In general, an icosidodecahedron is a 32-faced polyhedron.


polyhdron net "The" (regular) icosidodecahedron is the 32-faced Archimedean solid A_4 with faces 20{3}+12{5}. It is one of the two convex quasiregular polyhedra. It is also uniform polyhedron U_(24) and Wenninger model W_(12). It has Schläfli symbol {3; 5} and Wythoff symbol 2|35.

It is implemented in the Wolfram Language as PolyhedronData["Icosidodecahedron"].


Several symmetric projections of the icosidodecahedron are illustrated above. The dual polyhedron is the rhombic triacontahedron. The polyhedron vertices of an icosidodecahedron of polyhedron edge length 2phi^(-1) are (+/-2,0,0), (0,+/-2,0), (0,0,+/-2), (+/-phi,+/-phi^(-1),+/-1), (+/-1,+/-phi,+/-phi^(-1)), (+/-phi^(-1),+/-1,+/-phi). The 30 polyhedron vertices of an octahedron 5-compound form an icosidodecahedron (Ball and Coxeter 1987). Faceted versions include the small icosihemidodecahedron and small dodecahemidodecahedron.

Origami icosidodecahedron

The icosidodecahedron constructed in origami is illustrated above (Kasahara and Takahama 1987, pp. 48-49). This construction uses 120 sonobè units, each made from a single sheet of origami paper.

The faces of the icosidodecahedron consist of 20 triangles and 12 pentagons. Furthermore, its 60 edges are bisected perpendicularly by those of the reciprocal rhombic triacontahedron (Ball and Coxeter 1987).

Five octahedra inscribed in an icosidodecahedron

Five octahedra of edge length sqrt(3+sqrt(5)) can be inscribed on the vertices of an icosidodecahedron of unit edge length, resulting in the beautiful octahedron 5-compound.

The inradius r_d of the dual, midradius rho=rho_d of the solid and dual, and circumradius R of the solid for a=1 are

 approx 1.46353
 approx 1.53884
 approx 1.61803.

The surface area and volume for an icosidodecahedron are given by


The distance to the centers of the triangular and pentagonal faces are


The dihedral angle between triangular and pentagonal faces is

=142.62... degrees.

See also

Archimedean Solid, Equilateral Zonohedron, Great Icosidodecahedron, Icosidodecahedral Graph, Quasiregular Polyhedron, Small Icosihemidodecahedron, Small Dodecahemidodecahedron

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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.Cundy, H. and Rollett, A. "Icosidodecahedron. (3.5)^2." §3.7.8 in Mathematical Models, 3rd ed. Stradbroke, England: Tarquin Pub., p. 108, 1989.Kasahara, K. "From Regular to Semiregular Polyhedrons." Origami Omnibus: Paper-Folding for Everyone. Tokyo: Japan Publications, pp. 220-221, 1988.Kasahara, K. and Takahama, T. Origami for the Connoisseur. Tokyo: Japan Publications, 1987.Geometry Technologies. "Icosidodecahedron.", M. J. "The Icosidodecahedron." Model 12 in Polyhedron Models. Cambridge, England: Cambridge University Press, pp. 26 and 73, 1989.

Cite this as:

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

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