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        riddles >> medium >> Regular polyhedrons
        
(Message started by: Ghost Sniper on Nov 11^th, 2007, 12:34pm)

Title: Regular polyhedrons
Post by Ghost Sniper on Nov 11^th, 2007, 12:34pm

Prove, mathematically, that there exist only 5 regular polyhedrons.

Title: Re: Regular polyhedrons
Post by towr on Nov 11^th, 2007, 12:54pm

We know 5, and there can't be more; therefore there are only 5 :P

Well ok:
[hide]There can't be more, because we can only use regular triangles, squares, and pentagons. You always need at least 3 polygons to meet at a vertex, and you can't have more than 5 triangles, 3 squares or 3 pentagons meet at a vertex. So at most 3+1+1=5.[/hide]

Title: Re: Regular polyhedrons
Post by ecoist on Nov 12^th, 2007, 9:41am

A fun one is determing all convex deltahedra! A convex deltahedron is a convex polyhedron, all of whose faces are equilateral triangles. Can you imagine what my favorite, with 14 faces, looks like?

Title: Re: Regular polyhedrons
Post by towr on Nov 12^th, 2007, 10:09am

on 11/12/07 at 09:41:32, ecoist wrote:

A fun one is determing all convex deltahedra! A convex deltahedron is a convex polyhedron, all of whose faces are equilateral triangles.

Aren't there infinitely many? Or is it disallowed for (adjacent) triangles to lie in the same plane?

Title: Re: Regular polyhedrons
Post by Grimbal on Nov 13^th, 2007, 12:43am

on 11/12/07 at 09:41:32, ecoist wrote:

Can you imagine what my favorite, with 14 faces, looks like?

If it has 14 faces, there is only one, the triaugmented triangular prism.