Albion College
Mathematics and Computer Science
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).
All are welcome!
Palenske 227
October 28, 2010