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Small Rhombicosidodecahedron


SmallRhombicosidodecahedronSolidWireframeNet

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The (small) rhombicosidodecahedron (Cundy and Rowlett 1989, p. 111), sometimes simply called the rhombicosidodecahedron (Maeder 1997; Wenninger 1989, p. 27; Conway et al. 1999; Maeder 1997), is the 62-faced Archimedean solid with faces 20{3}+30{4}+12{5}. It is also the uniform polyhedron with Maeder index 10 (Maeder 1997), Wenninger index 13 (Wenninger 1989), Coxeter index 22 (Coxeter et al. 1954), and Har'El index 15 (Har'El 1993). It is illustrated above together with a wireframe version and a net that can be used for its construction.

It has Schläfli symbol r{3; 5} and Wythoff symbol 35|2. The small dodecicosidodecahedron and small rhombidodecahedron are faceted versions.

SmallRhombicosProjections

Some symmetric projections of the small rhombicosidodecahedron are illustrated above.

It is implemented in the Wolfram Language as UniformPolyhedron["Rhombicosidodecahedron"]. Precomputed properties are available as PolyhedronData["SmallRhombicosidodecahedron", prop].

SmallRhombicosidodecahedronConvexHulls

The small rhombicosidodecahedron is the convex hull of the small dodecicosidodecahedron, small rhombidodecahedron, and small stellated truncated dodecahedron.

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

r_d=1/(41)(15+2sqrt(5))sqrt(11+4sqrt(5))=2.12099...
(1)
rho=1/2sqrt(10+4sqrt(5))=2.17625...
(2)
R=1/2sqrt(11+4sqrt(5))=2.23295....
(3)

The unit small rhombicosidodecahedron has surface area

 S=5(6+sqrt(3))+3·5^(3/4)sqrt(2+sqrt(5))
(4)

and volume

 V=1/3(60+29sqrt(5)).
(5)

The unit small rhombicosidodecahedron Dehn invariant

D=60<3>_5-30<5>_1
(6)
=30sin^(-1)((5-4sqrt(5))/(15)),
(7)

where the first expression uses the basis of Conway et al. (1999). It can be dissected into the metabigyrate rhombicosidodecahedron, parabigyrate rhombicosidodecahedron, and trigyrate rhombicosidodecahedron, from which it differs only by the relative rotations of three cupola.

SmallRhombicosidodecahedronAndDual

The dual polyhedron of the small rhombicosidodecahedron is the deltoidal hexecontahedron, both of which are illustrated above together with their common midsphere.


See also

Archimedean Solid, Equilateral Zonohedron, Great Rhombicosidodecahedron, Hexecontahedron, Quasirhombicosidodecahedron, Rhombicosidodecahedron, Zome

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References

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. "(Small) Rhombicosidodecahedron. 3.4.5.4." §3.7.11 in Mathematical Models, 3rd ed. Stradbroke, England: Tarquin Pub., p. 111, 1989.Geometry Technologies. "Rhombicosidodecahedron." http://www.scienceu.com/geometry/facts/solids/rh_icosidodeca.html.Har'El, Z. "Uniform Solution for Uniform Polyhedra." Geometriae Dedicata 47, 57-110, 1993.Kasahara, K. "From Regular to Semiregular Polyhedrons." Origami Omnibus: Paper-Folding for Everyone. Tokyo: Japan Publications, pp. 220-221, 1988.Maeder, R. E. "10: Rhombicosidodecahedron." 1997. https://www.mathconsult.ch/static/unipoly/10.html.Wenninger, M. J. "The Rhombicosidodecahedron." Model 14 in Polyhedron Models. Cambridge, England: Cambridge University Press, p. 28, 1989.

Cite this as:

Weisstein, Eric W. "Small Rhombicosidodecahedron." From MathWorld--A Wolfram Web Resource. https://mathworld.wolfram.com/SmallRhombicosidodecahedron.html

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