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