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Rectified tesseractic honeycomb

## Summary

quarter cubic honeycomb
(No image)
Type Uniform 4-honeycomb
Family Quarter hypercubic honeycomb
Schläfli symbol r{4,3,3,4}
r{4,31,1}
r{4,31,1}
q{4,3,3,4}
Coxeter-Dynkin diagram

=
=
=
=

4-face type h{4,32},
h3{4,32},
Cell type {3,3},
t1{4,3},
Face type {3}
{4}
Edge figure
Square pyramid
Vertex figure
Elongated {3,4}×{}
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 rectified tesseractic honeycomb is a uniform space-filling tessellation (or honeycomb) in Euclidean 4-space. It is constructed by a rectification of a tesseractic honeycomb which creates new vertices on the middle of all the original edges, rectifying the cells into rectified tesseracts, and adding new 16-cell facets at the original vertices. Its vertex figure is an octahedral prism, {3,4}×{}.

It is also called a quarter tesseractic honeycomb since it has half the vertices of the 4-demicubic honeycomb, and a quarter of the vertices of a tesseractic honeycomb.[1]

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.

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

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 ${\displaystyle {\tilde {D}}_{4}}$  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:

D4 honeycombs
Extended
symmetry
Extended
diagram
Extended
group
Honeycombs
[31,1,1,1]       ${\displaystyle {\tilde {D}}_{4}}$  (none)
<[31,1,1,1]>
↔ [31,1,3,4]

${\displaystyle {\tilde {D}}_{4}}$ ×2 = ${\displaystyle {\tilde {B}}_{4}}$  (none)
<2[1,131,1]>
↔ [4,3,3,4]

${\displaystyle {\tilde {D}}_{4}}$ ×4 = ${\displaystyle {\tilde {C}}_{4}}$        1,       2
[3[3,31,1,1]]
↔ [3,3,4,3]

${\displaystyle {\tilde {D}}_{4}}$ ×6 = ${\displaystyle {\tilde {F}}_{4}}$        3,        4,        5,        6
[4[1,131,1]]
↔ [[4,3,3,4]]

${\displaystyle {\tilde {D}}_{4}}$ ×8 = ${\displaystyle {\tilde {C}}_{4}}$ ×2       7,       8,       9
[(3,3)[31,1,1,1]]
↔ [3,4,3,3]

${\displaystyle {\tilde {D}}_{4}}$ ×24 = ${\displaystyle {\tilde {F}}_{4}}$
[(3,3)[31,1,1,1]]+
↔ [3+,4,3,3]

½${\displaystyle {\tilde {D}}_{4}}$ ×24 = ½${\displaystyle {\tilde {F}}_{4}}$        10

## See also

Regular and uniform honeycombs in 4-space:

## Notes

1. ^ Coxeter, Regular and Semi-Regular Polytopes III, (1988), p318

## 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". o4x3o3o4o, o3o3o *b3x4o, x3o3x *b3o4o, x3o3x *b3o *b3o - rittit - O87
• Conway JH, Sloane NJH (1998). Sphere Packings, Lattices and Groups (3rd ed.). ISBN 0-387-98585-9.
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 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