Runcinated tesseractic honeycomb
| Runcinated tesseractic honeycomb | |
|---|---|
| (No image) | |
| Type | Uniform 4-honeycomb |
| Schläfli symbol | t0,3{4,3,3,4} t0,3{4,3,31,1} |
| Coxeter-Dynkin diagram |      
|
| 4-face type | runcinated tesseract  tesseract 40px rectified tesseract 40px cuboctahedral prism 
|
| Cell type | Cuboctahedron  Tetrahedron 20px Cube 20px Triangular prism 
|
| Face type | {3}, {4} |
| Vertex figure | triangular-antipodial antifastigium |
| 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 runcinated tesseractic honeycomb is a uniform space-filling tessellation (or honeycomb) in Euclidean 4-space. It is constructed by a runcination of a tesseractic honeycomb creating runcinated tesseracts, and new tesseract, rectified tesseract and cuboctahedral prism facets.
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 | |||
|---|---|---|---|
| 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 | |
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 *b3o4x, x4o3o3x4o - sidpitit - O91
- 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
| ||||||
|---|---|---|---|---|---|---|
| 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 |
 |  |
