Catalan solid

 ( Polyhedra ) 

The Catalan solids are the dual polyhedron of Archimedean solids. The Archimedean solids are thirteen highly-symmetric polyhedra with regular faces and symmetric vertices. The faces of the Catalan solids correspond by duality to the vertices of Archimedean solids, and vice versa.

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The solids The Catalan solids are face-transitive or isohedral meaning that their faces are symmetric to one another, but they are not vertex-transitive because their vertices are not symmetric. Their dual, the Archimedean solids, are vertex-transitive but not face-transitive. Each Catalan solid has constant , meaning the angle between any two adjacent faces is the same. Additionally, two Catalan solids, the rhombic dodecahedron and rhombic triacontahedron, are edge-transitive, meaning their edges are symmetric to each other. Some Catalan solids were discovered by Johannes Kepler during his study of zonohedron, and Eugene Catalan completed the list of the thirteen solids in 1865.

In general, each face of a dual uniform polyhedron (including the Catalan solid) can be constructed by using the Dorman Luke construction. Some of the Catalan solids can be constructed, starting from the set of Platonic solids, all faces of which are attached by pyramids. These examples are the Kleetope of Platonic solids: triakis tetrahedron, tetrakis hexahedron, triakis octahedron, triakis icosahedron, and pentakis dodecahedron.

Two Catalan solids, the pentagonal icositetrahedron and the pentagonal hexecontahedron, are chiral, meaning that these two solids are not their own mirror images. They are dual to the snub cube and snub dodecahedron respectively, which are also chiral.

Eleven of the thirteen Catalan solids are known to have the Rupert property that a copy of the same solid can be passed through a hole in the solid.

+ The thirteen Catalan solids ! Name ! Image ! Faces ! Edges ! Vertices ! Dihedral angle ! Point group
triakis tetrahedron		12 isosceles triangles	18	8	129.521°	Td
rhombic dodecahedron		12 Rhombus	24	14	120°	Oh
triakis octahedron		24 isosceles triangles	36	14	147.350°	Oh
tetrakis hexahedron		24 isosceles triangles	36	14	143.130°	Oh
deltoidal icositetrahedron		24 kites	48	26	138.118°	Oh
disdyakis dodecahedron		48 Scalene triangle	72	26	155.082°	Oh
pentagonal icositetrahedron		24 Pentagon	60	38	136.309°	O
rhombic triacontahedron		30 Rhombus	60	32	144°	Ih
triakis icosahedron		60 isosceles triangles	90	32	160.613°	Ih
pentakis dodecahedron		60 isosceles triangles	90	32	156.719°	Ih
deltoidal hexecontahedron		60 kites	120	62	154.121°	Ih
disdyakis triacontahedron		120 scalene triangles	180	62	164.888°	Ih
pentagonal hexecontahedron		60 pentagons	150	92	153.179°	I

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Footnotes

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Works cited

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• (Section 3-9)

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External links

• Catalan Solids – at Visual Polyhedra
• Archimedean duals – at Virtual Reality Polyhedra
• Interactive Catalan Solid in Java
• Download link for Catalan's original 1865 publication – with beautiful figures, PDF format

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