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February 5th, 2011, 12:37 AM  #1 
Member Joined: Jun 2009 Posts: 83 Thanks: 1  Platonic solids  existence proof
Platonic solids are explained here: http://en.wikipedia.org/wiki/Platonic_solid It can be eassily deduced what the neccessary conditions for the number of vertices, edges and faces are and that only 5 such polyhedrons can exist. But how to prove that such polyhedrons do really exist? I am interested in some "logical" proof. Of course it can be proved geometrically by constructing such polyhedrons but is there another way of proving this without constructing these polyhedrons? Thank you 
February 5th, 2011, 02:15 PM  #2 
Global Moderator Joined: Nov 2006 From: UTC 5 Posts: 16,046 Thanks: 938 Math Focus: Number theory, computational mathematics, combinatorics, FOM, symbolic logic, TCS, algorithms  Re: Platonic solids  existence proof
I look at it as calculating where is the measure (in radians) of the external angle of an ssided regular polygon. (You could express it in degrees if you like.) If you wanted to include tilings ("solids" of infinite volume and zero curvature) you'd compute instead which I think would give 7 instead of 5. 

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