The Catalan solids are a set of semiregular polyhedra first studied by the Belgian mathematician Eugene Catalan. There are 13 distinct polyhedra in the set, two of which come in chiral (left-handed and right-handed) forms. These solids are dual to the Archimedean solids and as such, have the following properties:
- All the faces of any given solid in the set are identical
- All the faces are tangent to one sphere (the insphere)
- There exists a rotational symmetry mapping each face to each other face
- The solids have (at least) the rotational symmetry group of one of the Platonic solids
- All the dihedral angles are equal
The Catalan solids are the following:
Rhombic dodecahedron: The dual of the cuboctahedron is the rhombic dodecahedron. It has 14 vertices, 24 edges and 12 faces. Each face is in the shape of a rhombus with interior angle α = cos-1(1/3). It is a space-filling polyhedron and can be formed by adding a square pyramid of height 1/2 to each face of a unit cube. It has the octahedral group of symmetries.
Disdyakis triacontahedron: The dual of the great rhombicosidodechahedron is the disdyakis triacontahedron. It has 62 vertices, 180 edges and 120 faces. Each face is in the shape of a triangle. It has the icosahedral group of symmetries.
Disdyakis dodecahedron: The dual of the great rhombicuboctahedron is the disdyakis dodecahedron. It has 26 vertices, 72 edges and 48 faces. Each face is in the shape of a triangle. It has the octahedral group of symmetries.
Rhombic triacontahedron: The dual of the icosidodecahedron is the rhombic triacontahedron. It has 32 vertices, 60 edges and 30 faces. Each face is in the shape of a rhombus. It can be formed by adding a pentagonal pyramid of the correct height to each face of a dodecahedron.
Deltoidal hexecontahedron: The dual of the small rhombicosidodecahedron is the deltoidal hexecontahedron. It has 62 vertices, 120 edges and 60 faces. Each face is in the shape of a kite or deltoid. It has the icosahedral group of symmetries.
Deltoidal icositetrahedron: The dual of the small rhombicuboctahedron is the deltoidal icositetrahedron. It has 26 vertices, 48 edges and 26 faces. Each face is in the shape of a kite or deltoid. It has the octahedral group of symmetries.
Pentagonal icositetrahedron (chiral): The dual of the (left-handed) snub cube is the (right-handed) pentagonal icositetrahedron. It has 38 vertices, 60 edges and 24 faces. Each face is in the shape of an irregular pentagon. It has the rotational octahedral group of symmetries.
Pentagonal hexecontahedron (chiral): The dual of the (left-handed) snub dodecahedron is the (right-handed) pentagonal hexecontahedron. It has 92 vertices, 150 edges and 60 faces. Each face is in the shape of an irregular pentagon. It has the rotational icosahedral group of symmetries.
Small triakis octahedron: The dual of the truncated cube is the small triakis octahedron. It has 14 vertices, 36 edges and 24 faces. Each face is in the shape of an isosceles triangle. It has the octahedral group of symmetries.
Triakis icosahedron: The dual of the truncated dodecahedron is the triakis icosahedron. It has 32 vertices, 90 edges and 60 faces. Each face is in the shape of an isosceles triangle. It has the icosahedral group of symmetries.
Pentakis dodecahedron: The dual of the truncated icosahedron is the pentakis dodecahedron. It has 32 vertices, 90 edges and 60 faces. Each face is in the shape of an isosceles triangle. It has the icosahedral group of symmetries.
Tetrakis hexahedron: The dual of the truncated octahedron is the tetrakis hexahedron. It has 14 vertices, 36 edges and 24 faces. Each face is in the shape of an isosceles triangle. It has the octahedral group of symmetries.
Triakis tetrahedron: The dual of the
truncated tetrahedron is the triakis tetrahedron. It has 8 vertices, 18 edges and 12 faces. Each face is in the shape of an isosceles triangle with middle angle α = cos
-1 (-7/18). It has the
tetrahedral group of symmetries.