Minimal definition of a Platonic Solid

Question

I searched in Google for the definition of Platonic solid, and I found:

A platonic solid is a polyhedron all of whose faces are congruent regular polygons, and where the same number of faces meet at every vertex

Why is "where the same number of faces meet at every vertex" needed? Is there an example of a polyhedron whose faces are congruent regular polygons where DIFFERENT number of faces meet at at least two vertices?

In the article in the Hebrew Wikipedia this is stated as an additional condition in the definition.

In the article in the English Wikipedia, the definition is listed as regular convex polyhedron, "regular" being defined in a separate article.

On the other hand, does this also imply that the same number of edges meet at every vertex? How difficult is it to prove it?

Answer 1

Join two regular tetrahedrons. The new polyhedron has two three-faced vertices and three four-faced vertices.

About the edges: the number of edges that join in a vertex is obviously the same that the number of faces because each edge is between two faces.

Answer 2

The most succinct definition is probably that a Platonic solid is a polyhedron which is both convex and transitive on its flags.

A figure is convex if a line may be drawn between any two interior points without intersecting the surface.

A flag is a connected set of a face, one of its edges, and one of that edge's vertices.

If a polyhedron is regular then under any of its symmetries, any flag may be transposed onto any other flag. We say that the flags are transitive under the symmetries of the polyhedron and, by extension, that the polyhedron is transitive on its flags.