Truncated cubic honeycomb
| Truncated cubic honeycomb | |
|---|---|
  | |
| Type | Uniform honeycomb |
| Schläfli symbol | t{4,3,4} t0,1{4,3,4} |
| Coxeter-Dynkin diagrams |     |
| Cell type | 3.8.8, {3,4} |
| Face type | {3}, {4}, {8} |
| Cells/edge | (3.8.8)4 {3,4}.(3.8.8)2 |
| Faces/edge | {8}4 {3}2.{8} |
| Cells/vertex | 3.8.8 (4) {3,4} (1) |
| Faces/vertex | {8}4+{3}4 |
| Edges/vertex | 5 |
| Euler characteristic | 0 |
| Vertex figure |  square pyramid |
| Space group Fibrifold notation | Pm3m (221) 4−:2 |
| Coxeter group |  , [4,3,4] |
| Dual | Pyramidille (Hexakis cubic honeycomb) |
| Properties | vertex-transitive |
The truncated cubic honeycomb is a uniform space-filling tessellation (or honeycomb) in Euclidean 3-space. It is composed of truncated cubes and octahedra in a ratio of 1:1.
John Horton Conway calls this honeycomb a truncated cubille, and its dual pyramidille.
Symmetry
There is a second uniform colorings by reflectional symmetry of the Coxeter groups, the second seen with alternately colored truncated cubic cells.
| Construction | Bicantellated alternate cubic | Truncated cubic honeycomb |
|---|---|---|
| Coxeter group | [4,31,1],  |
[4,3,4], =<[4,31,1]> |
| Space group | Fm3m | Pm3m |
| Coloring |  |
 |
| Coxeter-Dynkin diagram |   |
|
| Vertex figure |  |
|
Related honeycombs
The [4,3,4], , Coxeter group generates 15 permutations of uniform tessellations, 9 with distinct geometry including the alternated cubic honeycomb. The expanded cubic honeycomb (also known as the runcinated tesseractic honeycomb) is geometrically identical to the cubic honeycomb.
| Space group |
Fibrifold | Extended symmetry |
Extended diagram |
Order | Honeycombs |
|---|---|---|---|---|---|
| Pm3m (221) |
4−:2 | [4,3,4] |     |
×1 | 1, 2, 3, 4, 5, 6 |
| Fm3m (225) |
2−:2 | [1+,4,3,4] = [4,31,1] |
 =   |
Half | 7, 11, 12, 13 |
| I43m (217) |
4o:2 | [[(4,3,4,2+)]] |    |
Half × 2 | (7), |
| Fd3m (227) |
2+:2 | [[1+,4,3,4,1+]] = [[3[4]]] |
 =   |
Quarter × 2 | 10, |
| Im3m (229) |
8o:2 | [[4,3,4]] |   |
×2 |
The [4,31,1], , Coxeter group generates 9 permutations of uniform tessellations, 4 with distinct geometry including the alternated cubic honeycomb.
| Space group |
Fibrifold | Extended symmetry |
Extended diagram |
Order | Honeycombs |
|---|---|---|---|---|---|
| Fm3m (225) |
2−:2 | [4,31,1] = [4,3,4,1+] |
 = |
×1 | 1, 2, 3, 4 |
| Fm3m (225) |
2−:2 | <[1+,4,31,1]> = <[3[4]]> |
= |
×2 | (1),  (3) |
| Pm3m (221) |
4−:2 | <[4,31,1]> | |
×2 |
See also
References
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Wikimedia Commons has media related to Truncated cubic honeycomb. |
- John H. Conway, Heidi Burgiel, Chaim Goodman-Strauss, (2008) The Symmetries of Things, ISBN 978-1-56881-220-5 (Chapter 21, Naming the Archimedean and Catalan polyhedra and tilings, Architectonic and Catoptric tessellations, p 292-298, includes all the nonprismatic forms)
- George Olshevsky, Uniform Panoploid Tetracombs, Manuscript (2006) (Complete list of 11 convex uniform tilings, 28 convex uniform honeycombs, and 143 convex uniform tetracombs)
- Branko Grünbaum, Uniform tilings of 3-space. Geombinatorics 4(1994), 49 - 56.
- 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
- (Paper 22) H.S.M. Coxeter, Regular and Semi Regular Polytopes I, [Math. Zeit. 46 (1940) 380-407, MR 2,10] (1.9 Uniform space-fillings)
- A. Andreini, Sulle reti di poliedri regolari e semiregolari e sulle corrispondenti reti correlative (On the regular and semiregular nets of polyhedra and on the corresponding correlative nets), Mem. Società Italiana della Scienze, Ser.3, 14 (1905) 75–129.
- Richard Klitzing, 3D Euclidean Honeycombs, x4x3o4o - tich - O14
- Uniform Honeycombs in 3-Space: 03-Tich