Yes. There are already many explanations for those five platonic solids and many other semiregular polyhedra such as Archimedean and prisms and antiprisms. My question is why they all exists.For example, if we want to find a regular polyhedra with faces consist of equilateral triangles, we may first specify the number of triangles at each vertex, say 3, 4, 5 and then by means of many methods e.g. Euler characteristics and double counting, can find the total number of faces and so on. And analyzing the angles between faces, one can prove that those cases are all feasible. Now there arose a question. Why that all the cases are feasible? E.g. it may be the case that tetrahedron and octahedron exist but icocahedron, during actual construction we may met a kink at the last stage so that it may not be able to be formed. However, this kind of situation does not happen.For every combination of expected combinations of regular polygons, we have almost always a corresponding really existing semiregular polyhedra. Can we give a explanation for existance of those combinations without invastigating individual cases?At my opinion, this is not trivial. For example, suppose that we want a polygon with faces given non-regular triangle with prescribed rules at each vertex how they are linked. Can we expect resulting solid almost surely exists? At my opinion, for this case, the answer would be 'no'.
Semiregular polyhedrons are defined as polyhedrons having all faces regular polygons and all corners identical. The semiregular polyhedrons are often call the Achemedean solids of which there are exactly 13.
