I understand that there are $5$ possibilities of Platonic solids possible (see below) given the following values of $s$ and $m$ where $s$ denotes the number of sides at each face and $m$ denotes the number of faces at a given corner.
$\textbf{Question:}$ Why couldn't there be another construction for a given $s$ and $m$ other than those shapes depicted on the very far right of the table above?
I understand it is of a convention to define Platonic solids by the values of $s$ and $m$, but I was just curious why the shape couldn't look different given specified values for $s$ and $m$.
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Best Answer
Take $(s,m) = (3,3)$ as an example. A given corner is incident with 3 faces, each face has 3 sides. Because all angles and side lengths are unique, this gives 3 equilateral triangles, each pair sharing a side. If I give you 3 equilateral triangles all the same size, there is only one way to glue them together sharing a point. It only leaves one more place for another equilateral triangle, so you get a pyramid.
For $(s,m) = (3,4)$, when you glue together 4 equilateral triangles together at a point, with any two sharing an edge, there is only one way to do this (all angles in 3-space are forced), it will leave an empty square that needs to get filled in, if you take the next corner it will need two more equilateral triangles; if you actually try to do this, you will see that all of the angles between faces sharing a line are determined by the shapes you are combining (and how many are sharing a corner).
The same with all of the other examples. When you are gluing together regular polygons, all the same size, together at a point, there is only one way (at most) they will go together in 3-dimensional space. (In other words, everything is determined.)