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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.
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.
Rhombic triacontahedron (Dual of icosidodecahedron) — V(3.5.3.5) arccos (- √ 5 +1 / 4 ) = 4 π / 5 144° Medial rhombic triacontahedron (Dual of dodecadodecahedron) — V(5. 5 / 2 .5. 5 / 2 ) arccos (- 1 / 2 ) = 2 π / 3 120° Great rhombic triacontahedron (Dual of great icosidodecahedron) — V(3 ...
Medial disdyakis triacontahedron; Medial hexagonal hexecontahedron; Medial icosacronic hexecontahedron; Medial inverted pentagonal hexecontahedron; Medial pentagonal hexecontahedron; Medial rhombic triacontahedron; Hexahemioctacron; Hemipolyhedron; Octahemioctacron; Rhombicosacron; Small complex icosidodecahedron; Small ditrigonal dodecacronic ...
It slightly resembles an inflated rhombic triacontahedron: if one replaces each face of the rhombic triacontahedron with a single vertex and four triangles in a regular fashion, one ends up with a disdyakis triacontahedron. That is, the disdyakis triacontahedron is the Kleetope of the rhombic triacontahedron.
Icosidodecahedron (light) and rhombic triacontahedron (dark) Dual compounds of Archimedean and Catalan solids A dual compound is composed of a polyhedron and its dual, arranged reciprocally about a common midsphere , such that the edge of one polyhedron intersects the dual edge of the dual polyhedron.
[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.