Catalan polyhedron

The duals of the Archimedean solids are the Catalan polyhedra. They are named after the French-Belgian mathematician Eugene Charles Catalan, who first described them in 1862. Given the characteristics of their duals, discussed above, Catalan polyhedra have more than one type of vertices but all their faces are equivalent, noimegular polygons. Furthermore, each Catalan polyhedron has the same nnm-ber of edges and the same symmetry as its Archimedean dual. The names of the Catalan solids and their duals are given in Table 4. 

The number of edges and symmetry of each Catalan polyhedron can be found in the entry of the corresponding Archimedean dual in Table 3. 

Catalan polyhedron A polyhedron that has a duality relationship with an Archimedean polyhedron, with all faces of identical irregular polygons and more than one set of vertices. 

Classical stmctures inelude the regular and Arehimedean polyhedra Archimedean meaning only n, the polygonality, is fractional. In the 19 century, Catalan identified the reeiproeal polyhedra to the Archimedean polyhedra where n and p were exehanged with each other. Catalan polyhedra can be constmcted from their reeiproeals by joining the midpoints of each face of the Archimedean polyhedron to each other [21]. This reciprocal nature occurs also in two dimensions, one can easily visualize the reciprocals of (4,4), (3, 6) and (6,3), for example but is apparently absent in 3-dimensional nets. 

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