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3D model of a rhombic triacontahedron. The rhombic triacontahedron, sometimes simply called the triacontahedron as it is the most common thirty-faced polyhedron, is a convex polyhedron with 30 rhombic faces. It has 60 edges and 32 vertices of two types. It is a Catalan solid, and the dual polyhedron of the icosidodecahedron. It is a zonohedron.
The following 31 pages use this file: Alternation (geometry) Catalan solid; Conway polyhedron notation; Deltoidal hexecontahedron; Disdyakis triacontahedron
[1] [2] There are different truncations of a rhombic triacontahedron into a topological rhombicosidodecahedron: Prominently its rectification (left), the one that creates the uniform solid (center), and the rectification of the dual icosidodecahedron (right), which is the core of the dual compound.
It has 30 intersecting rhombic faces. It can also be called the great stellated triacontahedron. The great rhombic triacontahedron can be constructed by expanding the size of the faces of a rhombic triacontahedron by a factor of τ 3 = 1+2τ = 2+√5, where τ is the golden ratio.
The rhombic triacontahedron, with two types of alternating vertices, 20 with three rhombic faces, and 12 with five rhombic faces. In addition, by duality with the octahedron, the cube , which is usually regular , can be made quasiregular if alternate vertices are given different colors.
The rhombic hexecontahedron is a stellation of the rhombic triacontahedron. It is nonconvex with 60 golden rhombic faces with icosahedral symmetry. The rhombic enneacontahedron is a polyhedron composed of 90 rhombic faces, with three, five, or six rhombi meeting at each vertex. It has 60 broad rhombi and 30 slim ones.
In geometry, the great rhombic triacontahedron is a nonconvex isohedral, isotoxal polyhedron. It is the dual of the great icosidodecahedron (U54). Like the convex rhombic triacontahedron it has 30 rhombic faces, 60 edges and 32 vertices (also 20 on 3-fold and 12 on 5-fold axes).
3D model of a medial rhombic triacontahedron. In geometry, the medial rhombic triacontahedron (or midly rhombic triacontahedron) is a nonconvex isohedral polyhedron. It is a stellation of the rhombic triacontahedron, and can also be called small stellated triacontahedron. Its dual is the dodecadodecahedron.