##### Oct 2-4, 2023

Los Altos High School

# Simon Rubinstein-Salzedo, PhD

Mathematician

###### Stanford Euler Circle

#### Constructible Polygons and Related Questions

#### Math

The ancient Greeks were the first to consider the problem of constructions using compass and unmarked straightedge. Among many other things, they knew how to construct equilateral triangles, squares, regular pentagons, and regular hexagons using compass and straightedge. It wasn't until the late 18th century when further progress was made, and Gauss and Wantzel found an exact criterion describing the regular polygons that can be constructed using compass and straightedge; in particular, regular 17-gons, 257-gons, and 65537-gons can all be constructed with compass and straightedge, whereas a regular heptagon cannot.

One way of generalizing this result is to realize that constructing a regular n-gon is the same thing as dividing a circle into n arcs of equal length. It is then natural to ask when we can divide other closed curves into n pieces of equal length using a compass and straightedge. With my student Nitya Mani, who was a high-school student at the time, we solved this problem in the case of a family of curves called hypocycloids; these are essentially the shapes formed when playing with spirograph toys.

#### Meet the Speaker:

Simon Rubinstein-Salzedo received his PhD at Stanford University in 2012 under the direction of Akshay Venkatesh in algebraic number theory. After completing his PhD, he did postdocs at Dartmouth College and Stanford University, and then founded Euler Circle, where he teaches college-level mathematics classes to advanced high-school students. He has done research in many areas of mathematics, including number theory, algebraic geometry, combinatorics, probability, game theory, and complex analysis.

In addition to his teaching at universities and at Euler Circle, Simon has been teaching mathematics to advanced middle-school and high-school students for over a decade and is extremely popular among his students. He is currently the lecturer for Program II at the Stanford University Mathematics Camp (SUMaC), where he teaches algebraic topology. He has also worked at The Art of Problem Solving and has taught at many math events and run many math circles in the Bay Area. His greatest claim to fame in life is probably having a factoring trick named after him.