achille
Posts:
575
Registered:
2/10/09


Re: convex polyhedra with all faces regular
Posted:
Dec 23, 2012 4:58 AM


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 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.

