## Girls’ Angle Bulletin, Volume 11, Number 2

You could call this issue an ode to the triangle because half the content pertains to that most basic of geometric shapes, including the cover.

However, we open with an interview with UCLA Professor of Mathematics Deanna Needell. One of Prof. Needell’s mathematical interests is about how to get accurate views of things with less information, such as getting useful MRI scans in less scan time.

Next up is a nifty way to introduce vectors via partition functions by a group from Williams College, led by Pamela E. Harris. They motivate the whole concept by imagining a fictitious martian currency which behaves just like our dollar, except that there’s more than one unit of currency and they cannot be interchanged. The setup inevitably leads them to begin computing what is known as Kostant’s partition function for $A_3$. In fact, the authors dedicate the article to the memory of Bertram Kostant who passed away earlier this year.

Inscribed equilateral hexagons in an isosceles right triangle.

Milena Harned and Miriam Rittenberg are back explaining some results they found and proved this past summer about inscribing equilateral hexagons in triangles in such a way that every side of the triangle is flush with at least one side of the hexagon. In fact, for every triangle, they found a continuous family of such equilateral hexagons and they “rotate” about inside the triangle. You can explore these inscribed equilateral hexagons using a JavaScript app on the Girls’ Angle website.

Girls’ Angle mentor Ashley Wang pays tribute to triangles through her Math Buffet.

After an installment of Meditate to the Math (which asks you to interpret Sierpinski’s triangle as a sequence of binary numbers and contemplate its patterns), we present a self-referential test created by four Girls’ Angle members: Ghost Inthehouse, HolAnnHerKat, Katnis Everdeen, and Shark Inthepool. Earlier this fall, they solved Jim Propp’s self-referential multiple choice test and enjoyed it so much, they wanted to create their own. Their work was primarily overseen by Girls’ Angle mentor Rachel Burns.

Emily and Jasmine’s epic search for nice triangles comes to a heady conclusion. They succeed in putting all the pieces of the puzzle together to get a fairly complete classification of triangles with various numbers of integer side lengths and angles that measure rational multiples of $\pi$. Their journey took them through some lattice combinatorics, rational parameterizations of curves, and some basic Galois theory and a study of the values of cyclotomic polynomials at the square root of -1. And it all began with a fake triangle.

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!

## Get ready for a brand new math adventure – SUMIT 2018!

SUMIT 2018 is a fun, fully collaborative, math adventure for girls who like math in grades 6-10. Registration is open now. We’re running the event twice on February 3 and 4 at the Broad Institute in Cambridge.

SUMIT 2018 is designed not only to be a super fun, mathematically-interesting challenge for participants, but also to give participants a great opportunity to meet other girls who like math, build lasting friendships, and develop leadership skills. We’re creating the stage, but it’s up to the participants to take control of their destiny!

## 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!

## Girls’ Angle Bulletin, Volume 10, Number 6

10 years of the Girls’ Angle Bulletin!

With Volume 10, Number 6, we’ve published 42 interviews with women in mathematics, dozens of Summer Fun problem sets, and some 1500 pages of math and math educational content, authored by professional mathematicians and scientists, graduate students, math teachers, undergraduates, and Girls’ Angle members. There have been galleries of math related art, comic strips, dialogues, articles, brain teasers, math challenges, math games, and much more. Thank you to the over 150 people have contributed to Bulletin content over the last decade! And Thank You to Mathworks, whose continuing support for the Girls’ Angle Bulletin has made so much of this possible.

Volume 10, Number 6 opens with an interview with Kathryn Mann, assistant professor in the Department of Mathematics at Brown University. Prof. Mann received her doctoral degree from the University of Chicago under the supervision of Benson Farb and was previously an assistant professor and NSF postdoctoral researcher at UC Berkeley. Her research interests include geometry, topology, and geometric group theory.

Next, Emily and Jasmine make steady progress on their quest to classify “nice triangles”. Their journey has taken them into the realm of algebra where they have been evaluating the cyclotomic polynomials at i. Hopefully, their perseverance will pay off, but even if it doesn’t, they’ve learned a lot of neat facts about the cyclotomic polynomials.

In this issue’s Math In Your World, we explain the rationale behind so-called “geometric probability.” The reason for this article, like so much content in the Bulletin, is because there are some current Girls’ Angle members who may be about to begin an investigation that requires understanding continuous probability distributions. (To all members: your content requests are taken very seriously and given a high priority!)

In Anna’s Math Journal, Anna gets an idea for looking at special tilings of 1 by $\sqrt{2}$ rectangles that could potentially lead to a derivation of the formula for the number of such tilings that doesn’t use generating functions.

And, finally, we close with the solutions to this summer’s Summer Fun problem sets.

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!

## Happy YP Day, HCSSiM!

The yellow pig has a 2 sentence solution!

## Girls’ Angle Bulletin, Volume 10, Number 5

Volume 10, Number 5 opens with an interview with Ruth Charney, the Theodore and Evelyn Berenson Professor of Mathematics at Brandeis University. Prof. Charney studied geometric group theory and received her doctoral degree in mathematics from Princeton University. She was formerly a professor at Ohio State University.

Next comes the second half of the article by Milena Harned and Miriam Rittenberg on NIM Counting. They give a general formula for the number of ways a NIM game with 2 starting piles can be played out and they investigate the form of the formula.

Errorbusters! returns after a long absence! The column, which was originated by Cammie Smith-Barnes, has been revived by Hamilton College Assistant Professor of Mathematics Courtney Gibbons. In her first installment, she writes about “Errors of Apathy,” which include such dastardly errors as substituting $x^2 + y^2$ for $(x+y)^2$.

The cover is a homage to the 24th cyclotomic polynomial. Of late, Emily and Jasmine have been filling reams of scratch paper with computations involving cyclotomic polynomials as they continue their quest to classify all “nice triangles”. The computation is daunting and they don’t even know if the computation will prove useful in their quest, but they bravely press on! Such is math research…

Next up: This summer’s batch of Summer Fun problem sets. This year, we have problem sets on cyclotomic polynomials (as a kind of primer to Emily and Jasmine’s work), sets (by Debbie Seidell), the fourth dimension, and unsuspected appearances of geometry (by Matthew de Courcy-Ireland).

We conclude with a few problems from our traditional end-of-session math collaboration and a few chocolate Hasse diagrams from the mini-chocolate tasting of Meet 11.

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!