User:Tomruen/Catalan solid

A rhombic dodecahedron

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, Eugène Catalan, who first described them in 1865.

The Catalan solids are all convex. They are face-transitive but not vertex-transitive. This is because the dual Archimedean solids are vertex-transitive and not face-transitive. 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-transitive: the rhombic dodecahedron and the rhombic triacontahedron. These are the duals of the two quasi-regular Archimedean solids.

Two of the Catalan solids are chiral: the pentagonal icositetrahedron and the pentagonal hexecontahedron, dual to the chiral snub cube and snub dodecahedron. These each come in two enantiomorphs. Not counting the enantiomorphs there are a total of 13 Catalan solids.

Name(s) Coxeter-Dynkin diagram(s)	Picture Solid	Picture Transparent	Net	Dual (Archimedean solids)	Faces	Edges	Vertices	Face polygon	Symmetry
triakis tetrahedron		(Animation)		truncated tetrahedron	12	18	8	isosceles triangle V3.6.6	Td
rhombic dodecahedron		(Animation)		cuboctahedron	12	24	14	rhombus V3.4.3.4	Oh
triakis octahedron		(Animation)		truncated cube	24	36	14	isosceles triangle V3.8.8	Oh
tetrakis hexahedron (or disdyakis hexahedron or hexakis tetrahedron)		(Animation)		truncated octahedron	24	36	14	isosceles triangle V4.6.6	Oh
deltoidal icositetrahedron (or trapezoidal icositetrahedron)		(Animation)		rhombicuboctahedron	24	48	26	kite V3.4.4.4	Oh
disdyakis dodecahedron (or hexakis octahedron)		(Animation)		truncated cuboctahedron	48	72	26	scalene triangle V4.6.8	Oh
pentagonal icositetrahedron		(Anim.)(Anim.)		snub cube	24	60	38	irregular pentagon V3.3.3.3.4	O
rhombic triacontahedron		(Animation)		icosidodecahedron	30	60	32	rhombus V3.5.3.5	Ih
triakis icosahedron		(Animation)		truncated dodecahedron	60	90	32	isosceles triangle V3.10.10	Ih
pentakis dodecahedron		(Animation)		truncated icosahedron	60	90	32	isosceles triangle V5.6.6	Ih
deltoidal hexecontahedron (Or trapezoidal hexecontahedron)		(Animation)		rhombicosidodecahedron	60	120	62	kite V3.4.5.4	Ih
disdyakis triacontahedron (or hexakis icosahedron)		(Animation)		truncated icosidodecahedron	120	180	62	scalene triangle V4.6.10	Ih
pentagonal hexecontahedron		(Anim.)(Anim.)		snub dodecahedron	60	150	92	irregular pentagon V3.3.3.3.5	I

See also

• List of uniform tilings Shows dual uniform polygonal tilings similar to the Catalan solids
• Conway polyhedron notation A notational construction process
• Archimedean solid
• Johnson solid

References

• Eugène Catalan Mémoire sur la Théorie des Polyèdres. J. l'École Polytechnique (Paris) 41, 1-71, 1865.
• Alan Holden Shapes, Space, and Symmetry. New York: Dover, 1991.
• Wenninger, Magnus (1983), Dual Models, Cambridge University Press, ISBN 978-0-521-54325-5, MR 0730208 (The thirteen semiregular convex polyhedra and their duals)
• Williams, Robert (1979). The Geometrical Foundation of Natural Structure: A Source Book of Design. Dover Publications, Inc. ISBN 0-486-23729-X. (Section 3-9)

External links

• Archimedean duals – at Virtual Reality Polyhedra
• Interactive Catalan Solid in Java