Catalan Solids

Name: Catalan Solids - Geometry Calculator
Author: Jürgen Kummer

On Catalan solids: three-dimensional geometric objects with one type of irregular face

Catalan solids are the duals of the Archimedean solids. The dual of a polyhedron is formed by connecting the centers of its faces, such that these centers become the vertices of a new polyhedron. In this process, the edges connecting any two vertices of the original polyhedron correspond to the edges between two faces of its dual. The duals of the Platonic solids are themselves Platonic solids, either of the same or a different type. This principle had long been known when, in the mid-19th century, the Belgian mathematician Eugène Charles Catalan conceived the idea of applying it to the Archimedean solids. In doing so, he discovered, described, and classified in a single stroke thirteen new polyhedra. All of these possess faces that are congruent, non-regular polygons. The types of faces found among Catalan solids include triangles, rhombi, kites and axially symmetric pentagons. The dual of the dual solid is the original solid. Thus, the duals of the Catalan solids are the according Archimedean solids. However, the size has changed in the process.
The standard ordering of the Catalan solids follows that of the corresponding Archimedean solids. Since the latter are usually listed according to their number of faces, the Catalan solids are consequently ordered by their number of vertices. As with the Archimedean solids, there are two chiral Catalan solids, the pentagonal icositetrahedron and the pentagonal hexecontahedron. These exist in two mirror-image variants, though the calculation of their dimensions is identical for both. If these are counted as distinct Catalan solids, the total number rises to fifteen. However, it is more common to speak of thirteen Catalan solids.

The thirteen Catalan solids can be calculated on these pages:

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