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Catalan solids

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.

 

A rhombic dodecahedron

 

Name(s)Picture
Solid
Picture
Transparent
NetDual (Archimedean solids)FacesEdgesVerticesFace polygonSymmetry
triakis tetrahedron Triakis tetrahedron Triakis tetrahedron
(Animation)
Triakistetrahedron net.png truncated tetrahedron 12 18 8 isosceles triangle
V3.6.6
Td
rhombic dodecahedron Rhombic dodecahedron Rhombic dodecahedron
(Animation)
Rhombicdodecahedron net.svg cuboctahedron 12 24 14 rhombus
V3.4.3.4
Oh
triakis octahedron Triakis octahedron Triakis octahedron
(Animation)
Triakisoctahedron net.png truncated cube 24 36 14 isosceles triangle
V3.8.8
Oh
tetrakis hexahedron
(or disdyakis hexahedron or hexakis tetrahedron)
Tetrakis hexahedron Tetrakis hexahedron
(Animation)
Tetrakishexahedron net.png truncated octahedron 24 36 14 isosceles triangle
V4.6.6
Oh
deltoidal icositetrahedron
(or trapezoidal icositetrahedron)
Deltoidal icositetrahedron Deltoidal icositetrahedron
(Animation)
Deltoidalicositetrahedron net.png rhombicuboctahedron 24 48 26 kite
V3.4.4.4
Oh
disdyakis dodecahedron
(or hexakis octahedron)
Disdyakis dodecahedron Disdyakis dodecahedron
(Animation)
Disdyakisdodecahedron net.png truncated cuboctahedron 48 72 26 scalene triangle
V4.6.8
Oh
pentagonal icositetrahedron Pentagonal icositetrahedron Pentagonal icositetrahedron (Ccw)Pentagonal icositetrahedron (Cw)
(Anim.)(Anim.)
Pentagonalicositetrahedron net.png snub cube 24 60 38 irregular pentagon
V3.3.3.3.4
O
rhombic triacontahedron Rhombic triacontahedron Rhombic triacontahedron
(Animation)
Rhombictriacontahedron net.png icosidodecahedron 30 60 32 rhombus
V3.5.3.5
Ih
triakis icosahedron Triakis icosahedron Triakis icosahedron
(Animation)
Triakisicosahedron net.png truncated dodecahedron 60 90 32 isosceles triangle
V3.10.10
Ih
pentakis dodecahedron Pentakis dodecahedron Pentakis dodecahedron
(Animation)
Pentakisdodecahedron net.png truncated icosahedron 60 90 32 isosceles triangle
V5.6.6
Ih
deltoidal hexecontahedron
(Or trapezoidal hexecontahedron)
Deltoidal hexecontahedron Deltoidal hexecontahedron
(Animation)
Deltoidalhexecontahedron net.png rhombicosidodecahedron 60 120 62 kite
V3.4.5.4
Ih
disdyakis triacontahedron
(or hexakis icosahedron)
Disdyakis triacontahedron Disdyakis triacontahedron
(Animation)
Disdyakistriacontahedron net.png truncated icosidodecahedron 120 180 62 scalene triangle
V4.6.10
Ih
pentagonal hexecontahedron Pentagonal hexecontahedron Pentagonal hexecontahedron (Ccw)Pentagonal hexecontahedron (Cw)
(Anim.)(Anim.)
Pentagonalhexecontahedron net.png snub dodecahedron 60 150 92 irregular pentagon
V3.3.3.3.5
I

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