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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, Eugène Catalan who first described them in 1865.## See also

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The Catalan solids are all convex. They are face-transitive but not vertex-transitive. This is because the dual Archimedean solids are vertex-transitive and not face-transitive. 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-transitive: 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.

- List of uniform tilings Shows dual uniform polygonal tilings similar to the Catalan solids
- Conway polyhedron notation A notational construction process

- Eugène Catalan Mémoire sur la Théorie des Polyèdres. J. l'École Polytechnique (Paris) 41, 1-71, 1865.
- Alan Holden Shapes, Space, and Symmetry. New York: Dover, 1991.
- Wenninger, Magnus (1983).
*Dual Models*. Cambridge University Press. ISBN 0-521-54325-8. - Williams, Robert (1979).
*The Geometrical Foundation of Natural Structure: A Source Book of Design*. Dover Publications, Inc. ISBN 0-486-23729-X. (Section 3-9)

- Archimedean duals – at Virtual Reality Polyhedra
- Interactive Catalan Solid in Java
- Housing Construction using the Rhombic Dodecahedron

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Last updated on Wednesday September 24, 2008 at 00:11:09 PDT (GMT -0700)

View this article at Wikipedia.org - Edit this article at Wikipedia.org - Donate to the Wikimedia Foundation

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