Catalan solid
The Catalan solids are the dual polyhedra of Archimedean solids. The Archimedean solids are thirteen highly-symmetric polyhedra with regular faces and symmetric vertices. The faces of the Catalan solids correspond by duality to the vertices of Archimedean solids, and vice versa.
Description
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.File:DormanLuke.svg|thumb|upright|The rhombic dodecahedron's construction, the dual polyhedron of a cuboctahedron, by Dorman Luke construction
In general, each face of a dual uniform polyhedron 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.
| Name | Image | Faces | Edges | Vertices | Dihedral angle | Point group |
| triakis tetrahedron | Image:triakistetrahedron.svg|70px|Triakis tetrahedron | 12 isosceles triangles | 18 | 8 | 129.521° | Td |
| rhombic dodecahedron | Image:rhombicdodecahedron.jpg|70px|Rhombic dodecahedron | 12 rhombi | 24 | 14 | 120° | Oh |
| triakis octahedron | Image:triakisoctahedron.jpg|70px|Triakis octahedron | 24 isosceles triangles | 36 | 14 | 147.350° | Oh |
| tetrakis hexahedron | Image:tetrakishexahedron.jpg|70px|Tetrakis hexahedron | 24 isosceles triangles | 36 | 14 | 143.130° | Oh |
| deltoidal icositetrahedron | Image:deltoidalicositetrahedron.jpg|70px|Deltoidal icositetrahedron | 24 kites | 48 | 26 | 138.118° | Oh |
| disdyakis dodecahedron | Image:disdyakisdodecahedron.jpg|70px|Disdyakis dodecahedron | 48 scalene triangles | 72 | 26 | 155.082° | Oh |
| pentagonal icositetrahedron | Image:pentagonalicositetrahedronccw.jpg|70px|Pentagonal icositetrahedron | 24 pentagons | 60 | 38 | 136.309° | O |
| rhombic triacontahedron | Image:rhombictriacontahedron.svg|70px|Rhombic triacontahedron | 30 rhombi | 60 | 32 | 144° | Ih |
| triakis icosahedron | Image:triakisicosahedron.jpg|70px|Triakis icosahedron | 60 isosceles triangles | 90 | 32 | 160.613° | Ih |
| pentakis dodecahedron | Image:pentakisdodecahedron.jpg|70px|Pentakis dodecahedron | 60 isosceles triangles | 90 | 32 | 156.719° | Ih |
| deltoidal hexecontahedron | Image:deltoidalhexecontahedron.jpg|70px|Deltoidal hexecontahedron | 60 kites | 120 | 62 | 154.121° | Ih |
| disdyakis triacontahedron | Image:disdyakistriacontahedron.jpg|70px|Disdyakis triacontahedron | 120 scalene triangles | 180 | 62 | 164.888° | Ih |
| pentagonal hexecontahedron | Image:pentagonalhexecontahedronccw.jpg|70px|Pentagonal hexecontahedron | 60 pentagons | 150 | 92 | 153.179° | I |
Works cited
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