Albion College Mathematics and Computer Science Colloquium

Title: Spirals in Planes and Space

Speaker: Aaron Cinzori
Associate Professor and Chair
Department of Mathematics
Hope College
Holland, Michigan

Abstract: 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).

Location: Palenske 227

Date: 10/28/2010

Time: 3:10

@abstract{MCS:Colloquium:AaronCinzori:2010:10:28, author = "{Aaron Cinzori}", title = "{Spirals in Planes and Space}", address = "{Albion College Mathematics and Computer Science Colloquium}", month = "{28 October}", year = "{2010}" }

Title:	Spirals in Planes and Space
Speaker:	Aaron Cinzori Associate Professor and Chair Department of Mathematics Hope College Holland, Michigan
Abstract:	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).
Location:	Palenske 227
Date:	10/28/2010
Time:	3:10