Department of Mathematics and Statistics

Summer Project Descriptions


Sarah Gourlie, Geometric patterns in complex power series, Summer 04
Faculty Advisor: Dan Kalman

One way to visualize the convergence of a complex power series is to graph the successive terms vector fashion in the complex plane.  So, for the series
   a0  + a
1 + a2 + ...
draw a
0 as a vector starting at the origin, then draw a1 as a vector starting from the end of a0, and so on.  When this is done for the power series 1 + z + z2 + ... , the terms all depend on the selected value of z.  This can be viewed on a computer screen, with z varying as a mouse is dragged across the complex plane.  The results are visually quite stunning.  See some samples below.

series figures
The idea of this research project is to identify and classify various types of geometric patterns that arise in this fashion, and to determine the values of z that produce them. 

Sarah developed computer activities to explore this topic using software called Mathwright.  She identified and classified several different types of patterns, and then determined which values of z correspond to which patterns. Continuing her research in a capstone project for the honors program, Sarah extended her investigation to other infinite series.  Sarah presented the results of her summer research at a regional meeting of the Mathematical Association of American, where she won an award for an outstanding student paper.