## Girls’ Angle Bulletin, Volume 11, Number 1

We begin the second decade of the Bulletin with a fascinating image by Arnaud Chéritat, CNRS/Institut de Mathématiques de Toulouse, called Two mating polynomial Julia sets. For more about these images, visit his website. The two images by him (on the cover and inside) were included at the urging of Sarah Koch, this issue’s interview subject. Prof. Koch is Associate Professor of Mathematics at the University of Michigan. She studies complex analysis, Teichmüller theory, and complex dynamics. In her interview, she describes a game you can play called the “chaos game”. If you play the chaos game long enough, you will create fractals. We include some images of such fractals right after the interview.

Recently, Girls’ Angle member π has been learning about integrals and decided to set herself the task of computing the center of mass of a semicircle of uniform mass density. One question led to another, and before we knew it, we had stumbled upon Euler’s formula

$\displaystyle \frac{\sin x}{x} = \prod_{k=1}^{\infty} \cos(x/2^k)$.

We retrace our journey in Pac-Man Meets Euler.

In Volume 10, Number 3 of this Bulletin, Addie Summer explained how she found the quadratic formula. As it turns out, Lightning Factorial also figured out the quadratic formula without having to be taught it. She explains her method in The Quadratic Formula, Revisited.

In Anna’s Math Journal, Anna succeeds in finding a way to show that the number of special tilings of a rectangle are counted by the Catalan numbers without using generating functions. This represents the culmination of 6 installments’ investigation.

Next comes a special Math Buffet. We asked a number of mathematicians to contribute an excerpt from their scratch work to give us a window on what it looks like when they are in the act of creating mathematics. A huge and heartfelt Thank You to Timothy Chow, Brendan Creutz, Danijela Damjanović, Laura DeMarco, Ellen Eischen, Elisenda Grigsby, Kathryn Mann, Elizabeth Meckes, Maria Monks, Radmila Sazdanović, Marjorie Senechal, Bianca Viray, Fan Wei, Kirsten Wickelgren, Lauren Williams, and Helen Wong for allowing us a look into their personal process of doing math. Special thanks to Ashley Wang for doing the layout.

Emily and Jasmine are getting very close to resolving their long-standing search for nice triangles. In this installment, they succeed in computing the constant terms of all the minimum polynomials of cosines of rational multiples of π. By Vieta’s formulas, this is equivalent to computing

$\displaystyle \prod_{(k,n)=1, 0 < k \le n/2} \cos(2\pi\frac{k}{n})$

for all $n > 1$.

We conclude with some notes from the club.

We hope you enjoy it!

Finally, a reminder: when you subscribe to the Girls’ Angle Bulletin, you’re not just getting a subscription to a magazine. You are also gaining access to the Girls’ Angle mentors.  We urge all subscribers and members to write us with your math questions or anything else in the Bulletin or having to do with mathematics in general. We will respond. We want you to get active and do mathematics. Parts of the Bulletin are written to induce you to wonder and respond with more questions. Don’t let those questions fade away and become forgotten. Send them to us!

We continue to encourage people to subscribe to our print version, so we have removed some content from the electronic version.  Subscriptions are a great way to support Girls’ Angle while getting something concrete back in return.  We hope you subscribe!