Proof of the existance of Platonic Solids other than in Euclids 13 Elements

Question

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.

If I write down the coordinates of the vertices and check that each face is planar, and that they're all congruent and regular, is that enough? If not, why not? — John Hughes, Oct 22 '16 at 15:24
It's pretty easy to check using the distance formula in Euclidean $3$-space that the twelve points $$(0, \pm 1, \pm\gamma),\quad(\pm 1, \pm\gamma, 0),\quad( \pm\gamma, 0, \pm 1)$$lying on three golden rectangles form twenty equilateral triangles. Is that the type of argument you have in mind? — Andrew D. Hwang, Oct 22 '16 at 15:24
Related: Cleverest construction of a dodecahedron / icosahedron? — Andrew D. Hwang, Oct 22 '16 at 15:41
Thank you for the link. So often one finds constructions just say you end up with the required solid but don't actually prove it. It does look promising though. — Kevin Bowman, Oct 22 '16 at 16:36

score 2 · Answer 1 · answered Oct 22 '16 at 15:26

2

H.S.M. Coxeter, Introduction to Geometry. Chapter 10: The Five Platonic Solids. For you, 10.4: Radii and Angles

answered Oct 22 '16 at 15:26

GEdgar

117,296

Thanks for this reference. However you will notice that when Coxeter says in section 10.4 that (essentially) the existence was proved in section 10.1 you see that it is a heuristic argument and not a rigorous proof. – Kevin Bowman Oct 22 '16 at 16:12

Proof of the existance of Platonic Solids other than in Euclids 13 Elements

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