Catalan solid
In mathematics, a Catalan solid, or Archimedean dual, is a dual polyhedron to an Archimedean solid. There are 13 Catalan solids. They 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. Being face-transitive, Catalan solids are isohedra.
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 as prisms and antiprisms are generally not considered Archimedean solids, so bipyramids and trapezohedra are generally not considered Catalan solids, despite being face-transitive.
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, bipyramids, and trapezohedra, there are a total of 13 Catalan solids.
| Archimedean solid | Catalan solid | |
| 1 | truncated tetrahedron | triakis tetrahedron |
| 2 | truncated cube | triakis octahedron |
| 3 | truncated cuboctahedron | disdyakis dodecahedron |
| 4 | truncated octahedron | tetrakis hexahedron |
| 5 | truncated dodecahedron | triakis icosahedron |
| 6 | truncated icosidodecahedron | disdyakis triacontahedron |
| 7 | truncated icosahedron | pentakis dodecahedron |
| 8 | cuboctahedron | rhombic dodecahedron |
| 9 | icosidodecahedron | rhombic triacontahedron |
| 10 | rhombicuboctahedron | deltoidal icositetrahedron |
| 11 | rhombicosidodecahedron | deltoidal hexecontahedron |
| 12 | snub cube | pentagonal icositetrahedron |
| 13 | snub dodecahedron | pentagonal hexecontahedron |
Symmetry
The Catalan solids, along with their dual Archimedean solids, can be grouped in those with tetrahedral, octahedral and icosahedral symmetry.For both octahedral and icosahedral symmetry there are six forms. The only Catalan solid with genuine tetrahedral symmetry is the triakis tetrahedron. Rhombic dodecahedron and tetrakis hexahedron have octahedral symmetry, but they can be colored to have only tetrahedral symmetry. Rectification and snub also exist with tetrahedral symmetry, but they are Platonic instead of Archimedean, so their duals are Platonic instead of Catalan.
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List
Geometry
All dihedral angles of a Catalan solid are equal. Denoting their value by , and denoting the face angle at the vertices where faces meet by, we haveThis can be used to compute and,,..., from, ... only.
Triangular faces
Of the 13 Catalan solids, 7 have triangular faces. These are of the form Vp.q.r, where p, q and r take their values among 3, 4, 5, 6, 8 and 10. The angles, and can be computed in the following way. Put,, and putThen
For and the expressions are similar of course. The dihedral angle can be computed from
Applying this, for example, to the disdyakis triacontahedron gives and.
Quadrilateral faces
Of the 13 Catalan solids, 4 have quadrilateral faces. These are of the form Vp.q.p.r, where p, q and r take their values among 3, 4, and 5. The angle can be computed by the following formula:From this,, and the dihedral angle can be easily computed. The faces are kites, or, if, rhombi.
Applying this, for example, to the deltoidal icositetrahedron, we get.
Pentagonal faces
Of the 13 Catalan solids, 2 have pentagonal faces. These are of the form Vp.p.p.p.q, where p=3, and q=4 or 5. The angle can be computed by solving a degree three equation:Metric properties
For a Catalan solid let be the dual with respect to the midsphere of. Then is an Archimedean solid with the same midsphere. Denote the length of the edges of by. Let be the inradius of the faces of, the midradius of and, the inradius of, and the circumradius of. Then these quantities can be expressed in and the dihedral angle as follows:These quantities are related by, and.
As an example, let be a cuboctahedron with edge length. Then is a rhombic dodecahedron. Applying the formula for quadrilateral faces with and gives, hence,,,.
All vertices of of type lie on a sphere with radius given by
and similarly for.
Dually, there is a sphere which touches all faces of which are regular -gons in their center. The radius of this sphere is given by
These two radii are related by. Continuing the above example, we have and, which gives,, and.