Runcicantellated tesseractic honeycomb | |
---|---|
(No image) | |
Type | Uniform 4-honeycomb |
Schläfli symbol | t0,2,3{4,3,3,4} |
Coxeter-Dynkin diagram | |
4-face type |
bitruncated tesseract
|
Cell type | |
Face type | {4}, {6}, {8} |
Vertex figure | trapezoidal duopyramid |
Coxeter group | = [4,3,3,4] = [4,3,31,1] |
Dual | |
Properties | vertex-transitive |
In four-dimensional Euclidean geometry, the runcicantellated tesseractic honeycomb is a uniform space-filling tessellation (or honeycomb) in Euclidean 4-space.
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 | ||
[[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 |
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 | |||
<[4,3,31,1]>: ↔[4,3,3,4] |
↔ |
×2 | ||
[3[1+,4,3,31,1]] ↔ [3[3,31,1,1]] ↔ [3,3,4,3] |
↔ ↔ |
×3 | ||
[(3,3)[1+,4,3,31,1]] ↔ [(3,3)[31,1,1,1]] ↔ [3,4,3,3] |
↔ ↔ |
×12 |
Regular and uniform honeycombs in 4-space:
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 | 133 • 331 |
E8 | Uniform 8-honeycomb | {3[9]} | δ9 | hδ9 | qδ9 | 152 • 251 • 521 |
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 | 1k2 • 2k1 • k21 |
Runcicantellated tesseractic honeycomb | |
---|---|
(No image) | |
Type | Uniform 4-honeycomb |
Schläfli symbol | t0,2,3{4,3,3,4} |
Coxeter-Dynkin diagram | |
4-face type |
bitruncated tesseract
|
Cell type | |
Face type | {4}, {6}, {8} |
Vertex figure | trapezoidal duopyramid |
Coxeter group | = [4,3,3,4] = [4,3,31,1] |
Dual | |
Properties | vertex-transitive |
In four-dimensional Euclidean geometry, the runcicantellated tesseractic honeycomb is a uniform space-filling tessellation (or honeycomb) in Euclidean 4-space.
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 | ||
[[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 |
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 | |||
<[4,3,31,1]>: ↔[4,3,3,4] |
↔ |
×2 | ||
[3[1+,4,3,31,1]] ↔ [3[3,31,1,1]] ↔ [3,3,4,3] |
↔ ↔ |
×3 | ||
[(3,3)[1+,4,3,31,1]] ↔ [(3,3)[31,1,1,1]] ↔ [3,4,3,3] |
↔ ↔ |
×12 |
Regular and uniform honeycombs in 4-space:
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 | 133 • 331 |
E8 | Uniform 8-honeycomb | {3[9]} | δ9 | hδ9 | qδ9 | 152 • 251 • 521 |
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 | 1k2 • 2k1 • k21 |