Cantellated cubic honeycomb

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Cantellated cubic honeycomb
Cantellated cubic tiling.png HC A5-A3-P2.png
Type Uniform honeycomb
Schläfli symbol rr{4,3,4}
t0,2{4,3,4}
Coxeter-Dynkin diagram Template:CDD
Cells rr{4,3} Uniform polyhedron-43-t02.png
r{4,3} Uniform polyhedron-43-t1.png
{4,3} Uniform polyhedron-43-t0.png
Euler characteristic 0
Vertex figure Cantellated cubic honeycomb verf.png
(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

Cantellated cubic honeycomb.png Perovskite.jpg
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.

Vertex uniform colorings by cell
Construction Truncated cubic honeycomb Bicantellated alternate cubic
Coxeter group [4,3,4],
=<[4,31,1]>
[4,31,1],
Space group PmTemplate:Overlinem FmTemplate:Overlinem
Coxeter-Dynkin diagram Template:CDD Template:CDD
Coloring Cantellated cubic honeycomb.png Cantellated cubic honeycomb2.png
Vertex figure Cantellated cubic honeycomb verf.png Runcicantellated alternate cubic honeycomb verf.png
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]
= [4,31,1]
Template:CDD
= Template:CDD
Half Template:CDD 7, Template:CDD 11, Template:CDD 12, Template:CDD 13
ITemplate:Overline3m
(217)
4o: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)
8o:2 [[4,3,4]] Template:CDD ×2

Template:CDD (1), Template:CDD 8, Template:CDD 9

The [4,31,1], 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 [4,31,1]
= [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+,4,31,1]>
= <[3[4]]>
Template:CDD
= Template:CDD
×2 Template:CDD (1), Template:CDD (3)
PmTemplate:Overlinem
(221)
4:2 <[4,31,1]> 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

Template:Sister

  • 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

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