They are 13 convex semiregular polyhedrons, whose faces are either two or three different types of regular polygons (triangles, squares, pentagons, hexagons, octagons or decagons), unlike the Platonic solids which have them all of the same type.

All the edges, of each one of these polyhedrons, are equal, because all the types of the regular polygons it consists of, have equal edges. Also, at every vertice of the polyhedron, all multi hedral angles are equal. Every one of these polyhedrons is inscribable in a sphere, because all vertices have equal distance from its center. Similarly, all of the edges of polygon, have equal distance from the polyhedron’s center. As to the faces of the polyhedron, all of the same type of polygon, have equal distance from the polyhedron’s center.

The Archimedes’ solids are 13 and have multiple types of regular polygons for faces. All of them can come of the Platonic solids, through proper transformations, like severance of the

vertices or the edges. They were discovered by Archimides, who dealt with them in his lost treatise “On 13 Semiregular Polyhedrons” and are the following: a) truncated tetrahedron with 8 faces (4 triangles and 4 hexagons), 12 vertices and 18 edges. b) cuboctahedron with 14 faces (8 triangles and 6 squares), 12 vertices and 24 edges.c) truncated cube with 14 faces (8 triangles and 6 octagons), 24 vertices and 36 edges. d) truncated octahedron with 14 faces (6 squares and 8 octagons), 24 vertices and 36 edges. e) rhombicuboctahedron with 26 faces (8 triangles and 18 squares), 24 vertices and 48 edges.f) truncated cuboctahedron with 26 faces (12 squares, 8 hexagons and 6 octagons), 48 vertices and 72 edges. g) snub cube with 38 faces (32 triangles and 6 squares), 24 vertices and 60 edges. h) cosidodecahedron with 32 faces (20 triangles and 12 pentagons), 30 vertices and 60 edges. i) truncated dodecahedron with 32 faces (20 triangles and 12 pentagons), 60 vertices and 90 edges. j) truncated icosahedron with 32 faces (12 pentagons and 20 hexagons), 60 vertices and 90 edges.k) rhombicosidodecahedron with 62 faces (20 triangles, 30 squares and 12 pentagons), 60 vertices and 120 edges. l) truncated icosidodecahedron with 62 faces (30 squares, 20 hexagons and 12 decagons), 120 vertices and 180 edges. m) snub dodecahedron with 92 faces (80 triangles, 12 pentagons), 60 vertices and 150 edges.