Girls’ Angle Bulletin, Volume 14, Number 5

The electronic version of the latest issue of the Girls’ Angle Bulletin is now available on our website.

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!

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Girls’ Angle Bulletin, Volume 14, Number 4

The electronic version of the latest issue of the Girls’ Angle Bulletin is now available on our website.

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!

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Girls’ Angle Bulletin, Volume 14, Number 3

The electronic version of the latest issue of the Girls’ Angle Bulletin is now available on our website.

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!

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Happy New Year 2021!

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Girls’ Angle Bulletin, Volume 14, Number 2

The electronic version of the latest issue of the Girls’ Angle Bulletin is now available on our website.

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.

Political scheming in the Kingdom of ABBABA. Illustration by Esmé Krom.

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!

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Girls’ Angle Bulletin, Volume 14, Number 1

The electronic version of the latest issue of the Girls’ Angle Bulletin is now available on our website.

We open Volume 14 with the first half of a two-part interview with Williams College Associate Professor of Mathematics Pamela E. Harris. Prof. Harris’s journey into mathematics is extraordinary and quite unique. She also coauthored a wonderful contribution to the Bulletin in Volume 11, Number 2, entitled “Partitions from Mars”. Prof. Harris is actively involved in promoting minorities in mathematics. Among her many talents, Prof. Harris is expert at involving undergraduates in mathematical research.

Next, Deanna Needell gives us a survey of the various ways in which she and her team has studied Lyme Disease using machine learning. With the vast amounts of data being produced every second today, there is no hope for humans to analyze it all without the aide of computers. But computers are only as good as the algorithms that run on them. Data Science is a burgeoning field full of opportunities and promise.

Have you ever noticed how difficult it can be to keep track of multiple characteristics, even if the characteristics only come in two flavors, especially if the various characteristics influence each other? In Switcheroo!, Lightning Factorial tests your ability to stay organized in the face of many influencing, two-valued, switches. All you have to do is determine whether a light bulb is on or off. Good luck!

We enjoy turning activities that lend themselves to collaboration into mathematical activities. For example, crossword puzzles and jigsaw puzzles work really well as group activities, and at Girls’ Angle, we’ve “mathefied” both. (The first instance of a “mathefied” jigsaw puzzle is due to Girls’ Angle mentor Elise McCormack-Kuhman.) Word searches also lend themselves to collaboration, so we “mathefied” that this fall to create an activity that works well in the virtual world. Addie Summer provides a few example for you to try your hand at, as well as posing a number of math questions pertaining to these “number searches”.

Emily and Jasmine spent an afternoon building an icosahedron out of toothpicks. If you’re interested, they provide detailed instructions. They also analyzed the shape of the column of water that falls from a tap in a new installment of Emily and Jasmine’s adventures in The Water Column.

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!

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Girls’ Angle Bulletin, Volume 13, Number 6

Cover of Volume 13, Number 6 of the Girls' Angle BulletinThe electronic version of the latest issue of the Girls’ Angle Bulletin is now available on our website.

We close up Volume 13 with an interview with Middlesex Community College Associate Professor of Mathematics Aisha Arroyo. We frequently meet girls who like math but are not interested in math competitions. However, there certainly are girls who enjoy math competitions, and Prof. Arroyo was one of them in her student years. Today, she is actively involved in college-level math education.

The Summer Fun Solutions take up most of this issue, but we do squeeze in a Meditate to the Math on barycentric coordinates. Always try to understand the mathematical facts that you encounter. Don’t settle for memorizing them. The point of Meditate to the Math is exactly to approach mathematics through conceptual understanding.

The Summer Fun Solutions are quite detailed this year, and I’d like to thank Jasmine Zou and Matthew de Courcy-Ireland for their complete solutions that explain a lot of math in the process. For example, how quickly can you compute, with paper and pencil, the sum of the first 1,000 perfect 10th powers? Matthew shows us the way in complete detail.

These solutions also make heavy use of the Pell equation x^2 - 2y^2 = \pm 1, and its integer solutions are covered in quite a bit of depth.

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!

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Thirst For Firsts – A Girls’ Angle Raffle

For a PDF version, please click here.

(We will not use your contact information for any purpose other than to deliver your prize, should you win. After the winner has been selected, all emails received will be promptly deleted. At the winner’s discretion, we will let you know who won. Anyone who makes more than one submission will be disqualified! Sorry! Also, this offer is only valid in those states in the United States where such things are legal. There is no fee to enter this puzzle contest.)

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Girls’ Angle Bulletin, Volume 13, Number 5

The electronic version of the latest issue of the Girls’ Angle Bulletin is now available on our website.

At Girls’ Angle, we frequently have members who express interest in making their drawings look more realistic, and this often leads to a study of perspective drawing, which is a fabulous way to get into geometry. In this issue, we’re fortunate to feature an interview with Franklin & Marshall College Professor Annalisa Crannell, who recently wrote an entire book on perspective drawing together with Marc Frantz and Fumiko Futamura. And, special thanks to Princeton University Press, we include a two-page excerpt from their book, “Perspective and Projective Geometry.” Their book is highly engaging and offers a series of excellent perspective drawing exercises that are not only mathematically interesting, but also a lot of fun to do.

We have been incredibly fortunate to include a regular column, Needell in the Haystack, authored by Professor Deanna Needell on big data techniques. She is an amazing, benevolent force, and today, she’s also helping to fight the pandemic. However, in this issue, she writes about how bias creeps into mathematical analyses of data, which is an incredibly important and apt topic for our times. Normally, we remove her column from the electronic version, but we’re including the full article this time.

The cover might look like a perspective drawing, but it isn’t. It’s a design that Liliana Smolen and Isabel Wood created while the were playing around coming up with designs that illustrate mathematical identities. This issue’s Mathematical Buffet features four more of their designs. While Liliana and Isabel dreamed up these designs entirely from a blank white board, these particular designs have been seen before, and many more are collected in the books Proof Without Words: Exercises in Visual Thinking, by Roger Nelsen. The cover is a triangular version of the Fibonacci spiral. They even came up with a spiral that represents the mathematical constant e, but, unfortunately, there wasn’t enough space to include it. Perhaps you can come up with a way?

And we have our traditional Summer Fun Problem Sets. This summer, we present five: Cannonballs and Combinatorics by Girls’ Angle mentor Annie Yun, Tetrahedra with Congruent Faces by Ken Fan, Bernoulli Numbers by Matthew de Courcy-Ireland, Matrix Expedition by Girls’ Angle mentor Jasmine Zou, and Two Whole Squares by Ken Fan and Girls’ Angle Head Mentor Grace Work. As always, members and subscribers are encouraged to send us any questions and/or solutions.

We conclude with some Notes from the Club, which are authored by our Head Mentor Grace Work.

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!

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LCM Optimal Sequences

The best way to learn math is to do math, which is one of the reasons I’m thrilled about the work of Antonella Catanzaro, Jaemin Feldman, Mika Higgins, Bradford Kimball, Henry Kirk, Ana Chrysa Maravelias, and Darius Sinha, who were all eighth graders at the Buckingham, Browne, and Nichols Middle School when they embarked on their own personal mathematical adventure and discovered some curious results which I’d like to draw more attention to, especially because they left behind a number of conjectures which hopefully someone might be interested in pursuing.

Their adventure began when Ms. Higgins wondered aloud whether algorithms and number theory could be combined. After some brainstorming, the group came up with the following question:

What is the least expensive path through n cities, labeled 1 through n, starting at city 1 and allowing multiple visits to a city, if the cost to travel between city a and b is the least common multiple of a and b?

Here’s a table of some examples of optimal paths.

n Cost Sample Optimal Path
1 0 1
2 2 1, 2
3 7 1, 2, 1, 3
4 12 1, 3, 1, 2, 4
5 21 1, 3, 1, 2, 4, 1, 5
6 28 1, 3, 6, 2, 4, 1, 5
7 40 1, 3, 6, 2, 4, 1, 5, 1, 7
8 51 1, 5, 1, 7, 1, 3, 6, 2, 4, 8
9 65 1, 5, 1, 7, 1, 4, 8, 2, 6, 3, 9
10 79 1, 4, 8, 2, 6, 3, 9, 1, 7, 1, 5, 10

In each optimal path, note that in every pair of adjacent numbers, one is always a multiple of the other. This is something that the seven students proved, and they proved it by showing that for positive integers x and y,

LCM(x, y) < x + y if and only if x divides y or y divides x.

In other words, if x is not a factor or multiple of y, then it is cheaper to travel from x to y via 1 than it is to travel directly from x to y, because the cost of traveling from x to 1 to y is xy, whereas the cost of traveling from x straight to y is LCM(xy). A corollary of this fact is that prime numbers greater than n/2 will always be sandwiched by ones in an optimal path, unless the prime occurs at the very end of the path.

They showed that there always exists an optimal path that visits numbers greater than 1 exactly once, as all of the optimal paths in the table do. And they showed that when n is prime, all optimal paths end in n, but this is not necessarily true when n is composite as the example for n = 6 in the table above shows. Through their math department chair, the seven submitted the sequence of optimal costs to the Online Encyclopedia of Integer Sequences, and it was recently approved as sequence A333354. For details, see pages 13-19 of Volume 13, Number 4 of the Girls’ Angle Bulletin which can be accessed for free at the Girls’ Angle website.

Here are some conjectures and avenues for further investigation:

  • For a given n, do all optimal paths have the same length?
  • What are good upper and lower bounds on the minimal cost sequence?
  • What can be said if you replace the cost of travel between a and b with the highest power of 2 that divides one of the two numbers?
  • What is an efficient way to generate optimal paths for n > 20?

They determined the minimal cost for n up to 20, and for prime n below 20, the costs are:

n 2 3 5 7 11 13 17 19
Cost 2 7 21 40 100 138 238 295

In every prime case listed above, the minimal cost is equal to

\displaystyle n + 2(\sum_{k=(n+1)/2}^{n-1} k) + (\sum_{k=\lceil n/3 \rceil}^{(n-1)/2} k).

Is that true for all prime numbers n?

If you find anything, please do let us know at girlsangle “at” gmail.com!

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