Birectified 16-cell honeycomb

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.

Birectified 16-cell honeycomb
(No image)
TypeUniform honeycomb
Schläfli symbolt2{3,3,4,3}
Coxeter-Dynkin diagram
=
4-face typeRectified tesseract
Rectified 24-cell
Cell typeCube
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?
Propertiesvertex-transitive

Symmetry constructions

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






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
OrderHoneycombs
[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

Regular and uniform honeycombs in 4-space:

Notes

    References

    • 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 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 {3[3]} δ3 hδ3 qδ3 Hexagonal
    E3 Uniform convex honeycomb {3[4]} δ4 hδ4 qδ4
    E4 Uniform 4-honeycomb {3[5]} δ5 hδ5 qδ5 24-cell honeycomb
    E5 Uniform 5-honeycomb {3[6]} δ6 hδ6 qδ6
    E6 Uniform 6-honeycomb {3[7]} δ7 hδ7 qδ7 222
    E7 Uniform 7-honeycomb {3[8]} δ8 hδ8 qδ8 133331
    E8 Uniform 8-honeycomb {3[9]} δ9 hδ9 qδ9 152251521
    E9 Uniform 9-honeycomb {3[10]} δ10 hδ10 qδ10
    E10 Uniform 10-honeycomb {3[11]} δ11 hδ11 qδ11
    En-1 Uniform (n-1)-honeycomb {3[n]} δn hδn qδn 1k22k1k21
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