Cantitruncated cubic honeycomb

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Cantitruncated cubic honeycomb

 

TypeUniform honeycomb
Schläfli symboltr{4,3,4}
t0,1,2{4,3,4}
Coxeter-Dynkin diagram
Vertex figure
(Irreg. tetrahedron)
Coxeter group[4,3,4], {{\tilde {C}}}_{3}
Space group
Fibrifold notation		Pm3m (221)
4−:2
Dualtriangular pyramidille
Propertiesvertex-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

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], {{\tilde {C}}}_{3}
=<[4,31,1]>		 [4,31,1], {{\tilde {B}}}_{3}
Space groupPm3m (221)Fm3m (225)
Fibrifold4−:22−:2
Coloring
Coxeter-Dynkin diagram
Vertex figure
Vertex
figure
symmetry		 [ ]
order 2		 [ ]+
order 1

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

(1), 8, 9

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

5, 6, 7, (6), 9, 10, 11

Alternation

Vertex figure for alternated bitruncated cubic honeycomb, with 5 tetrahedral, one icosahedral, and two snub cube cells, but edge-lengths can't be made equal.

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.
This honeycomb exists in two mirror image forms. Although it is not uniform, constructionally it can be given as Coxeter-Dynkin diagrams or .

See also

Wikimedia Commons has media related to Cantitruncated cubic honeycomb.

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)
  • Williams, Robert (1979). The Geometrical Foundation of Natural Structure: A Source Book of Design. Dover Publications, Inc. p. 197. ISBN 0-486-23729-X.  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
    • (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, x4x3x4o - grich - O18
  • Uniform Honeycombs in 3-Space: 06-Grich
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