## Girls’ Angle Bulletin, Volume 15, Number 4

It’s hard to imagine getting very far in mathematics without stumbling upon Pascal’s triangle. According to Wikipedia, the concept occurred even as far back as the 2nd century BC in India. The cover is a graphical depiction of the triangle, and in this issue’s Mathematical Buffet, you can find more graphic depictions of various aspects of this famous triangle of numbers, including images created by Sofia Egan and Yancheng Zhao, two 8th graders at the Buckingham, Browne, and Nichols Middle School. The triangle features prominently in Robert Donley’s article Pascal’s Triangle, the Binomial Theorem, and Chu-Vandermonde Convolution, where he presents several ways of interpreting the entries in Pascal’s triangle. If you were to create an image of Pascal’s triangle, how would you depict it?

In the last issue, we concluded our 3-part interview with University of Wisconsin, Madison Associate Professor of Mathematics Tullia Dymarz. One of the reasons we interviewed Prof. Dymarz is because she runs a wonderful program for high school girls called Girls Night Out! However, in her interview, she mentioned that the program was actually founded by her colleague Prof. Gloria Marí Beffa. So in this issue, we are pleased to present an interview with Gloria Marí Beffa. In addition to asking about her motivations for creating Girls Night Out!, we also discuss problem solving strategies and her unique route into mathematics.

Anna Ma and her cat return for an installment of Needell in the Haystack about data collection, which has changed dramatically with the advent of modern technology. Emily and Jasmine further their understanding of magic grids, and 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!

Also, the Girls’ Angle Bulletin is a venue for students who wish to showcase their mathematical achievements that go above and beyond the curriculum. If you’re a student and have discovered something nifty in math, considering submitting it to the Bulletin.

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 15, Number 3

On the cover, a kite with three congruent acute angles is seen at last, thanks to Milena Harned, who discovered that the only convex quadrilaterals whose angle bisectors are also perimeter bisectors are the rhombi, and such kites. To prove it, she analyzed the envelopes of the perimeter bisectors of polygons. For details, read her article in the peer-reviewed International Journal of Geometry, Volume 10, Number 4. For more, see this issue’s Member’s Thoughts and check out this article about Milena in the Notices of the American Mathematical Society by Scott Hershberger. I have a feeling we’ll be learning more math from Milena! (The photograph is fitting: It is the view from the mathematics department at UC Berkeley, courtesy of UC Berkeley Math Department Chair Michael Hutchings.)

We conclude our 3-part interview with University of Wisconsin, Madison Associate Professor of Mathematics Tullia Dymarz. Thank you, Tullia, for this wonderful opportunity to learn so much from you. In these interviews, we learned about quasi-isometry, the lamplighter group, and, in this concluding part, a fantastic model for involving high school girls in mathematics embodied in the program Girls Night Out!, which continues the brainchild of Gloria Marí Beffa.

What do cats and matrices have to do with each other? Read about one such connection in Anna Ma’s latest installment of Needell in the Haystack. Actually, what do matrices have to do with almost anything and everything? The field of data science could almost be described as a the mathematics of matrix manipulation.

We welcome back Robert Donley (aka Math Doctor Bob), who lays the groundwork for lattice path counting in Shortcuts to Counting. In Magic Grids, Emily and Jasmine become intrigued by a problem in the most recent Math Prize for Girls contest, and Addie Summer demonstrates that math accommodates many ways of thinking by presenting three different proofs of the angle sum formula for sine in Different Angles.

We conclude with a few 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!

Also, the Girls’ Angle Bulletin is a venue for students who wish to showcase their mathematical achievements that go above and beyond the curriculum. If you’re a student and have discovered something nifty in math, considering submitting it to the Bulletin.

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 15, Number 2

The cover shows all 35 ways to tile a 4 by 4 rectangle with 1 by 1 and 2 by 2 tiles, as organized according to a scheme dreamed up by a group of 8th graders at the Buckingham, Browne, and Nichols Middle School in Cambridge, MA. The number of such tilings of a 4 by N rectangle has been well-studied and a number of recursion formulas are known. But one recursion formula was noted by Schneider without proof or reference:

$S_n = 2S_{n-1} + 3S_{n-2} - 2S_{n-3}$,

where $S_n$ is the number of ways to tile a 4 by n rectangle with 1 by 1 and 2 by 2 tiles. These 8th graders succeeded in proving it, and in just a few clever lines! Details are in their article on page 8.

In Part 1 of our interview with Professor Tullia Dymarz, Tullia describes the lamplighter group and a pipe cleaner model she made of one of its Cayley graphs. If you were interested in trying your hand at building the pipe cleaner model but haven’t learned about Cayley graphs yet, check out this issue’s Learn by Doing to get going. In Part 2 of Tullia’s interview, she tells us more about her research process and describes one of her own discoveries which concerns which Cayley graphs of different lamplighter groups are quasi-isometric.

One point we try to emphasize in the Bulletin is that doing math is just a matter of being curious, coming up with questions that can be analyzed rationally and abstractly, and trying to answer them. The mathematical universe is so enormous with so much unexplored territory that if you start doing this, it will not be long before you come upon something new. It is an astonishing journey, full of surprises and awesome beauty. For example, there was the time when Emily and Jasmine stumbled upon their “Zigzag” theorem. (See pages 16-20 of Volume 13, Number 3 and pages 20-26 of Volume 13, Number 4, or read this earlier blog post. Although a fictionalized account, the story does closely describe how the theorem was actually born.) In our ongoing attempt to further this point, we follow Lightning Factorial’s thoughts on 2022 and point out the math that oozes out from thinking about that number’s properties in All That Math.

Anna Ma, who has taken over The Needell in the Haystack from originator Deanna Needell, addresses a super important question in data science: What do you do if your data is incomplete? It is an extremely common predicament to find oneself in. She begins the discussion, but it is an active area of research.

We conclude with a few 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!

Also, the Girls’ Angle Bulletin is a venue for students who wish to showcase their mathematical achievements that go above and beyond the curriculum. If you’re a student and have discovered something nifty in math, considering submitting it to the Bulletin.

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 15, Number 1

Our 15th year of the Girls’ Angle Bulletin begins with Part 1 of a multi-part interview with University of Wisconsin, Madison Associate Professor of Mathematics Tullia Dymarz. This was our first interview conducted over Zoom. In this first part, Tullia beautifully explains the notion of quasi-isometry and one of her favorite objects of study, the Diestel-Leader graph as the Cayley graph of the lamplighter group.

For this interview, we benefitted greatly from a biographical sketch of Tullia written by Isa Barth, and Isa’s essay follows the interview, although we urge all readers to read Isa’s essay first, as our interview takes off from there.

If you’ve been looking for an application of pipe cleaners to mathematics, look no further… Tullia provides a super interesting one!

Next, University of Oregon Associate Professor of Mathematics Ellen Eischen presents a selection of images from a mathematical art show that she curated and organized called Creativity Counts and which was on display at the Jordan Schnitzer Museum of Art in Eugene, Oregon. The cover features a contribution by Ellen herself. As Ellen note, “Aesthetic aspects of number theory, an area illustrated in most of the pieces in the gallery, have enthralled mathematicians since antiquity.”

Many topics in mathematics lend themselves well to visual imagery. If you create mathematically inspired visual art, we’d love to see it!

Anna Ma shows us that the Kaczmarz algorithm is an instance of a much more general minimization algorithm called “gradient descent.” If you have a real-valued function defined on some n-dimensional real space, and it is differentiable, then at each point in the n-dimensional space, it will have a vector, called the gradient, which points in the direction that the function locally grows the fastest. Gradient descent algorithms attempt to find minima by moving in a direction opposite to where the gradient points.

Emily and Jasmine analyze the Valentine heart equation from the Valentine heart app that launched their Valentine heart journey. They identify the 2D cross section which corresponds to the heart shape and compare and contrast it with the equation that they concocted.

We conclude with a few 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!

Also, the Girls’ Angle Bulletin is a venue for students who wish to showcase their mathematical achievements that go above and beyond the curriculum. If you’re a student and have discovered something nifty in math, considering submitting it to the Bulletin.

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 14, Number 6

Volume 14, Number 6 opens with an interview with Sarah Bryant, a lecturer at Gettysburg College. Sarah brings a valuable and unique perspective to the mathematics profession and to becoming a mathematician. She is involved with many activities that draw people into mathematics, such as by creating the Shippensburg Area Math Circle for 4th and 5th graders. She has also applied mathematics to the study of questions in biology, specifically, she studied nematocysts in jellyfish. Normally, we truncate the interview for the electronic version, but you’ll find the complete interview with Sarah online.

Following the interview, three Girls’ Angle members, Eva Arneman, Altea Catanzaro, and Saideh Danison, present a game they created at the Girls’ Angle club and beautifully explain their winning first player strategy when this game is played on the edges of a tetrahedron. There are many follow-up questions that one can ask about this game, such as, on what graphs does the first player have a winning strategy? We hope some readers will have as much fun thinking about the possibilities as these three! Our cover is inspired by the game and created by Juliette Majid.

Next, we welcome Anna Ma of UC Irving, who authors our latest installment of The Needell In The Haystack. Anna earned her PhD under the supervision of series creator Deanna Needell. The importance of Data Science just grows and grows as the world becomes more digitized. We consider ourselves very fortunate to have this ongoing series. Deanna earlier wrote about the Kaczmarz algorithm, and in this issue, Anna Ma gives her own take on it.

The concluding half of Jovana Andrejevic’s article on paper crumpling comes next. She compares and contrasts crumpling with deliberate paper folder (such as paper folding as practiced by origami enthusiasts). Although crumpling doesn’t enjoy certain precise theorems that origami folding does, there are hints that there is some unrecognized hidden geometric structure to paper crumpling. Perhaps you can find it?

We conclude with solutions to the Summer Fun problem sets by Laura Pierson (on Wythoff’s game) and AnaMaria Perez and Josh Josephy-Zack (on Fibonacci partitions). We’re delaying solutions to Fan Wei’s probability problem set to honor a reader request.

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!

Also, the Girls’ Angle Bulletin is a venue for students who wish to showcase their mathematical achievements that go above and beyond the curriculum. If you’re a student and have discovered something nifty in math, considering submitting it to the Bulletin.

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 14, Number 5

Volume 14, Number 5 begins with an interview with Petronela Radu, Olson Professor of Mathematics and Undergraduate Chair at the University of Nebraska-Lincoln. Petronela created a remarkable interdisciplinary course called “Math In The City” where students apply mathematics to solving and understanding real-world problems. In this interview, we discuss Math In The City, Peridynamics, and how she became interested in mathematics.

There are so many ways that math can be applied to gain understanding of real-world problems that Petronela’s course seems like something that could and should be replicated everywhere.

Prof. Radu is our 60th interviewee and there is no doubt that the opportunity to interview all these remarkable women in mathematics has been one of the biggest highlights of Girls’ Angle’s history. It is fortunate that we live in a time where there are several extraordinary women in mathematics. Still, the numbers are far short of what they could be. Among girls, far more mathematical talent is lost than developed.

The cover, Just a Crumple?, is, indeed, just a crumpled piece of paper. And yet, Jovana Andrejevic, a graduate student at Harvard’s School of Engineering and Applied Sciences, and her colleagues found order in the seeming disorder of repeatedly crumpled paper. This is a great example that shows how mathematics can spring forth from something that we might not normally pay any attention to at all. In the first half of Order In Disorder, Jovana reveals a beautiful pattern in the crease lengths of these crumples.

I learned about Jovana’s work by reading The Latest Wrinkle in Crumple Theory in the New York Times. There is always something particularly valuable when we are given the gift of an explanation from the researcher herself because she has first-hand understanding of the material and writes with a telling nuance.

Pamela Harris and Maria Rodriguez Hertz conclude their expository article on the mathematics of juggling, which is also a first-hand account from the researchers, by indicating the connection between Kostant’s partition function and certain types of juggling patterns. Some of the most beautiful theorems in mathematics are about explicit bijections between two, apparently unrelated, sets. That is the subject of this article.

Next, we present this summer’s batch of Summer Fun problem sets. This year, we have contributions from AnaMaria Perez and Josh Josephy-Zack, Laura Pierson, and Fan Wei. AnaMaria, Laura, and Fan have all served as absolutely marvelous mentors at the Girls’ Angle Club. Fan Wei is now a postdoc in the mathematics department at Princeton University. The three problem sets cover Fibonacci partitions, Wythoff’s game, and random variables. Members are welcome to send us their solutions to any of the Summer Fun problems.

We conclude with a brief summary of our end-of-session math collaboration which was created by mentors Vievie Romanelli, Rachel Zheng, and Head Mentor Grace Work. To give members a welcome break from sitting in front of their computers for all these virtual meets, part of this math collaboration sent members rummaging through their homes in search of objects that fit various mathematical prescriptions.

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!

Also, the Girls’ Angle Bulletin is a venue for students who wish to showcase their mathematical achievements that go above and beyond the curriculum. If you’re a student and have discovered something nifty in math, considering submitting it to the Bulletin.

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 14, Number 4

We open Volume 14, Number 4 with an interview with Candice Price, Assistant Professor of Mathematics and Statistics at Smith College. Candice is a cofounder of the website Mathematically Gifted and Black. In this interview, Candice touches on many important topics in the mathematics community that concern making the field more inclusive. She also tells us about her remarkable journey into mathematics and about some of the math that fascinates her.

The cover of this issue relates to Emily and Jasmine’s quest for an equation whose graph is a Valentine heart. In this installment, they succeed in finding one. There is a lot of subjectivity in what is acceptable as a Valentine heart, but I think most would agree that the circles in the top row of the cover cannot be considered Valentine hearts and nor can the clover-like shapes along the bottom row. This cover graphic features hundreds of graphs of equations and could not have been created without a computer. Special thanks to Juliett Bennett, Violet Freimark, Bridget Li, and Kate Pearce for their assistance with the coding!

By the way, if you are uncomfortable with graphs of equations, fear not, there is a Learn by Doing especially for you in this issue. After working through it, you’ll have no trouble writing down equations whose graphs are all manner of shapes.

The last two issues of the Bulletin featured a wonderful expository article on parking functions by Williams College professor Pamela E. Harris and her student Kimberly Hadaway. In this issue, Pamela joins forces with Maria Rodriguez Hertz, a professor at SUSTC to bring us the first part of another wonderful expository article, this time on the mathematics of juggling.

During the pandemic, we pretty much shut down our Support Network visiting program. However, thanks to Nooks.in, we were able to bring in Daina Taimina, a pioneer in knitting hyperbolic space and an emeritus adjunct professor at Cornell, to present on her work and career. You can read about that in this issue’s 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!

Also, the Girls’ Angle Bulletin is a venue for students who wish to showcase their mathematical achievements that go above and beyond the curriculum. If you’re a student and have discovered something nifty in math, considering submitting it to the Bulletin.

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 14, Number 3

We open Volume 14, Number 3 with an interview with Meike Ackveld, Senior Scientist at ETH Zürich. Almost exactly 9 years ago, Girls’ Angle had the good fortune of an in-person visit from Dr. Akveld. Today, we’re excited to present this interview with her. Dr. Akveld is also the President of the Association Kangourou sans Frontières, which creates the international Math Kangaroo Competition. In this interview, Dr. Akveld discusses how she became interested in knot theory, what Math Kangaroo is all about, and more.

Next, we present Deanna Needell’s latest installment of Needell in the Haystack, which is all about big data. In this installment, Prof. Needell explains one way to solve a system of linear equations when the coefficient matrix is so enormous, that it cannot be held in computer memory at the same time. The method she discusses, the Kaczmarz method, also serves as the inspiration for this issue’s cover graphic.

Kimberly Hadaway and Pamela Harris’s concluding half of their Parking Functions expository paper comes next. Here, they prove the closed formula for the number of parking functions. We do hope that before you read this, you make a serious attempt to prove the formula yourself.

Emily and Jasmine embark on a new math adventure, this time seeking an equation that describes a Valentine’s heart. There starting point is this neat web app written by Aaron Montag which was recently featured in the New York Times. In typical Emily and Jasmine style, they opt to try to figure out an equation on their own, before studying the equation used by Mr. Montag. We hope that readers attempt to come up with their own equation for a Valentine’s heart. If you come up with something you like, please share it with us! There is not definitive Valentine’s heart shape, so there’s a lot of room for creativity and artistic license here.

For those of you who are unfamiliar with matrices, we give a very quick introduction to matrix notation as it pertains to systems of linear equations so that you can follow Prof. Needell’s article.

We conclude with 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!

Also, the Girls’ Angle Bulletin is a venue for students who wish to showcase their mathematical achievements that go above and beyond the curriculum. If you’re a student and have discovered something nifty in math, considering submitting it to the Bulletin.

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 14, Number 2

We open Volume 14 with the concluding half of our two-part interview with Williams College Associate Professor of Mathematics Pamela E. Harris. Here, she discusses work/life balance and aspects of learning mathematics, as well as describing some of the math she discovered.

Further on in this issue, Prof. Harris coauthors a wonderfully written introduction to parking functions with her student Kimberly Hadaway in Honk! Honk!, Part 1. If you ever wanted to learn about parking functions, this article is a great place to start. Parking functions are a neat example of turning an everyday real life problem into interesting mathematics.

Also, included is the concluding half of Deanna Needell’s survey of the various ways in which she and her team has studied Lyme Disease using machine learning. Interspersed throughout are insights about how to handle the various tools.

In Toblerone Game, four students give a complete analysis of a combinatorial game involving the sharing of a Toblerone candy bar. The idea is you have any number of Toblerone bars before you, and they can even be of different lengths. You wish to share it with your friend, though you still want to eat as much of the candy as you can. But, being polite, you divvy up the bar using the following civilized rules: You take turns. On any given turn, If there happens to be an isolated triangular piece, then you are free to eat that piece. If not, then you must split a bar and let your friend go. What is the optimal strategy? If you have to split a bar, which bar should you split, and where? You can find all the answers in this paper.

Another all-student paper, or, I should say, saga, is The Saga of Fran & Fred. Ostensibly about high politics in the Kingdom of ABBABA, it is actually a probability paper in disguise.

We conclude with 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!

Also, the Girls’ Angle Bulletin is a venue for students who wish to showcase their mathematical achievements that go above and beyond the curriculum. If you’re a student and have discovered something nifty in math, considering submitting it to the Bulletin.

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!