Catalan body
A Catalan body or dual-Archimedean body is a body that is dual to an Archimedean body . For example, the rhombic dodecahedron is dual to the cuboctahedron . The Catalan bodies - of which there are 13 - are named after the Belgian mathematician Eugène Charles Catalan . The Catalan solids are convex polyhedra .
A Catalan body has only one type of side surface, i.e. H. all side surfaces are congruent to one another . The side faces are non-regular polygons. On the other hand, there are at least two different types of corners (the diamond dodecahedron, for example, has corners adjoined by three rhombuses and corners adjoined by four rhombuses). It is the other way around with Archimedes' solids: they have one type of corner and several types of side faces.
All Catalan bodies have in common that they have an incugel that touches all surfaces from the inside. There is also an edge sphere that touches all edges from the inside. All dihedral angles of a Catalan body are the same.
A characteristic feature of the Catalan bodies is the uniformity of the surfaces. That means: If A, B are any two side surfaces, then the body can be rotated or mirrored in such a way that the body is transformed into itself and side A into side B. This property follows from the uniformity of the corners for Archimedean solids.
