On Friday, November 6, 2009, Professor Andrew F. Rich delivered the Annual Pi Mu Epsilon Lecture - *The Amazing "Hilda Function"* at 4:00 pm in Avery 115.

## The Amazing "Hilda Function"

### Abstract

Count the number of base 2 representations of a natural number, allowing digits 0, 1 and 2. This sequence has the surprising property that every positive rational occurs exactly once as a ratio of consecutive terms. The bijection between integers and rationals is explained using the Euclidean algorithm and continued fraction expansions. Flipping the bijection "across the decimal point" leads to an amazing continuous function with wonderful set-theoretic, analytic and fractal properties, first discovered by Minkowski.

### About the Speaker

Andrew Rich received his undergraduate degree at Bethel College, a small Mennonite college in Kansas, and was on a top ten Putnam team. He went to the University of Chicago with a National Science Foundation Fellowship, studied differential geometry, dropped out and went to Indonesia, where he taught at a Christian university. He returned to the States and attended graduate school. He received a Ph.D. for work on differential operators on foliated manifolds. Since 1992, he has been teaching the full range of undergraduate mathematics at Manchester College in North Manchester, Indiana. He has also spent time or taught at the University of Illinois at Chicago, the University of Colorado, Kansas State University, Handong University in South Korea and St. Andrews University in Scotland. In addition to other duties at Manchester, he coaches the college math team and supervises junior high and high school math contests. He enjoys being a husband and father, reading, traveling, running, science, working with students and, of course, thinking about interesting mathematics problems.