```Date: Dec 23, 2012 6:20 AM
Author: quasi
Subject: Re: convex polyhedra with all faces regular

achille wrote:>quasi wrote:>>>> Prove or disprove:>> >> For each positive integer n, there are only finitely many >> convex polyhedra, up to similarity, such that all faces are  >> regular polygons (not necessarily of the same type) with at  >> most n edges.>>Yes, it is finite.>>It is known that the strictly convex regular-faced polyhedra>comprises >>     2 infinite families (the prisms and antiprisms)>     5 Platonic solids,>    13 Archimedian solids>and 92 Johnson solids>  >Let N(n) be the number of convex polyhedra with regular polygons>up to n sides as faces. One has:>>   N(n) <= 2n+104>>Actually, it is pretty simple to prove N(n) < oo directly.>WOLOG, let us fix the sides of the regular polygons to has >length 1.>>Let's pick any convex polyhedron and one of its vertex v. >Let say's v is connected to k edges >e_0, e_1, e_2, ... e_k = e_0 >and a_i ( i = 1..k ) is the angle between e_(i-1) and e_i.>For this v, let >>     A(v) := 2 pi - sum_{i=1..k} a_i>>Being a convex polyhedron, we have A(v) > 0. It is also easy>to see if we sum over all vertices of the convex polyhedron, >we get:> >   sum_v A(v) = 4 pi >>If one build a convex polyhedron using regular polygons up to>n sides, it is easy to see 3 <= k <= 5 and there are only>finitely many possible choices of a_i:>>     (1 - 2/3) pi, (1 - 2/4) pi, ... ( 1 - 2/n) pi>>This mean there are finitely many possible choices of >a_1,.., a_k which satisfy:>>(*)     2 pi - sum_{i=1..k} a_i > 0 >>Let M(n) be the smallest possible value of L.H.S of (*) for >given n. On any vertex v of any convex polyhedron build from >regular polygons up to n sides, A(v) >= M(n) and hence the >convex polyhedron has at most 4 pi / M(n) vertices.>>Since the number of vertices is bounded, there are finitely >many ways to connect them to build a polyhedron. Using Cauchy >theorem of convex polytopes, each way of connecting the >vertices to from a polyhedron corresponds to at most 1 convex>polyhedron in Euclidean space. (since the length of all edges >has been fixed to 1).>>As a result, there are only finitely many convex polyhedra one>can build using regular polygons up to n sides.When I get a chance, I'll try to digest the details, but at first glance, it looks very good. Thanks.quasi
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