On Archimedean solids: three-dimensional geometric objects with regular polygons as faces
Archimedes was an ancient Greek mathematician, physicist, and engineer and in all these fields, one of the most significant figures in world history. One of his works, now lost, though its content has been preserved, dealt with polyhedra that have regular polygons as faces and identical vertices, yet are neither Platonic solids, nor prisms, nor antiprisms. There are 13 such solids in total. Archimedes likely described all of them, and they were subsequently named after him.
Archimedean solids are generally categorized by their number of faces, ranging from the truncated tetrahedron with eight faces to the snub dodecahedron with 92 faces. Like the snub cube, the snub dodecahedron exists in two distinct, mirror-image variants. The calculations for the dimensions of both variants are identical. Sometimes these variants are counted as separate Archimedean solids, bringing the total count to 15. Another borderline case is the pseudo-rhombicuboctahedron. Its vertices are locally, but not globally identical, and its dimensions match those of the rhombicuboctahedron. It is usually classified not as an Archimedean solid, but as a Johnson solid.
Archimedean solids occasionally appear in approximate forms within molecules and crystals, and they play a role in architecture and design. Generally, their frequency in these fields decreases as the number of faces increases. A notable exception is the ninth form according to standard counting, the truncated icosahedron, which is sometimes referred to as football solid. This name alludes to a characteristic associated with an all around increasing number of faces, edges, and vertices. According solids increasingly resemble a sphere. However, while regular polygons can approximate a circle to any desired degree of precision, the limited number of Archimedean solids prevents a similar level of approximation to a sphere.
To every Archimedean solid, there is a dual Catalan solid.