Does anyone know where there is a complete proof of the existence of the Platonic solids, particularly the Dodecahedron and the Icosahedron (other than amongst Euclids 13 elements)?
I do not mean a proof that a regular polyheron with each side a $p$-gon with $q$ meeting at each vertex must satisfy $\frac{1}{p}+\frac{1}{q} > 2$.
Nor a proof that the only solutions to the above are $\{ 3,3 \} , \{ 4,3 \}, \{ 3,4 \}, \{ 3,5 \}, \{ 5,3 \}$ (where $p$ is the first element listed in the set and $q$ is the second).
Nor a proof that if there exists a regular polyhedron satisfying one of the above then it is unique.
But rather a (mathematical) proof that there does exist a solution to (in particular) $\{ 5,3 \}$ and $\{ 3,5 \}$. Of course model building is not valid since you may be just making (essentially) a near miss Johnson solid. Another way to put the question is "... a proof that the platonic solids are not near miss Johnson solids ...".
Apologies if this has already been asked but I could not find it.
Thank you for your time and effort in considering this question.
H.S.M. Coxeter, Introduction to Geometry. Chapter 10: The Five Platonic Solids. For you, 10.4: Radii and Angles