”platonic solids are solids where all the sides are equal and all the faces are the same regular polygon” is true, but the reverse isn’t.
I think you also want all vertices to lie on a single sphere.
Without that, you can have various stellated polyhedra, constructions of 6 or 10 equilateral triangles, an infinite number of constructs that only contain squares, and probably others.
I also think “all the faces are the same regular polygon” implies “all the sides are equal” for the ‘normal’ definitions of “polyhedron”, but that definition is surprisingly hard. One typically wants to exclude faces crossing each other and constricts that aren’t connected, such as “two cubes”, so who knows?
I think you also want all vertices to lie on a single sphere.
Without that, you can have various stellated polyhedra, constructions of 6 or 10 equilateral triangles, an infinite number of constructs that only contain squares, and probably others.
I also think “all the faces are the same regular polygon” implies “all the sides are equal” for the ‘normal’ definitions of “polyhedron”, but that definition is surprisingly hard. One typically wants to exclude faces crossing each other and constricts that aren’t connected, such as “two cubes”, so who knows?