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In three-dimensional space, a Platonic solid is a regular, convex polyhedron. It is constructed by congruent (identical in shape and size) regular (all angles equal and all sides equal) polygonal faces with the same number of faces meeting at each vertex. Five solids meet those criteria:
Tetrahedron Four faces
Cube Six faces
Octahedron Eight faces
Dodecahedron Twelve faces
Icosahedron Twenty faces
Geometers have studied the mathematical beauty and symmetry of the Platonic solids for thousands of years. They are named for the ancient Greek philosopher Plato who hypothesized in his dialogue, the Timaeus, that the classical elements were made of these regular solids
In geometry, an Archimedean solid is one of the 13 solids first enumerated by Archimedes. They are the semi-regular convex polyhedra composed of regular polygons meeting in identical vertices, excluding the 5 Platonic solids (which are composed of only one type of polygon) and excluding the prisms and antiprisms. They differ from the Johnson solids, whose regular polygonal faces do not meet in identical vertices.
2 cuboctahedron(rhombitetratetrahedron)
4 truncated octahedron(truncated tetratetrahedron)
5rhombicuboctahedron(smallrhombicuboctahedron)
6truncatedcuboctahedron(greatrhombicuoctahedron)
7 snub cube(snub cuboctahedron) 8icosidodecahedron
10 truncated icosahedron c60 Bucky ball
c60 domes gmtrx c60 fractal gmtrx c60 y joint gmtrx 260mm concrete c60 gmtrx 392mm concrete c60 gmtrx
11rhombicosidodecahedron(small rhombicosidodecahedron)
12truncatedicosidodecahedron(great rhombicosidodecahedron)
13snub dodecahedron(snub icosidodecahedron)
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. 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.
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