Great Rhombicuboctahedron


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The great rhombicuboctahedron (Cundy and Rowlett 1989, p. 106) is the 26-faced Archimedean solid consisting of faces 12{4}+8{6}+6{8}. It is sometimes called the rhombitruncated cuboctahedron (Wenninger 1971, p. 29) or (improperly) the truncated cuboctahedron (Ball and Coxeter 1987, p. 143; Cundy and Rowlett 1989, p. 106; Maeder 1997; Conway et al. 1999). It is illustrated above together with a wireframe version and a net that can be used for its construction.

It is also the uniform polyhedron with Maeder index 11 (Maeder 1997), Wenninger index 15 (Wenninger 1989), Coxeter index 23 (Coxeter et al. 1954), and Har'El index 16 (Har'El 1993). It has Schläfli symbol t{3; 4} and Wythoff symbol 234|.


Some symmetric projections of the great rhombicuboctahedron are illustrated above.

The great rhombicuboctahedron is implemented in the Wolfram Language as UniformPolyhedron["GreatRhombicuboctahedron"]. Precomputed properties are available as PolyhedronData["GreatRhombicuboctahedron", prop].

Confusingly, the term "great rhombicuboctahedron" is also used by various authors (e.g., Maeder 1997) to refer to the distinct unform polyhedron with Maeder index 17 and Wenninger index 85. For reasons of clarity, that solid is better referred to using Wenninger's term quasirhombicuboctahedron (Wenninger 1971, p. 132).

The great rhombicuboctahedron is an equilateral zonohedron and the Minkowski sum of three cubes. It has Dehn invariant 0 (Conway et al. 1999) but is not a space-filling polyhedron. However, it can be combined with cubes and truncated octahedra into a regular space-filling pattern.

The small cubicuboctahedron is a faceted version of the great rhombicuboctahedron.


The skeleton of the great rhombicuboctahedron is the great rhombicuboctahedral graph, illustrated above in a number of embeddings.


The dual polyhedron of the great rhombicuboctahedron is the disdyakis dodecahedron, both of which are illustrated above together with their common midsphere. The inradius r of the dual, midradius rho of the solid and dual, and circumradius R of the solid for a=1 are

 approx 2.20974
 approx 2.26303
 approx 2.31761.

Additional quantities are


The distances between the solid center and centroids of the square and octagonal faces are


The surface area and volume are


See also

Archimedean Solid, Equilateral Zonohedron, Great Truncated Cuboctahedron, Small Rhombicuboctahedron, Octahemioctahedron, Quasirhombicuboctahedron, Rhombicuboctahedron

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Ball, W. W. R. and Coxeter, H. S. M. Mathematical Recreations and Essays, 13th ed. New York: Dover, p. 138, 1987.Conway, J. H.; Radin, C.; and Sadun, L. "On Angles Whose Squared Trigonometric Functions Are Rational." Discr. Computat. Geom. 22, 321-332, 1999.Coxeter, H. S. M.; Longuet-Higgins, M. S.; and Miller, J. C. P. "Uniform Polyhedra." Phil. Trans. Roy. Soc. London Ser. A 246, 401-450, 1954.Cundy, H. and Rollett, A. "Great Rhombicuboctahedron or Truncated Cuboctahedron. 4.6.8." §3.7.6 in Mathematical Models, 3rd ed. Stradbroke, England: Tarquin Pub., p. 106, 1989.Geometry Technologies. "Rhombitruncated Cubeoctahedron."'El, Z. "Uniform Solution for Uniform Polyhedra." Geometriae Dedicata 47, 57-110, 1993.Kasahara, K. "Two New Semiregular Polyhedrons." Origami Omnibus: Paper-Folding for Everyone. Tokyo: Japan Publications, p. 227, 1988.Maeder, R. E. "11: Truncated Cuboctahedron." 1997., M. J. "The Rhombitruncated Cuboctahedron." Model 15 in Polyhedron Models. Cambridge, England: Cambridge University Press, p. 29, 1989.

Cite this as:

Weisstein, Eric W. "Great Rhombicuboctahedron." From MathWorld--A Wolfram Web Resource.

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