Polygon Iterations » |

Submitted by the Institute for Computational and Experimental Research in Mathematics (ICERM)

In the Fall of 2013, ICERM had a semester program on experiental geometry, topology, and dynamics. One of the many projects undertaken during this semester was the study of polygon iterations - geometrically defined maps on the space of polygons.

Perhaps the simplest polygon iteration is the *midpoint map*. Starting
with an n-gon *P*_{1}, we let *P*_{2} be the
n-gon whose
vertices lie at the centers of
the edges of *P*_{1}. The construction can be done over and over again, producing
a sequence {*P*_{n}} of polygons which shrink to a point.
If these polygons are rescaled (and suitably rotated) so as to have unit diameter, then, for almost every choice, they
converge to an affinely regular n-gon.
That is, the limit has the form *T*(*R*_{n}),
where *T* is an affine transformation and *R*_{n} is a regular polygon.

The midpoint map is very close to the heat equation, in that the map
averages out the coordinates of the polygon. The proof of the convergence
properties of the midpoint map leads quickly into a discussion of finite Fourier
series. Thought of as a map on * C^{n}*, the midpoint map is a linear map whose
eigenvalues have the form

Almost every point in * C^{n}* is attracted to a linear combination of

Figure 2 shows a more exotic example. One starts with the blue polygon
*P*_{1} (in this case a pentagon) and then constructs the new yellow polygon
*P*_{2}. The rest of the picture is just scaffolding for the construction. Like the
midpoint map, the vertices of *P*_{2} lie on the edges of *P*_{1}. This map is even
more symmetric than the midpoint map. If you pick up the paper and look
at the picture in perspective, the midpoint map will not appear to be picking
out the midpoints. However, the new map does the same thing from every
perspective.

Figure 2:

We call this map the projective heat map, because it shares some features
of the midpoint map, but is natural with respect to the group of projective
transformations of the projective plane. The projective heat map induces
a map on the space *P _{n}* of projective equivalence classes of (not necessarily
convex) n-gons in the projective plane. Unlike the midpoint map, the projective heat map
is nonlinear; in general it is some rational map in 2n − 8
variables. The space

We were able to prove this result when n = 5.
When n = 5, we get a
rational map of the plane. We showed that the set *J*_{5} of pentagon classes
which do not converge to the origin has area 0. The set *J*_{5} has an intricate
fractal structure. Figure 3 shows a close-up of part of *J*_{5}.

Figure 3: (Part of) the set

The method of proof is to partition the plane into small polygons on which the action of the map is fairly easy to understand. The action of the map on these small polygons is then estimated by computer-aided manipulation of huge integer-valued polynomials.