If anyone cares, the proof is not too hard. Clearly one cannot have faces
that have more sides than five, because there no room angularly: three
regular hexagons at a point already fill up 360 degrees and seven or more
fill up more than 360, which is impossible(note that the solid has to be
convex at the vertices because every finite extent solid has one convex
vertex --shrink a big sphere down until it touches first time--and a
regular solid has by definition all vertices congruent. So they are all
convex). Euler showed that no of faces-no of edges+ no of vertices has to
be 2(for any dissection of the sphere into polygonal figures, regular or
not).
So one just looks at the possiblities for numbers of faces, number
of faces meeting at each vertex, and number of edges in each face,
and mess about with arithmetic to see that the only possiblities
are the cube , octahedron, tetrahedron, dodecahedron, and icosahedron.
(eg tetrahedron has 4 faces which are triangles, each edge belongs to two
faces--in all cases!----so there are 4 x3/2 = 6 edges. Each edge has 2
vertices and since three faces meet at each vertex, the number of vertices
has to be 6x2 / 3= 4 . And indeed 4-6 +4 = 2 as requires).
One cute part of all this is to see that the dodecahedron exists! It
exists "combinatorially--that is the numbers work in the above, but
it is not so clear that it exists in reality, that such a figure can
be realized. I leave that to you to think about.(Suggestion: Think
about the fact that when you fit three pentagons together at a
vertex , the result is rigid--no flexing is possible).
The icosahedron exits if the dodecahedron does (and it does!) since it can
be obtained
by using the centers of the faces of the dodecahedron as vertices. All the
others have an obvious existence.
Robert
On Thu, 11 Jul 2013, Michael Chapman wrote:
Robert Greene wrote :
...
If you need more points, then
there is no "canonical" choice(and no one is going to "discover"
any more Platonic solids--there aren't any more!).
...
Sorry to start that one ... it was basically a joke (I say basically as
like perpetual motion machines I had the impression that this was a field
that had (too many) claims;-)>
(Where "too many" >= 1.)
Martin Gardner had a proof (of "no more") that was very elegant, very
short and in normal prose ... its only negative feature is that it was
(for me, at least) highly unmemorable ...
Happy etiolating,
Michael
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