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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, Eugene 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 enantimorphs there are a total of 13 Catalan solids.

Name and picture	Dual Archimedean solid	Faces	Edges	Vertices	Face Polygon	SymmetryThe symmetry group of a geometric figure is the group of congruencies under which it is invariant, with composition as the operation. The article on group theory also contains an explanation of the concept. In Euclidean geometry, discrete symmetry groups
rhombic dodecahedron ()	cuboctahedronA cuboctahedron is a polyhedron with eight triangular faces and six square faces. A cuboctahedron has 12 identical vertices, with two triangles and two squares meeting at each, and 24 identical edges, each separating a triangle from a square. As such it i	12	24	14	rhombusIn geometry, a rhombus (also known as a rhomb is a parallelogram in which all of the sides are of equal length. More colloquially it may be described as a diamond or lozenge shape. In any rhombus, opposite sides will be parallel. Thus, the rhombus is a sp	O_h
rhombic triacontahedron ()	icosidodecahedronAn icosidodecahedron is a polyhedron with twenty triangular faces and twelve pentagonal faces. An icosidodecahedron has 30 identical vertices, with two triangles and two pentagons meeting at each, and 60 identical edges, each separating a triangle from a	30	60	32	rhombusIn geometry, a rhombus (also known as a rhomb is a parallelogram in which all of the sides are of equal length. More colloquially it may be described as a diamond or lozenge shape. In any rhombus, opposite sides will be parallel. Thus, the rhombus is a sp	I_h
triakis tetrahedronA triakis tetrahedron is a catalan solid which looks a bit like an overinflated tetrahedron. Catalan solids. ()	truncated tetrahedron	12	18	8	isosceles triangle	T_d
triakis octahedron ()	truncated cube	24	36	14	isosceles triangle	O_h
tetrakis hexahedron ()	truncated octahedron	24	36	14	isosceles triangle	O_h
triakis icosahedron ()	truncated dodecahedron	60	90	32	isosceles triangle	I_h
pentakis dodecahedron ()	truncated icosahedron	60	90	32	isosceles triangle	I_h
deltoidal icositetrahedron ()	rhombicuboctahedron	24	48	26	kite	O_h
disdyakis dodecahedron ()	truncated cuboctahedron	48	72	26	scalene triangle	O_h
deltoidal hexecontahedron ()	rhombicosidodecahedron	60	120	62	kite	I_h
disdyakis triacontahedron ()	truncated icosidodecahedron	120	180	62	scalene triangle	I_h
pentagonal icositetrahedron () ()	snub cube	24	60	38	irregular pentagon	O
pentagonal hexecontahedron () ()	snub dodecahedron	60	150	92	irregular pentagon	I

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Catalan languageCatalan Catal Valenci is a Romance language spoken by approximately 12 million people in a territory called the Catalan countries which spans portions of Spain, France, Andorra and Italy, although the majority of Catalan speakers are in Spain. Classificat	CatalanCatalan can mean: the Catalan language Eugene Charles Catalan the mathematician inhabitant of Catalonia There is also information on Catalan names.	Catalan's constantCatalan's constant ''K which occasionally appears in estimates in combinatorics, is defined by : or equivalently : Its numerical value is approximately K . 915 965 594 177 219 015 054 603 514 932 384 110 774. It is not known whether K is rational or irrat
Catalan's conjectureCatalan's conjecture is a simple conjecture in number theory that was proposed by the mathematician Eugene Charles Catalan. To understand the conjecture notice that 23 8 and 32 9 are two consecutive powers of natural numbers. Catalan's conjecture states t	Catalan numberThe Catalan numbers named after the Belgian mathematician Eugene Charles Catalan (1814—1894), form a sequence of natural numbers that occur in various counting problems in combinatorics. The n''th Catalan number is defined by : (see binomial coefficient).	Catalan CompanyThe Oriental Catalan Company or the Grand Company, was founded by Roger de Flor (who inspired the medieval tale of Tirant lo Blanc) after the Peace of Caltabellotta in 1302 had left jobless the soldiers from Catalonia and Aragon fighting against the Frenc
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