Catalan solid
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.
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.
*
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.
*
Magnus Wenninger Dual Models Cambridge, England: Cambridge University Press, 1983.
*
Catalan Solid – MathWorld site
*
Archimedean duals – at Virtual Reality Polyhedra
*
Interactive Catalan Solid in Java
