Albion College

Mathematics and Computer Science

Mathematics and Computer Science

COLLOQUIUM

Spirals in Planes and Space

Aaron Cinzori

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).

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).