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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.

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    $\begingroup$ 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? $\endgroup$ – John Hughes Oct 22 '16 at 15:24
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    $\begingroup$ 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? $\endgroup$ – Andrew D. Hwang Oct 22 '16 at 15:24
  • $\begingroup$ Yes I think that does it $\endgroup$ – Kevin Bowman Oct 22 '16 at 15:34
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    $\begingroup$ Related: Cleverest construction of a dodecahedron / icosahedron? $\endgroup$ – Andrew D. Hwang Oct 22 '16 at 15:41
  • $\begingroup$ 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. $\endgroup$ – Kevin Bowman Oct 22 '16 at 16:36
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H.S.M. Coxeter, Introduction to Geometry. Chapter 10: The Five Platonic Solids. For you, 10.4: Radii and Angles

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  • $\begingroup$ 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. $\endgroup$ – Kevin Bowman Oct 22 '16 at 16:12

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