```Date: Dec 23, 2012 4:58 AM
Author: achille
Subject: Re: convex polyhedra with all faces regular

On Thursday, December 6, 2012 12:28:40 PM UTC+8, 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.> > > > quasiYes, it is finite.It is known that the strictly convex regular-faced polyhedracomprises      2 infinite families (the prisms and antiprisms)     5 Platonic solids,    13 Archimedian solidsand 92 Johnson solids  Let N(n) be the number of convex polyhedra with regular polygonsup to n sides as faces. One has:   N(n) <= 2n+104Actually, 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_iBeing a convex polyhedron, we have A(v) > 0. It is also easyto 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 ton sides, it is easy to see 3 <= k <= 5 and there are onlyfinitely many possible choices of a_i:     (1 - 2/3) pi, (1 - 2/4) pi, ... ( 1 - 2/n) piThis 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 polygonsup to n sides, A(v) >= M(n) and hence the convex polyhedron has atmost 4 pi / M(n) vertices.Since the number of vertices is bounded, there are finitely many waysto connect them to build a polyhedron. Using Cauchy theorem of convexpolytopes, each way of connecting the vertices to from a polyhedron corresponds to at most 1 convex polyhedron in Euclidean space. (sincethe length of all edges has been fixed to 1).As a result, there are only finitely many convex polyhedra one can buildusing regular polygons up to n sides.
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