triacontahedron

The triacontahedron is a polyhedron with thirty faces. The term is most commonly associated with the rhombic triacontahedron, a Catalan solid whose faces are congruent rhombi.

The rhombic triacontahedron has 30 rhombic faces, 60 edges, and 32 vertices. It is the dual of the Archimedean solid called the icosidodecahedron, and it thus inherits icosahedral symmetry (the same symmetry as the icosahedron and dodecahedron). Each face is a rhombus, with all faces congruent and opposite faces parallel. In the vertex arrangement, four rhombi meet at each vertex.

Construction and realization can be described in terms of duality: the vertices of the rhombic triacontahedron correspond to the faces of the icosidodecahedron, and its edges connect centers of neighboring faces of that solid. Equivalently, it can be realized with high symmetry inside a sphere, with coordinates often given in relation to the golden ratio through the dual relationship with the icosidodecahedron.

In broader usage, triacontahedron simply denotes any polyhedron with thirty faces, though most references refer specifically to the rhombic form with its distinctive rhombic faces and Catalan-solid status. The rhombic triacontahedron is notable for its uniform faces, its relation to the icosidodecahedron, and its role as a classic example of a dual polyhedron with high symmetry.