Runcitruncated cubic honeycomb
Runcitruncated cubic honeycomb | |
---|---|
Type | Uniform honeycomb |
Schläfli symbol | t_{0,1,3}{4,3,4} |
Coxeter-Dynkin diagrams | Template:CDD |
Vertex figure | (Trapezoidal pyramid) |
Coxeter group | [4,3,4], |
Space group Fibrifold notation |
PmTemplate:Overlinem (221) 4^{−}:2 |
Dual | square quarter pyramidille |
Properties | vertex-transitive |
The runcitruncated cubic honeycomb is a uniform space-filling tessellation (or honeycomb) in Euclidean 3-space. It is composed of rhombicuboctahedra, truncated cubes, octagonal prisms, and cubes in a ratio of 1:1:3:3.
Its name is derived from its Coxeter-Dynkin diagram, Template:CDD with three ringed nodes representing 3 active mirrors in the Wythoff construction from its relation to the regular cubic honeycomb.
John Horton Conway calls this honeycomb a 1-RCO-trille, and its dual square quarter pyramidille.
Images
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 |
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.
- Norman Johnson Uniform Polytopes, Manuscript (1991)
- Template:The Geometrical Foundation of Natural Structure (book)
- {{#invoke:citation/CS1|citation
|CitationClass=book }}
- Template:KlitzingPolytopes
- Uniform Honeycombs in 3-Space: 07-Prich
- 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.
- D. M. Y. Sommerville, An Introduction to the Geometry of n Dimensions. New York, E. P. Dutton, 1930. 196 pp. (Dover Publications edition, 1958) Chapter X: The Regular Polytopes