About 10 yeras ago I had a long conversation with Daniel
Allcock about proving the existence of the icosahedron and/or
dodecahedron "by continuity" -- in particular, without any
calculation. Yesterday I was talking to Igor Pak and mentioned
the proof of the existence of the icosahedron that we'd come
up with, and Igor was hoping to find a reference to it in print, so
as to give proper credit. (I'm sure the idea long predates my and
Daniel's conversation.)
Does anyone know the pedigree of the following line of reasoning?
I haven't defined "by continuity," but I think the intent will be clear
from the proof.
1) In 2d, all the regular n-gons certainly exist (by continuity).
Take a circle and a chord thereof, and repeat the chord around
the circle, end-to-end, n times. By adjusting the length of the chord,
you can make the nth copy's endpoint hit the starting point, and you
get a regular n-gon. [In particular, by this construction, all edges
are
identical and all vertices lie on a sphere.]
2) Make a pentagonal antiprism. That is, make a polyhedron
whose top and bottom faces are pentagons lying in parallel
planes such that the line joining their centers is perpendicular
to those planes. If the pentagons are rotated relative to one
another, we need ten triangles to close this polyhedron around
the equator. Rotating one pentagon, we can make the triangles
isosceles (by continuity); then by adjusting the distance between
the two planes, we can make them equilateral (by continuity).
3) On each of the pentagonal faces of the antiprism, erect a
five-sided pyramid whose apex is over the center of the pentagon.
Adjusting the altitude, you can make those five triangles equilateral
as well (by continuity).
There you have it: the polyhedron so constructed clearly has
twenty equilateral triangular faces meeting five at each vertex.
(Mind you, this proof is still not entirely satisfying: it only endows
the figure with dihedral, not icosahedral, symmetry. And indeed
if you start with a 3-gon or 4-gon instead of a 5-gon, everything
still works, but you don't magically get extra symmetry at the end.)
Can anyone point to an origin for this?
--Michael Kleber
kleber(a)brandeis.edu
