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Topic: convex polyhedra with all faces regular
Replies: 7   Last Post: Dec 23, 2012 6:20 AM

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quasi

Posts: 10,225
Registered: 7/15/05
Re: convex polyhedra with all faces regular
Posted: Dec 23, 2012 6:20 AM
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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



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