Cubic honeycomb

Cubic honeycomb
Cubic honeycomb
Partial cubic honeycomb.png
Type Regular honeycomb
Family Hypercube honeycomb
Schläfli symbol {4,3,4}
Coxeter-Dynkin diagram CDel node 1.pngCDel 4.pngCDel node.pngCDel 3.pngCDel node.pngCDel 4.pngCDel node.png
Cell type {4,3}
Face type {4}
Vertex figure Cubic honeycomb verf.png
(octahedron)
Cells/edge {4,3}4
Faces/edge 44
Cells/vertex {4,3}8
Faces/vertex 412
Edges/vertex 6
Euler characteristic 0
Coxeter groups {\tilde{C}}_3, [4,3,4]
Dual self-dual
Properties vertex-transitive
edge framework

The cubic honeycomb is the only regular space-filling tessellation (or honeycomb) in Euclidean 3-space, made up of cubic cells. It has 4 cubes around every edge, and 8 cubes around each vertex. Its vertex figure is a regular octahedron.

It is a self-dual tessellation with Schläfli symbol {4,3,4}. It is part of a multidimensional family of hypercube honeycombs, with Schläfli symbols of the form {4,3,...,3,4}, starting with the square tiling, {4,4} in the plane.

It is one of 28 uniform honeycombs using convex uniform polyhedral cells.

Contents

Related polytopes and tesellations

It is related to the regular 4-polytope tesseract, Schläfli symbol {4,3,3}, which exists in 4-space, and only has 3 cubes around each edge. It's also related to the order-5 cubic honeycomb, Schläfli symbol {4,3,5}, of hyperbolic space with 5 cubes around each edge.

Uniform colorings

There is a large number of uniform colorings, derived from different symmetries. These include:

Coxeter-Dynkin diagram Schläfli symbol Partial
honeycomb
Colors by letters
CDel node 1.pngCDel 4.pngCDel node.pngCDel 3.pngCDel node.pngCDel 4.pngCDel node.png {4,3,4} Partial cubic honeycomb.png 1: aaaa/aaaa
CDel node 1.pngCDel 4.pngCDel node.pngCDel 4.pngCDel node.pngCDel 2.pngCDel node 1.pngCDel infin.pngCDel node 1.png {4,4}xt{∞} Square prismatic honeycomb.png 2: aaaa/bbbb
CDel node.pngCDel 4.pngCDel node 1.pngCDel 4.pngCDel node.pngCDel 2.pngCDel node 1.pngCDel infin.pngCDel node.png t1{4,4}x{∞} Square prismatic 2-color honeycomb.png 2: abba/abba
CDel node 1.pngCDel 4.pngCDel node.pngCDel split1.pngCDel nodes.png {4,31,1} Bicolor cubic honeycomb.png 2: abba/baab
CDel node 1.pngCDel infin.pngCDel node 1.pngCDel 2.pngCDel node 1.pngCDel infin.pngCDel node 1.pngCDel 2.pngCDel node 1.pngCDel infin.pngCDel node.png t{∞}xt{∞}x{∞} Square 4-color prismatic honeycomb.png 4: abcd/abcd
CDel node 1.pngCDel 4.pngCDel node.pngCDel 3.pngCDel node.pngCDel 4.pngCDel node 1.png t0,3{4,3,4} Runcinated cubic honeycomb.png 4: abbc/bccd
CDel node 1.pngCDel infin.pngCDel node 1.pngCDel 2.pngCDel node 1.pngCDel infin.pngCDel node 1.pngCDel 2.pngCDel node 1.pngCDel infin.pngCDel node 1.png t{∞}xt{∞}xt{∞} Cubic 8-color honeycomb.png 8: abcd/efgh

See also

References

  • Coxeter, H.S.M. Regular Polytopes, (3rd edition, 1973), Dover edition, ISBN 0-486-61480-8 p.296, Table II: Regular honeycombs
  • 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.
  • Richard Klitzing, 3D Euclidean Honeycombs, x4o3o4o - chon - O1
  • Uniform Honeycombs in 3-Space: 01-Chon

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