3,3)[1+,4,3,31,1]]
↔ [(3,3)[31,1,1,1]]
↔ [3,4,3,3]
       
     
         
×12

        20,         21,         22,         23

There are ten uniform honeycombs constructed by the   Coxeter group, all repeated in other families by extended symmetry, seen in the graph symmetry of rings in the Coxeter–Dynkin diagrams. The 10th is constructed as an alternation. As subgroups in Coxeter notation: [3,4,(3,3)*] (index 24), [3,3,4,3*] (index 6), [1+,4,3,3,4,1+] (index 4), [31,1,3,4,1+] (index 2) are all isomorphic to [31,1,1,1].

The ten permutations are listed with its highest extended symmetry relation:

D4 honeycombs
Extended
symmetry
Extended
diagram
Extended
group
Honeycombs
[31,1,1,1]         (none)
<[31,1,1,1]>
↔ [31,1,3,4]
     
       
 ×2 =   (none)
<2[1,131,1]>
↔ [4,3,3,4]
     
         
 ×4 =         1,       2
[3[3,31,1,1]]
↔ [3,3,4,3]
      
         
 ×6 =         3,        4,        5,        6
[4[1,131,1]]
↔ [[4,3,3,4]]
     
         
 ×8 =  ×2       7,       8,       9
[(3,3)[31,1,1,1]]
↔ [3,4,3,3]
     
         
 ×24 =  
[(3,3)[31,1,1,1]]+
↔ [3+,4,3,3]
     
         
½ ×24 = ½        10

See also

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Regular and uniform honeycombs in 4-space:

Notes

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References

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  • 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". x3o3x *b3x *b3o, x3o3o *b3x4o, o3o3x4o3o - bricot - O106
Space Family           /   /  
E2 Uniform tiling 0[3] δ3 3 3 Hexagonal
E3 Uniform convex honeycomb 0[4] δ4 4 4
E4 Uniform 4-honeycomb 0[5] δ5 5 5 24-cell honeycomb
E5 Uniform 5-honeycomb 0[6] δ6 6 6
E6 Uniform 6-honeycomb 0[7] δ7 7 7 222
E7 Uniform 7-honeycomb 0[8] δ8 8 8 133331
E8 Uniform 8-honeycomb 0[9] δ9 9 9 152251521
E9 Uniform 9-honeycomb 0[10] δ10 10 10
E10 Uniform 10-honeycomb 0[11] δ11 11 11
En-1 Uniform (n-1)-honeycomb 0[n] δn n n 1k22k1k21