Great Rhombicosidodecahedron


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The great rhombicosidodecahedron is the 62-faced Archimedean solid with faces 30{4}+20{6}+12{10}. It is also known as the rhombitruncated icosidodecahedron, and is sometimes improperly called the truncated icosidodecahedron (Ball and Coxeter 1987, p. 143; Maeder 1997; Conway et al. 1999), a name which is inappropriate since truncation would yield rectangular instead of square. 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 28 (Maeder 1997), Wenninger index 16 (Wenninger 1989), Coxeter index 31 (Coxeter et al. 1954), and Har'El index 33 (Har'El 1993). It has Schläfli symbol t{3; 5} and Wythoff symbol 235|.


Some symmetric projections of the great rhombicosidodecahedron are illustrated above.

The great rhombicosidodecahedron is an equilateral zonohedron and is the Minkowski sum of five cubes. It has Dehn invariant 0 (Conway et al. 1999) but is not space-filling.


Its skeleton is the great rhombicosidodecahedral graph, illustrated above.

The inradius of the dual, midradius of the solid and dual, and circumradius of the solid for a=1 are

r=3sqrt(5/(241)(39+16sqrt(5))) approx 3.73665
rho=sqrt((15)/2+3sqrt(5)) approx 3.76938
R=1/2sqrt(31+12sqrt(5)) approx 3.80239.

The great rhombicosidodecahedron has surface area


and volume

Origami great rhombicosidodecahedron

The great rhombicosidodecahedron constructed by E. K. Herrstrom in origami is illustrated above (Kasahara and Takahama 1987, pp. 46-49). This construction uses 900 sonobè units, each made from a single sheet of origami paper.


The dual polyhedron of the great rhombicosidodecahedron is the disdyakis triacontahedron, both of which are illustrated above together with their common midsphere.

See also

Equilateral Zonohedron, Great Rhombicosidodecahedral Graph, Quasirhombicosidodecahedron, Rhombicosidodecahedron, Small Rhombicosidodecahedron

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Ball, W. W. R. and Coxeter, H. S. M. Mathematical Recreations and Essays, 13th ed. New York: Dover, p. 137, 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 Rhombicosidodecahedron or Truncated Icosidodecahedron. 4.6.10." §3.7.12 in Mathematical Models, 3rd ed. Stradbroke, England: Tarquin Pub., pp. 112-113, 1989.Geometry Technologies. "Rhombitruncated Icosidodecahedron."'El, Z. "Uniform Solution for Uniform Polyhedra." Geometriae Dedicata 47, 57-110, 1993.Kasahara, K. and Takahama, T. Origami for the Connoisseur. Tokyo: Japan Publications, 1987.Kasahara, K. "The Final Semiregular Polyhedron." Origami Omnibus: Paper-Folding for Everyone. Tokyo: Japan Publications, p. 229, 1988.Maeder, R. E. "28: Truncated Icosidodecahedron." 1997., M. J. "The Rhombitruncated Icosidodecahedron." Model 16 in Polyhedron Models. Cambridge, England: Cambridge University Press, p. 30, 1989.

Cite this as:

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

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