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

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.