Cantitruncated cubic honeycomb
Cantitruncated cubic honeycomb | |
---|---|
Type | Uniform honeycomb |
Schläfli symbol | tr{4,3,4} t_{0,1,2}{4,3,4} |
Coxeter-Dynkin diagram | Template:CDD |
Vertex figure | (Irreg. tetrahedron) |
Coxeter group | [4,3,4], |
Space group Fibrifold notation |
PmTemplate:Overlinem (221) 4^{−}:2 |
Dual | triangular pyramidille |
Properties | vertex-transitive |
The cantitruncated cubic honeycomb is a uniform space-filling tessellation (or honeycomb) in Euclidean 3-space, made up of truncated cuboctahedra, truncated octahedra, and cubes in a ratio of 1:1:3.
John Horton Conway calls this honeycomb a n-tCO-trille, and its dual triangular pyramidille.
Images
Four cells exist around each vertex:
Symmetry
Cells can be shown in two different symmetries. The linear Coxeter-Dynkin diagram form can be drawn with one color for each cell type. The bifurcating diagram form can be drawn with two types (colors) of truncated cuboctahedron cells alternating.
Construction | Cantitruncated cubic | Omnitruncated alternate cubic |
---|---|---|
Coxeter group | [4,3,4], =<[4,3^{1,1}]> |
[4,3^{1,1}], |
Space group | PmTemplate:Overlinem (221) | FmTemplate:Overlinem (225) |
Fibrifold | 4^{−}:2 | 2^{−}:2 |
Coloring | ||
Coxeter-Dynkin diagram | Template:CDD | Template:CDD |
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} |
Alternation
This image shows a partial honeycomb of the alternation of the cantitruncated cubic honeycomb. It contains three types of cells: snub cubes, icosahedra (snub tetrahedron), and tetrahedra. In addition the gaps created at the alternated vertices form tetrahedral cells. |
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)
- Template:The Geometrical Foundation of Natural Structure (book) Chapter 5 (Polyhedral packing and spacing filling): Fig. 5-13, p.176 shows this honeycomb. Fig. 5-34 shows a partial honeycomb of the alternation with only snub cube cells show.
- 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: 06-Grich