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) | Picture Solid |
Picture Transparent |
Net | Dual (Archimedean solids) | Faces | Edges | Vertices | Face polygon | Symmetry |
|---|---|---|---|---|---|---|---|---|---|
| triakis tetrahedron | truncated tetrahedron | 12 | 18 | 8 | isosceles triangle V3.6.6 |
Td | |||
| rhombic dodecahedron | cuboctahedron | 12 | 24 | 14 | rhombus V3.4.3.4 |
Oh | |||
| triakis octahedron | truncated cube | 24 | 36 | 14 | isosceles triangle V3.8.8 |
Oh | |||
| tetrakis hexahedron (or disdyakis hexahedron or hexakis tetrahedron) |
truncated octahedron | 24 | 36 | 14 | isosceles triangle V4.6.6 |
Oh | |||
| deltoidal icositetrahedron (or trapezoidal icositetrahedron) |
rhombicuboctahedron | 24 | 48 | 26 | kite V3.4.4.4 |
Oh | |||
| disdyakis dodecahedron (or hexakis octahedron) |
truncated cuboctahedron | 48 | 72 | 26 | scalene triangle V4.6.8 |
Oh | |||
| pentagonal icositetrahedron | snub cube | 24 | 60 | 38 | irregular pentagon V3.3.3.3.4 |
O | |||
| rhombic triacontahedron | icosidodecahedron | 30 | 60 | 32 | rhombus V3.5.3.5 |
Ih | |||
| triakis icosahedron | truncated dodecahedron | 60 | 90 | 32 | isosceles triangle V3.10.10 |
Ih | |||
| pentakis dodecahedron | truncated icosahedron | 60 | 90 | 32 | isosceles triangle V5.6.6 |
Ih | |||
| deltoidal hexecontahedron (Or trapezoidal hexecontahedron) |
rhombicosidodecahedron | 60 | 120 | 62 | kite V3.4.5.4 |
Ih | |||
| disdyakis triacontahedron (or hexakis icosahedron) |
truncated icosidodecahedron | 120 | 180 | 62 | scalene triangle V4.6.10 |
Ih | |||
| pentagonal hexecontahedron | snub dodecahedron | 60 | 150 | 92 | irregular pentagon V3.3.3.3.5 |
I |
Famous quotes containing the words catalan and/or solid:
“It’s better that it should make you sick than that you don’t eat it at all.”
—Catalan proverb, quoted in Colman Andrews, Catalan Cuisine.
“It is a sign of our times, conspicuous to the coarsest observer, that many intelligent and religious persons withdraw themselves from the common labors and competitions of the market and the caucus, and betake themselves to a certain solitary and critical way of living, from which no solid fruit has yet appeared to justify their separation.”
—Ralph Waldo Emerson (1803–1882)