Runcitruncated tesseractic honeycomb
Runcitruncated tesseractic honeycomb | |
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(No image) | |
Type | Uniform 4-honeycomb |
Schläfli symbol | t0,1,3{4,3,3,4} |
Coxeter-Dynkin diagram | ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
4-face type | t0,1,3{4,3,3} ![]() t1{3,4,3} 40px t1{3,4}×{} 40px 4-8 duoprism ![]() |
Cell type | Cuboctahedron ![]() |
Face type | {3}, {4}, {8} |
Vertex figure | triangular prism-based tilted pyramid |
Coxeter group | [math]\displaystyle{ {\tilde{C}}_4 }[/math] = [4,3,3,4] [math]\displaystyle{ {\tilde{B}}_4 }[/math] = [4,3,31,1] |
Dual | |
Properties | vertex-transitive |
In four-dimensional Euclidean geometry, the runcitruncated tesseractic honeycomb is a uniform space-filling tessellation (or honeycomb) in Euclidean 4-space.
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.
C4 honeycombs | |||
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Extended symmetry |
Extended diagram |
Order | Honeycombs |
[4,3,3,4]: | ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
×1 | |
[[4,3,3,4]] | ![]() ![]() ![]() ![]() ![]() |
×2 | ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
[(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 |
See also
Regular and uniform honeycombs in 4-space:
- Tesseractic honeycomb
- Demitesseractic honeycomb
- 24-cell honeycomb
- Truncated 24-cell honeycomb
- Snub 24-cell honeycomb
- 5-cell honeycomb
- Truncated 5-cell honeycomb
- Omnitruncated 5-cell honeycomb
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 [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". https://bendwavy.org/klitzing/dimensions/flat.htm. x3o3x *b3x4x, x4x3o3x4o - potatit - O95
- Conway JH, Sloane NJH (1998). Sphere Packings, Lattices and Groups (3rd ed.). ISBN 0-387-98585-9. https://archive.org/details/spherepackingsla0000conw_b8u0.
Fundamental convex regular and uniform honeycombs in dimensions 2-9
| ||||||
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Space | Family | [math]\displaystyle{ {\tilde{A}}_{n-1} }[/math] | [math]\displaystyle{ {\tilde{C}}_{n-1} }[/math] | [math]\displaystyle{ {\tilde{B}}_{n-1} }[/math] | [math]\displaystyle{ {\tilde{D}}_{n-1} }[/math] | [math]\displaystyle{ {\tilde{G}}_2 }[/math] / [math]\displaystyle{ {\tilde{F}}_4 }[/math] / [math]\displaystyle{ {\tilde{E}}_{n-1} }[/math] |
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 | |
En-1 | Uniform (n-1)-honeycomb | {3[n]} | δn | hδn | qδn | 1k2 • 2k1 • k21 |
![]() | Original source: https://en.wikipedia.org/wiki/Runcitruncated tesseractic honeycomb.
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