# Catalan solid

A rhombic dodecahedron

In mathematics, a Catalan solid, or Archimedean dual, is a dual polyhedron to an Archimedean solid. The Catalan solids are named for the Belgian mathematician, Eugene Catalan who first described them in 1865.

The Catalan solids are all convex. They are face-uniform but not vertex-uniform. This is because the dual Archimedean solids are vertex-uniform and not face uniform. Note that unlike Platonic solids and Archimedean solids, the faces of Catalan solids are not regular polygons. However, the vertex figures of Catalan solids are regular, and they have constant dihedral angles. Additionally, two of the Catalan solids are edge-uniform: the rhombic dodecahedron and the rhombic triacontahedron. These are the duals of the two quasi-regular Archimedean solids.

Just like their dual Archimedean partners there are two chiral Catalan solids: the pentagonal icositetrahedron and the pentagonal hexecontahedron. These each come in two enantiomorphs. Not counting the enantimorphs there are a total of 13 Catalan solids.

Name and picture Dual Archimedean solid Faces Edges Vertices Face Polygon Symmetry
rhombic dodecahedron

(Video)
cuboctahedron 12 24 14 rhombus Oh
rhombic triacontahedron

(Video)
icosidodecahedron 30 60 32 rhombus Ih
triakis tetrahedron

(Video)
truncated tetrahedron 12 18 8  isosceles triangle Td
triakis octahedron

(Video)
truncated cube  24 36 14 isosceles triangle Oh
tetrakis hexahedron

(Video)
truncated octahedron  24 36 14 isosceles triangle Oh
triakis icosahedron

(Video)
truncated dodecahedron  60 90 32 isosceles triangle Ih
pentakis dodecahedron

(Video)
truncated icosahedron  60 90 32 isosceles triangle Ih
deltoidal icositetrahedron

(Video)
rhombicuboctahedron  24 48 26 kite Oh
disdyakis dodecahedron
or hexakis octahedron

(Video)
truncated cuboctahedron  48 72 26 scalene triangle Oh
deltoidal hexecontahedron

(Video)
rhombicosidodecahedron  60 120 62 kite Ih
disdyakis triacontahedron
or hexakis icosahedron

(Video)
truncated icosidodecahedron  120 180 62 scalene triangle Ih
pentagonal icositetrahedron

(Video)

(Video)
snub cube  24 60 38 irregular pentagon O
pentagonal hexecontahedron

(Video)

(Video)
snub dodecahedron 60 150 92 irregular pentagon I