Spirals in Planes and Space
Aaron Cinzori
Associate Professor and Chair
Department of Mathematics
Hope College
We'll explore an algorithm that takes $n$ points in
$\mathbb{R}^2$ or $\mathbb{R}^3$ and produces a piecewise-linear
spiral that uses the given points as its initial nodes. We generate
further points in the spiral by repeatedly taking a convex
combination of $m \le n$ (existing) points at a time. In particular,
let $P_0,\ldots,P_{n-1}$ be the initial points, and let $0\le t_1,
t_2, \ldots, t_m \le 1$ be fixed parameters with
$t_1+t_2+\cdots+t_m=1$. Produce more points by using the formula
$P_{k+n} = t_1P_k + t_2P_{k+1}+ \cdots + t_mP_{k+m-1}$ for each
$k\ge 0$.
We can then ask a lot of questions: Where does the spiral end up?,
How long is it? When and how can we arrange things so that the
segment lengths are a geometric series? What is the general
behavior of the spiral as it approaches its limit? The tools we'll
use will come from linear algebra, complex analysis, infinite
series, and linear recurrences. We'll also talk a bit about how this
problem evolved from a Problem of the Week to several REU projects
and papers (including one in the Spring 2010 $\Pi$ME Journal).