Catalan solid

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File:Rhombic dodecahedron.jpg
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-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 enantiomorphs there are a total of 13 Catalan solids.

Name picture Dual Archimedean solid Faces Edges Vertices Face Polygon Symmetry
triakis tetrahedron Triakis tetrahedron
truncated tetrahedron 12 18 8  isosceles triangle Td
rhombic dodecahedron Rhombic dodecahedron
cuboctahedron 12 24 14 rhombus Oh
triakis octahedron Triakis octahedron
truncated cube  24 36 14 isosceles triangle Oh
tetrakis hexahedron Tetrakis hexahedron
truncated octahedron  24 36 14 isosceles triangle Oh
deltoidal icositetrahedron Deltoidal icositetrahedron
rhombicuboctahedron  24 48 26 kite Oh
disdyakis dodecahedron
or hexakis octahedron
Disdyakis dodecahedron
truncated cuboctahedron  48 72 26 scalene triangle Oh
pentagonal icositetrahedron Pentagonal icositetrahedron (Ccw)Pentagonal icositetrahedron (Cw)
snub cube  24 60 38 irregular pentagon O
rhombic triacontahedron Rhombic triacontahedron
icosidodecahedron 30 60 32 rhombus Ih
triakis icosahedron Triakis icosahedron
truncated dodecahedron  60 90 32 isosceles triangle Ih
pentakis dodecahedron Pentakis dodecahedron
truncated icosahedron  60 90 32 isosceles triangle Ih
deltoidal hexecontahedron Deltoidal hexecontahedron
rhombicosidodecahedron  60 120 62 kite Ih
disdyakis triacontahedron
or hexakis icosahedron
Disdyakis triacontahedron
truncated icosidodecahedron  120 180 62 scalene triangle Ih
pentagonal hexecontahedron Pentagonal hexecontahedron (Ccw)Pentagonal hexecontahedron (Cw)
snub dodecahedron 60 150 92 irregular pentagon I

External links

de:Catalanischer Körper it:Solidi di Catalan zh:卡塔蘭立體 ja:双対多面体