five regular polyhedron is a direct-nonexistence proof; #180; 2nd ed; Correcting Math
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From: Archimedes Plutonium on 4 Oct 2009 01:53 I am surprized Euclid proved there were 5 and only 5 regular polyhedron. He must have been a busy man back then. The Wikipedia on regular polyhedron gives what Euclid performed for a proof and it appears to be direct-nonexistence as well as this topological proof: --- quoting Wikipedia --- A purely topological proof can be made using only combinatorial information about the solids. The key is Euler's observation that V - E + F = 2, and the fact that pF = 2E = qV. Combining these equations one obtains the equation 2E/q -E + 2E/p = 2 Simple algebraic manipulation then gives 1/q + 1/p = 1/2 + 1/E. Since E is strictly positive we must have 1/q + 1/p > 1/2. Using the fact that p and q must both be at least 3, one can easily see that there are only five possibilities for {p, q}: {3,3}, {4,3}, {3,4}, {5,3}, {3,5}. --- end quoting Wikipedia --- Archimedes Plutonium www.iw.net/~a_plutonium whole entire Universe is just one big atom where dots of the electron-dot-cloud are galaxies |