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.

>

>

>

> quasi

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.