Catalan solid
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Rhombic_dodecahedron.jpg
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 enantiomorphs there are a total of 13 Catalan solids.
| Name and picture | Dual Archimedean solid | Faces | Edges | Vertices | Face Polygon | Symmetry |
|---|---|---|---|---|---|---|
| rhombic dodecahedron Missing image Rhombicdodecahedron.jpg Rhombic dodecahedron (Video) |
cuboctahedron | 12 | 24 | 14 | rhombus | Oh |
rhombic triacontahedron (Video) |
icosidodecahedron | 30 | 60 | 32 | rhombus | Ih |
| triakis tetrahedron Missing image Triakistetrahedron.jpg Triakis tetrahedron (Video) |
truncated tetrahedron | 12 | 18 | 8 | isosceles triangle | Td |
| triakis octahedron Missing image Triakisoctahedron.jpg Triakis octahedron (Video) |
truncated cube | 24 | 36 | 14 | isosceles triangle | Oh |
tetrakis hexahedron (Video) |
truncated octahedron | 24 | 36 | 14 | isosceles triangle | Oh |
| triakis icosahedron Missing image Triakisicosahedron.jpg Triakis icosahedron (Video) |
truncated dodecahedron | 60 | 90 | 32 | isosceles triangle | Ih |
pentakis dodecahedron (Video) |
truncated icosahedron | 60 | 90 | 32 | isosceles triangle | Ih |
| deltoidal icositetrahedron Missing image Deltoidalicositetrahedron.jpg 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 Missing image Deltoidalhexecontahedron.jpg Deltoidal hexecontahedron (Video) |
rhombicosidodecahedron | 60 | 120 | 62 | kite | Ih |
| disdyakis triacontahedron or hexakis icosahedron Missing image Disdyakistriacontahedron.jpg Disdyakis triacontahedron (Video) |
truncated icosidodecahedron | 120 | 180 | 62 | scalene triangle | Ih |
| pentagonal icositetrahedron Missing image Pentagonalicositetrahedronccw.jpg Pentagonal icositetrahedron (Ccw) (Video)  (Video) |
snub cube | 24 | 60 | 38 | irregular pentagon | O |
pentagonal hexecontahedron (Video) Missing image Pentagonalhexecontahedroncw.jpg Pentagonal hexecontahedron (Cw) (Video) |
snub dodecahedron | 60 | 150 | 92 | irregular pentagon | I |
External links
- Catalan Solid (http://mathworld.wolfram.com/CatalanSolid.html) – MathWorld site
- Archimedean duals (http://www.georgehart.com/virtual-polyhedra/archimedean-duals-info.html) – at Virtual Reality Polyhedra
- Interactive Catalan Solid (http://ibiblio.org/e-notes/3Dapp/Catalan.htm) in Javade:Catalanischer Körper