Bitruncated tesseractic honeycomb

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In four-dimensional Euclidean geometry, the bitruncated tesseractic honeycomb is a uniform space-filling tessellation (or honeycomb) in Euclidean 4-space. It is constructed by a bitruncation of a tesseractic honeycomb. It is also called a cantic quarter tesseractic honeycomb from its q2{4,3,3,4} construction.

Bitruncated tesseractic honeycomb
(No image)
TypeUniform 4-honeycomb
Schläfli symbolt1,2{4,3,3,4} or 2t{4,3,3,4}
t1,2{4,31,1} or 2t{4,31,1}
t2,3{4,31,1}
q2{4,3,3,3,4}
Coxeter-Dynkin diagram




=

4-face typeBitruncated tesseract
Truncated 16-cell
Cell typeOctahedron
Truncated tetrahedron
Truncated octahedron
Face type{3}, {4}, {6}
Vertex figure
Square-pyramidal pyramid
Coxeter group = [4,3,3,4]
= [4,31,1]
= [31,1,1,1]
Dual
Propertiesvertex-transitive

Other names

  • Bitruncated tesseractic tetracomb (batitit)

The [4,3,3,4], , Coxeter group generates 31 permutations of uniform tessellations, 21 with distinct symmetry and 20 with distinct geometry. The expanded tesseractic honeycomb (also known as the stericated tesseractic honeycomb) is geometrically identical to the tesseractic honeycomb. Three of the symmetric honeycombs are shared in the [3,4,3,3] family. Two alternations (13) and (17), and the quarter tesseractic (2) are repeated in other families.

More information C4 honeycombs, Extendedsymmetry ...
C4 honeycombs
Extended
symmetry
Extended
diagram
Order Honeycombs
[4,3,3,4]: ×1

1, 2, 3, 4,
5, 6, 7, 8,
9, 10, 11, 12,
13

[[4,3,3,4]] ×2 (1), (2), (13), 18
(6), 19, 20
[(3,3)[1+,4,3,3,4,1+]]
↔ [(3,3)[31,1,1,1]]
↔ [3,4,3,3]


×6

14, 15, 16, 17

Close

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.

More information B4 honeycombs, Extendedsymmetry ...
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

Close

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:

More information , ...
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
Close

See also

Regular and uniform honeycombs in 4-space:

Notes

References

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