# Bitruncated tesseractic honeycomb

Bitruncated tesseractic honeycomb
(No image)
Type Uniform 4-honeycomb
Schläfli symbol t1,2{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 type Bitruncated tesseract
Truncated 16-cell
Cell type Octahedron
Truncated tetrahedron 20px
Truncated octahedron
Face type {3}, {4}, {6}
Vertex figure
Square-pyramidal pyramid
Coxeter group $\displaystyle{ {\tilde{C}}_4 }$ = [4,3,3,4]
$\displaystyle{ {\tilde{B}}_4 }$ = [4,31,1]
$\displaystyle{ {\tilde{D}}_4 }$ = [31,1,1,1]
Dual
Properties vertex-transitive

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.

## Other names

• Bitruncated tesseractic tetracomb (batitit)

## Related honeycombs

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.

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.

There are ten uniform honeycombs constructed by the $\displaystyle{ {\tilde{D}}_4 }$ 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:

Regular and uniform honeycombs in 4-space:

## 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 [1]
• (Paper 24) H.S.M. Coxeter, Regular and Semi-Regular Polytopes III, [Math. Zeit. 200 (1988) 3-45] See p318 [2]
• 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#4D".  x3x3x *b3o *b3o, x3x3x *b3o4o, o3x3o *b3x4o, o4x3x3o4o - batitit - O92
• Conway JH, Sloane NJH (1998). Sphere Packings, Lattices and Groups (3rd ed.). ISBN 0-387-98585-9.
Fundamental convex regular and uniform honeycombs in dimensions 2-9
Space Family $\displaystyle{ {\tilde{A}}_{n-1} }$ $\displaystyle{ {\tilde{C}}_{n-1} }$ $\displaystyle{ {\tilde{B}}_{n-1} }$ $\displaystyle{ {\tilde{D}}_{n-1} }$ $\displaystyle{ {\tilde{G}}_2 }$ / $\displaystyle{ {\tilde{F}}_4 }$ / $\displaystyle{ {\tilde{E}}_{n-1} }$
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