The Catalan solids are
face-transitive or
isohedral, meaning that their faces are symmetric to one another, but they are not
vertex-transitive because their vertices are not symmetric. Their duals, the Archimedean solids, are vertex-transitive but not face-transitive. Each Catalan solid has constant
dihedral angles, meaning the angle between any two adjacent faces is the same. Additionally, two Catalan solids, the
rhombic dodecahedron and
rhombic triacontahedron, are
edge-transitive, meaning their edges are symmetric to each other. Some Catalan solids were discovered by
Johannes Kepler during his study of
zonohedra, and
Eugene Catalan completed the list of the thirteen solids in 1865. 's construction, the dual polyhedron of a
cuboctahedron, by
Dorman Luke construction In general, each face of a dual uniform polyhedron (including the Catalan solid) can be constructed by using the
Dorman Luke construction. Some of the Catalan solids can be constructed by adding pyramids to the faces of Platonic solids. These examples are
Kleetopes of Platonic solids:
triakis tetrahedron,
tetrakis hexahedron,
triakis octahedron,
triakis icosahedron, and
pentakis dodecahedron. Two Catalan solids, the
pentagonal icositetrahedron and the
pentagonal hexecontahedron, are
chiral, meaning that these two solids are not their own mirror images. They are dual to the
snub cube and
snub dodecahedron, respectively, which are also chiral. Eleven of the thirteen Catalan solids are known to have the
Rupert property, which means that a copy of the same solid can be passed through a hole in the solid. == References ==