Mathematics

Modular multiplication circlesSpring 2026
with Prof. Allison Pacelli

Place $n$ evenly spaced points around a circle, number them $0$ to $n-1$, and draw a chord from each point $a$ to the point $ka \bmod n$. Call the result $G(n, k)$. With $n = 100$ and $k = 2$, the straight chords outline a cardioid; that’s the mark at the top of every page on this site. People have known about pictures like this since at least the nineteenth century, and they’re popular in recreational mathematics, but surprisingly little about them had been carefully proved.

Last spring a classmate and I worked with Prof. Allison Pacelli at Williams College to explain what these pictures show. Some of what we proved:

• The chords sit at only $\lfloor n/2g \rfloor + 1$ different distances from the center, where $g = \gcd(n, k-1)$. When $g$ is large, many chords touch the same few circles, and you see concentric rings.$G(100, 21)$: three rings, as the distance formula predicts.
• The grid-like pictures come from the chords pointing in only a few directions: at most $n/h$ of them, where $h = \gcd(n, k+1)$.$G(60, 29)$: every chord points one of two ways.
• As $n$ grows, $G(n, k)$ traces out an epicycloid, the curve drawn by a point on a circle rolling around another circle, with $k-1$ cusps. When $k$ is congruent to a fraction $\ell/m$ modulo $n$, it traces two epicycloids at once.$G(200, 3)$: the epicycloid with two cusps.

We also showed that some constructions that look different, like chords reflecting around the inside of a circle (as explored by the mathematician Simon Plouffe), produce the same pictures as $G(n, k)$.

We presented our results at the Hudson River Undergraduate Mathematics Conference on April 25, 2026, and have drafted a paper.

Teaching

This fall I’m the only teaching assistant for Topology, which has 19 students. I hold help hours and grade problem sets, about ten hours a week.

Last fall I was a TA for Abstract Algebra: weekly office hours and grading proofs for 25 students. Over the semester I kept improving an Excel workbook that anonymized the grading, calculated scores, and generated feedback for each student.

How I got here

Before college I took Art of Problem Solving courses and then, all through high school, university-level courses through Stanford Pre-Collegiate Studies, from multivariable calculus through real and complex analysis and differential equations. At Williams I’ve mostly followed algebra and number theory, through commutative algebra, algebraic number theory, and Galois theory, along with topology.

Lately I’ve been turning toward probability and data. I passed the first actuarial exam, on probability, in July 2026. This fall I’m taking statistics, and next semester I hope to take regression theory. I’m also taking a tutorial on harmonic analysis through music.

Outside of classes, I regularly participate in the math department’s weekly teas, book club, and talks.