Catalan solid

English

A disdyakis triacontahedron, one of the Catalan solids, being the dual of the truncated icosidodecahedron

Etymology

From Catalan (“a surname”) + solid, named for Belgian mathematician Eugène Charles Catalan.

Noun

Catalan solid (plural Catalan solids)

1. (geometry) The dual polyhedron of any Archimedean solid.
   • 1993, E. Heil, H. Martini, 1.11: Special convex bodies, P. M. Gruber, J. M> Wills (editors), Handbook of Convex Geometry, Volume A, page 352,

     These duals are also called Catalan solids, because Catalan was the first mathematician who described all of them, cf. Brückner (1900, p. 160). Some of the Catalan solids are much older, e.g., the rhombic dodecahedron, which plays a role in crystallography (cf. chapter 37), and the rhombic triacontahedron, which is of some importance also in the theory of quasicrystals (cf. chapter 3.5).

   • 2003, Rona Gurkewitz, Bennett Arnstein, Multimodular Origami Polyhedra: Archimedeans, Buckyballs and Duality, page 4:

     The set of thirteen such duals is historically designated the Catalan solids. Perhaps the most striking feature of a Catalan solid is that it is composed of just one kind of face that is not a regular polygon. A face of a Catalan solid can be simply constructed by the Dorman-Luke construction [3, 4, 6, 16], which is based on a dualization process known as polar reciprocation [6], with respect to the midsphere.

   • 2009, Stephen M. Phillips, The Mathematical Connection Between Religion and Science‎^[1], page 270:

     As the most complex of the Catalan solids, the disdyakis triacontahedron has 1680 geometrical elements surrounding its axis, according to Table 1, whilst 2400 elements surround it when its faces are divided into their sectors, i.e., 720 elements are added by this division. These numbers are ten times the corresponding numbers for the simplest Catalan solid.

Synonyms

• (dual polyhedron of an Archimedean solid): Archimedean dual

Translations

dual polyhedron of an Archimedean solid

• Catalan: sòlid de Catalan m, sòlid arquimedià dual m
• Finnish: Catalanin kappale, Catalanin monitahokas, Arkhimedeen duaali
• German: catalanischer Körper m, dual-archimedischer Körper m
• Italian: solido di Catalan m, solido archimedeo duale m

Further reading

• Catalan Solid on Wolfram MathWorld