quasi
Posts:
12,049
Registered:
7/15/05


Re: convex polyhedra with all faces regular
Posted:
Dec 23, 2012 6:20 AM


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 regularfaced 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_(i1) 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

