The Sublime Perfection and Beauty of Imperfection
Kurt Gödel and the loneliness and joy of being a Platonist in a smug world convinced of Human Omnipotence and that Man is the Measure of all things …
Proving Impossibility Through The Universe of Possibility
How do you prove that something is impossible? Often it’s not as hard as it sounds, and in fact most proofs of impossibility in mathematics result from studies of the universe of the possible: we characterize all possibilities, then we check whether some proposed attibute or procedure lies within those possibilities by testing whether it fulfils our characterization.
Probably the simplest proof of this kind is the impossibility of constructing a Platonic Solid (a convex polyhedron in three dimensional Euclidean space such that all the faces are the same polygon) with, say fifty sides, or a thousand sides: using the Euler Characteristic that must equal 2 for such a solid (a convex polyhedron can be thought of as networks on a sphere), and adding a linear equation relating a unique number of edges to the number of regular faces to express regularity, we can write down a Diophantine equation characterizing all possible such solids. It’s then a simple matter of algebra to…
