Cantellated cubic honeycomb
Cantellated cubic honeycomb | |
---|---|
Type | Uniform honeycomb |
Schläfli symbol | rr{4,3,4} t_{0,2}{4,3,4} |
Coxeter-Dynkin diagram | Template:CDD |
Cells | rr{4,3} r{4,3} {4,3} |
Euler characteristic | 0 |
Vertex figure | (Wedge) |
Space group Fibrifold notation |
PmTemplate:Overlinem (221) 4^{−}:2 |
Coxeter group | [4,3,4], |
Dual | quarter oblate octahedrille |
Properties | vertex-transitive |
The cantellated cubic honeycomb is a uniform space-filling tessellation (or honeycomb) in Euclidean 3-space. It is composed of rhombicuboctahedra, cuboctahedra, and cubes in a ratio of 1:1:3.
John Horton Conway calls this honeycomb a 2-RCO-trille, and its dual quarter oblate octahedrille.
Images
It is closely related to the perovskite structure, shown here with cubic symmetry, with atoms placed at the center of the cells of this honeycomb. |
Symmetry
There is a second uniform colorings by reflectional symmetry of the Coxeter groups, the second seen with alternately colored rhombicuboctahedral cells.
Construction | Truncated cubic honeycomb | Bicantellated alternate cubic |
---|---|---|
Coxeter group | [4,3,4], =<[4,3^{1,1}]> |
[4,3^{1,1}], |
Space group | PmTemplate:Overlinem | FmTemplate:Overlinem |
Coxeter-Dynkin diagram | Template:CDD | Template:CDD |
Coloring | ||
Vertex figure | ||
Vertex figure symmetry |
[ ] order 2 |
[ ]^{+} order 1 |
Related honeycombs
The [4,3,4], Template:CDD, 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 |
---|---|---|---|---|---|
PmTemplate:Overlinem (221) |
4^{−}:2 | [4,3,4] | Template:CDD | ×1 | Template:CDD _{1}, Template:CDD _{2}, Template:CDD _{3}, Template:CDD _{4}, Template:CDD _{5}, Template:CDD _{6} |
FmTemplate:Overlinem (225) |
2^{−}:2 | [1^{+},4,3,4] = [4,3^{1,1}] |
Template:CDD = Template:CDD |
Half | Template:CDD _{7}, Template:CDD _{11}, Template:CDD _{12}, Template:CDD _{13} |
ITemplate:Overline3m (217) |
4^{o}:2 | [[(4,3,4,2^{+})]] | Template:CDD | Half × 2 | Template:CDD _{(7)}, |
FdTemplate:Overlinem (227) |
2^{+}:2 | [[1^{+},4,3,4,1^{+}]] = [[3^{[4]}]] |
Template:CDD = Template:CDD |
Quarter × 2 | Template:CDD _{10}, |
ImTemplate:Overlinem (229) |
8^{o}:2 | [[4,3,4]] | Template:CDD | ×2 |
The [4,3^{1,1}], Template:CDD, 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 |
---|---|---|---|---|---|
FmTemplate:Overlinem (225) |
2^{−}:2 | [4,3^{1,1}] = [4,3,4,1^{+}] |
Template:CDD = Template:CDD |
×1 | Template:CDD _{1}, Template:CDD _{2}, Template:CDD _{3}, Template:CDD _{4} |
FmTemplate:Overlinem (225) |
2^{−}:2 | <[1^{+},4,3^{1,1}]> = <[3^{[4]}]> |
Template:CDD = Template:CDD |
×2 | Template:CDD _{(1)}, Template:CDD _{(3)} |
PmTemplate:Overlinem (221) |
4^{−}:2 | <[4,3^{1,1}]> | Template:CDD | ×2 |
Template:CDD _{5}, Template:CDD _{6}, Template:CDD _{7}, Template:CDD _{(6)}, Template:CDD _{9}, Template:CDD _{10}, Template:CDD _{11} |
See also
References
- 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 [1]
- (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.
- Template:KlitzingPolytopes
- Uniform Honeycombs in 3-Space: 04-Srich