I. THE PROJECTIVE HEAT MAP
I'll describe a polygon iteration which starts with
a polygon and does a straight line construction to produce
a new one with the same number of sides. The construction
is especially natural when it acts on the 2 dimensional
space of pentagons modulo projective equivalence. I'll
explain my theorem that the map converges almost every
(class of) pentagon to the projectively regular class, except for
a connected "Julia set" of planar measure 0.
The highlight of my result is something like
a coarse topological model for the Julia set. The
talk will have a lot of computer demos in it, and
it will start from scratch.
II. THE PLAID MODEL
I will introduce a construction which produces
embedded lattice polygons in the plane. The model
is highly structured and has both a combinatorial
and number-theoretic feel to it. I call the model
the plaid model because one of its descriptions
is in terms of grids of parallel lines. The plaid
model exhibits a hierarchical multi-scale kind of
structure. It is closely related to outer billiards
on kites, and also seems similar in spirit to Pat
Hooper's Truchet tile system. Mainly I will show
off the model with computer demos and explain what
theory about it I have worked out so far.