Definitions

# Catalan solid

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.

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 enantiomorphs there are a total of 13 Catalan solids.

Name(s) Picture Net Dual (Archimedean solids) Faces Edges Vertices Face polygon Symmetry
triakis tetrahedron
(triakistetrahedron.gif)
truncated tetrahedron 12 18 8 isosceles triangle
V3.6.6
Td
rhombic dodecahedron
(rhombicdodecahedron.gif)
cuboctahedron 12 24 14 rhombus
V3.4.3.4
Oh
triakis octahedron
(triakisoctahedron.gif)
truncated cube 24 36 14 isosceles triangle
V3.8.8
Oh
tetrakis hexahedron
(tetrakishexahedron.gif)
truncated octahedron 24 36 14 isosceles triangle
V4.6.6
Oh
deltoidal icositetrahedron
(deltoidalicositetrahedron.gif)
rhombicuboctahedron 24 48 26 kite
V3.4.4.4
Oh
disdyakis dodecahedron
or hexakis octahedron

(disdyakisdodecahedron.gif)
truncated cuboctahedron 48 72 26 scalene triangle
V4.6.8
Oh
pentagonal icositetrahedron
(pentagonalicositetrahedronccw.gif)(pentagonalicositetrahedroncw.gif)
snub cube 24 60 38 irregular pentagon
V3.3.3.3.4
O
rhombic triacontahedron
(rhombictriacontahedron.gif)
icosidodecahedron 30 60 32 rhombus
V3.5.3.5
Ih
triakis icosahedron
(triakisicosahedron.gif)
truncated dodecahedron 60 90 32 isosceles triangle
V3.10.10
Ih
pentakis dodecahedron
(pentakisdodecahedron.gif)
truncated icosahedron 60 90 32 isosceles triangle
V5.6.6
Ih
deltoidal hexecontahedron
(deltoidalhexecontahedron.gif)
rhombicosidodecahedron 60 120 62 kite
V3.4.5.4
Ih
disdyakis triacontahedron
or hexakis icosahedron

(disdyakistriacontahedron.gif)
truncated icosidodecahedron 120 180 62 scalene triangle
V4.6.10
Ih
pentagonal hexecontahedron
(pentagonalhexecontahedronccw.gif)(pentagonalhexecontahedroncw.gif)
snub dodecahedron 60 150 92 irregular pentagon
V3.3.3.3.5
I

## 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 0-521-54325-8.
• 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)