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February 5th, 2011, 12:37 AM   #1
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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
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February 5th, 2011, 02:15 PM   #2
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Re: Platonic solids - existence proof

I look at it as calculating

where is the measure (in radians) of the external angle of an s-sided 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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