Years ago, I helped MoMath create its Math Midway. We strove to deliver visceral experiences where visitors could gain increased appreciation for and understanding of mathematics. Yet, the implementation of every exhibition concept was plagued by one omnipresent challenge. Mathematics is an abstraction of the human mind, but museum exhibits live in the real world, and the real world is encumbered with physical constraints.

Science museums have an immediate advantage over math museums because science is about the real world. Science exhibits can show lightning bolts, tornadoes, flowers, pumping hearts, geodes, chemical reactions, and so on. But math lives inside the human mind. The portion of math that translates well to physical exhibition is but a small fraction of the mathematical universe. This small portion would include a lot of applied math, much finite discrete math, and quite a bit of 2 and 3-dimensional topology and geometry, but enormous swaths of important mathematics can only be hinted at in a physical exhibit.

Take the concept of infinity, for instance. Infinity is one of the most fascinating notions in mathematics and so much math consists of ideas for handling infinity rigorously. But you can’t put Hilbert’s Hotel Infinity in any physical museum. You can’t make a true model of the infinite number line and you can’t build a true Koch curve, with its infinitely many spikes.

There’s so much more one can’t actually do in a real museum. You can’t exhibit a true Klein bottle. You can’t accelerate people to beyond the speed of light. You can’t build a non-Euclidean room.

But even within the tiny portion of mathematics that seems amenable to real world presentation, physical reality would constantly throw hurdles in our way. For example, we wanted to come up with a way to illustrate cross-sections of 3D shapes. If you use solid objects precut in various ways, then you’re limited to those specific precut sections. We ended up using light (the Math Midway’s “Ring of Fire”), but that approach only works well for certain shapes, and then there’s the interference caused by one’s body or whatever one is using to hold the object in the light.

Other ideas were muddled by the mechanics of machines. Things break, but it’s not just that things break. The complexity of the real world can obscure the basic truths of mathematics. For instance, Newton (and others) famously showed that a cycloid is the path a frictionless bead should travel to get from point A to point B fastest (in a uniform gravitational field). But the real world is not frictionless and different real-world beads will behave differently from each other.

Indeed, as a math exhibit constructor for a real-world math museum, you have to be concerned that you’ll convey a fundamentally incorrect understanding of what math is about whenever one compromises on an idealization for the sake of having some kind of manipulative.

Virtual Reality washes these problems away.

With Virtual Reality, you can make people believe they’re looking at objects that are infinite in extent. You can show all the cross sections of a 3D shape unfettered by supports. You can illustrate the theorems of mathematics with their hypotheses uncompromised. In a Virtual Reality Math Museum, the cycloid will correctly win the brachistochrone race.

Here are two detailed examples of how Virtual Reality comes to the rescue of museum exhibitors.

A beautiful illustration of the Pythagorean theorem, frequently filmed, is a water demo. A square erected on the hypotenuse of a right triangle is filled with water. When the contraption is turned upside-down, the water flows into two squares erected on the legs of the right triangle. The Pythagorean theorem implies that the water will perfectly fill these two squares and it’s quite satisfying to see it happen.

A serious problem with the museum exhibit is that it only shows a specific instance of the theorem, though the theorem is valid for all right triangles. A better exhibit would allow the visitor to modify the right triangle as desired. An even better exhibit would allow the visitor to modify the shape of the water receptacles subject only to the condition that the 3 receptacles are all geometrically similar to each other with the sides of the triangle corresponding to each other under the similarity relationship.

Creating such an exhibit would be quite a challenge. Aside from the challenge of leaky containers, how could one allow the visitor to modify the receptacle shape while simultaneously adding or removing water as the volume of the receptacle changes?

In Virtual Reality, it’s a snap!

For the second example, consider a multiplication sculpture. These are created by erecting columns over the entries of a multiplication table. The height of the column in row *R* and column *C* is the product *RC*. Physical multiplication sculptures are necessarily finite, and they cannot cover much of the multiplication table before growing impractically high. Wood worker Jane Kostick built a 15 by 15 one for Girls’ Angle, with each unit represented by 3/8 inch. While the footprint is roughly a square, 6 inches on a side, it soars higher than most humans at over 7 feet!

But what is a true multiplication sculpture really like? What would it be like to stand before one? For one thing, it isn’t finite. It is a cliff face like nothing in this world. No matter how high you look down the diagonal of the multiplication table, the multiplication sculpture will be looming before you (unless, of course, you look straight up or backwards).

We can get a hint of this using a kind of “poor man’s” version of Virtual Reality: stereo pair videos. (By the way, if this were an educational post, I’d make a point of encouraging students to first imagine what they would see before watching the videos, and use the videos only as a confirmation or correction to what they imagined.)

In this stereo pair video, the viewer travels along a horizontal circle at height 100 units:

In this stereo pair video, the viewer rides along a curved escalator that moves farther and farther away and faster and faster up:

(In both videos, purple columns correspond to those with an odd height. The purple unit cube in the front of the cover image of the second video corresponds to the first row and column in the multiplication table.)

What these stereo pair movies lack is interactive freedom and the feeling of actually standing before a multiplication sculpture. Virtual Reality gives us the freedom to explore what our minds feel the need to understand in the moment and the feeling of “being there”.

One might argue that an interactive stereo pair would be just as effective, but the difference is virtually as great as that between watching a documentary on the Pantheon and being there, looking all around. The United Nations has a powerful use of Virtual Reality. By using Virtual Reality, the UN has created experiences that enable us to empathize more fully with people living in extreme circumstances. These Virtual Reality experiences induce empathy like no photojournalism has been able to do before.

Insofar as math can be experienced in a museum, Virtual Reality is the ideal way to experience it.

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Volume 9, Number 6 begins with an interview with Northeastern University’s Associate Professor of Mathematics Ana-Maria Castravet.

As has been our practice, we truncate the interview in the electronic version. For the full version, please subscribe!

Next, comes the first part of a scrumptious installment of Math In Your World by University College London Professor of Mathematics Helen Wilson, who explains the mathematics of chocolate flow, hence, also, the cover which features a picture of the chocolate fountain at the Langham Hotel in Boston (special thanks to the Langham Hotel for this wonderful pic!).

Anna B. has been on a long journey, investigating polynomials over a finite field, all launched by an exercise that Prof. Judy Walker suggested in her interview with Girls’ Angle in Volume 8, Number 6 of the Bulletin. But last issue, she reached a good stopping point, so this issue, she decided to tackle one of the Summer Fun problems: Problem 2 from Similar Tiles.

We conclude with solutions to last issue’s Summer Fun problem sets. Of special note is Matthew de Courcy-Ireland’s solution set which went into some very nice details, so much so, that we decided to break off his solution to the “Thirty Bird Problem” of Fibonacci into a separate article.

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!

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Volume 9, Number 5 kicks off with an interview with the ADVANCE Professor of Computing at Georgia Institute of Technology, Dana Randall. Prof. Randall is also the Director of the Algorithms and Randomness Center and an Adjunct Professor of Mathematics.

Next, comes the concluding half of Heidi Hurst’s article on how she used math to help find optimal locations for FEMA Disaster Recovery Centers.

Emily and Jasmine continue their exploration of nice triangles by commencing with an exploration of which triangles have 3 “nice” angles and 2 sides of integer length. This is the last major case in their investigation of “nice” triangles and involves the most advanced mathematics.

Anna manages to generalize the result she found in the last issue to the finite fields , where is prime. This result was known to Carl Friedrich Gauss.

We conclude with this summer’s batch of Summer Fun problem sets! This summer, we have contributions from Matthew de Courcy-Ireland, Aaron Levy, Lauren McGough (who is also a Girls’ Angle Advisory Board member and the Girls’ Angle Treasurer), Long Nguyen, and Zachary Sethna. Matthew, Aaron, Lauren, and Zachary are all graduate students at Princeton University. Long is an undergraduate at MIT.

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!

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Volume 9, Number 4 begins with an interview with Oberlin’s Andrew and Pauline Delaney Professor of Mathematics Susan Jane Colley, although as has been our practice, we truncate the interview in the electronic version. For the full version, please subscribe!

After a meditation on Miquel’s theorem, find out how a frog fish, dragon, and mathematician end up in the same story with the concluding half of *The Mountain Clock*, a fictional math story about an ancient city that adopted a 32-hour day.

Anna manages to prove a theorem she has been leading up to ever since Prof. Judy Walker suggested some exercises involving polynomials over the finite field with 2 elements.

Next comes a neat application of mathematics to disaster recovery. Harvard University undergraduate Heidi Hurst spent a summer using math to figure out how to find optimal locations for Disaster Recovery Centers. She explains her work in a two-part *Math In Your World* series.

Emily and Jasmine continue their exploration of nice triangles and establish a direct link between integer-sided triangles with perimeter *p* and partitions of *p* – 3 into parts with sizes 2, 3, and 4.

We conclude by unraveling some secrets of a marvelous recurrence relation created by one of our 11-year-old members.

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!

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Volume 9, Number 3 begins with the concluding half of an interview with French mathematician Alice Guionnet. Prof. Guionnet is a professor of mathematics at MIT and an expert on random matrices.

The cover features an illustration by Julia Zimmerman for a fictional math story entitled *The Mountain Clock*. The Mountain Clock tells the story of an ancient King who decreed a 32-hour day.

Katherine Cliff, a graduate teaching fellow at the University of Colorado, Colorado Springs, contributes this issue’s *Math In Your World*. She brings the rational world of mathematics to the computation of probabilities for the Powerball lottery. If you work through this installment, you’ll be able to compute lottery probabilities and the value of a lottery ticket from scratch.

Anna continues her investigation of irreducible polynomials over the finite field with 2 elements. In this installment, she formulates 2 conjectures, works out implications of them, and proves one of them. Anna’s entire investigation traces back to an exercise suggested by Prof. Judy Walker in her interview from Volume 8, Number 6. Can you prove the conjecture she doesn’t prove?

Emily and Jasmine continue their quest for “nice” triangles. This time, they explore integer-sided triangles with no conditions at all on their angles. They determine the number of such triangles that have 2 given side lengths and begin to study the question of how many integer-sided triangles there are of a given perimeter.

In this issue’s *Learn By Doing*, you can try your hand at coming up with counterexamples. There are few things that so decisively prove a conjecture wrong than coming up with a counterexample!

There’s a lot more inside and we hope you enjoy it!

*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!

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Thank you so much to Ken for allowing me to contribute a guest blog! I am a middle school math teacher at the Pollard Middle School in Needham. I have had the extreme pleasure of having Ken come to Pollard for the past four years to host a Math Collaboration. The events have been thoroughly amazing. From a teacher’s perspective, there is nothing I enjoy more than seeing students working tirelessly and collaboratively towards a common goal. The puzzles and problems that Ken puts together are the perfect balance of stretching students’ thinking while also making them attainable, and his gentle manner of prodding students without guiding them is something that I try to emulate on a daily basis in my own classroom. I cannot do justice to how exceptional these events are, so I asked two students who participated in the latest Collaboration to write their thoughts on the event. Here is what they wrote.

“Recently we participated in the math collaboration treasure hunt hosted by Girls’ Angle. This event was after school on an early release day, and beforehand it didn’t seem like the best way to spend our time out of school. We were surprised by the complexity of the puzzles and impressed at how fast we solved it. It was a change to not be given detailed instructions or told exactly what to do. Instead we had to figure out how to be organized and efficient, without relying on adults. At first everything was chaotic and nobody knew exactly what to do. But then we established a system with small groups working on each problem, and everybody helping those who needed it. This activity taught us to work together and help others to achieve a common goal. In the end, we were only able to solve the puzzle because every single person contributed in some way. We had to be independent, work together, use leadership skills, be organized and listen to everybody’s ideas in order to complete the treasure hunt and receive the prize. This collaboration was an amazing learning experience and built leadership and teamwork skills, but more importantly a fun way to spend an afternoon.”

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Volume 9, Number 2 begins with the first half of an interview with French mathematician Alice Guionnet. Prof. Guionnet is a professor of mathematics at MIT and an expert on random matrices.

In addition to the interview, there are 2 more articles in this issue that pertain to probability and statistics. One is the concluding half of Prof. Elizabeth Meckes articles on the laws of probability. This time, she pulls up the curtain on the central limit theorem. The other is this issue’s Math In Your World, which describes an activity led by Girls’ Angle Support Network visitor Jinger Zhao. Jinger is a financial modeler who works at TwoSigma, a hedge fund based in New York City. At the club, Jinger uses statistics to model the connection between wingspan and height.

Anna continues her investigation of irreducible polynomials over the finite field with 2 elements. In this installment, she works out the roots of all the irreducible polynomials of degrees 4 and 5. Anna’s entire investigation traces back to an exercise suggested by Prof. Judy Walker in her interview from Volume 8, Number 6. Do you think you can see where Anna might be headed? If you do, follow your thoughts and see where they lead. You’re invited to tell us about it; we’d love to hear from you. If you’re falling in love with polynomials over , check out this proposal for a new PolyMath project and the comments that follow.

Multiplication and exponentials are fundamental concepts in mathematics. For anyone working on learning these concepts, we hope Addie Summer’s *Thoughts on Multiplication* and our Learn by Doing on exponentials will be of use. The cover, which was created using MATLAB by MathWorks, honors multiplication.

Emily and Jasmine continue their quest for “nice” triangles. This time, they explore integer-sided triangles that have a 120 degree angle and establish a beautiful bridge between these and integer-sided triangles with a 60 degree angle.

We conclude with some notes from the club. If you’re a girl, aged roughly 10-18 in the Greater Boston Area, you’re welcome to join. Our next session begins January 28, 2016.

We hope you enjoy the Bulletin!

*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!

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Volume 9, Number 1 features material from two mathematicians: an extensive interview with Elizabeth Munch, Assistant Professor of Mathematics at the University of Albany and the first of a two-part article on the laws of probability by Elizabeth Meckes, Associate Professor of Mathematics at Case Western Reserve University.

Prof. Munch’s road to mathematics is interesting in that music plays a significant role in her life. In addition to her mathematical degrees, she also holds a degree in Harp performance from the Eastman School of Music. Find out how her musical studies affected her discovery of mathematics. In her interview, she mentions Takens embeddings, and the cover features a kind of Takens embedding. The embedding is actually a 4-dimensional Takens embedding consisting of two spatial dimensions and two that extend into the color space.

Statistics plays an important role in so many sciences. Research would grind to a halt without it. One of the most important results in statistics is the central limit theorem. Prof. Meckes has contributed an eloquent two-part article that explains the meaning of this theorem. In part 1, she explains the laws of probability.

In Math In Your World, Lightning Factorial uses statistics to improve at darts, following the lead of statisticians Ryan Tibshirani, Andrew Price, and Jonathan Taylor, who showed in their paper *A Statistician Plays Darts*, that depending on your dart throwing prowess, you might be better off not aiming for the bull’s-eye. Skeptical? Read Lightning’s article!

In Anna’s Math Journal, Anna continues her investigation of irreducible polynomials over the finite field with 2 elements. This investigation traces back to suggested exercises made by Prof. Judy Walker in the previous issue. If you’re unfamiliar with finite fields but want to follow along with Anna on her journey, this issue’s Learn by Doing is just for you. In it, finite fields are introduced assuming very little by way of prerequisites.

Meanwhile, Emily and Jasmine continue their quest for “nice” triangles. This time, they apply a technique that is often used to find Pythagorean triples to find formulas that yield the sides of all primitive triangles that contain a 60 degree angle, such as the 5-7-8 and 16-19-21 triangles.

We conclude with some notes from the club! We’ve got a wonderful group of members this semester and if you’re a girl in grades 5-12 who lives near Cambridge, MA, you’re welcome to attend!

We hope you enjoy it!

*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!

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(Added October 1, 2015: This raffle is now closed. Thank you to all who entered. Congratulations to Fran M. who won the general draw and Iris L. who won the member draw!)

Can you reconstruct Serena and Katherine’s path through the museum?

They start at the museum entrance. They choose a random direction, north, south, east, or west and walk together in that direction until they come upon the first exhibit that they had not yet visited and stop. When done looking at that exhibit, they again choose a random direction, north, south, east, or west and walk together in that direction until they come upon an exhibit that they had not yet visited and stop. They continue wandering in this manner. Miraculously, they manage to visit all 17 exhibits!

Figure out all 17 directions Serena and Katherine took and the order that they took them. Send your answer to girlsanglepuzzler “at” gmail.com by **midnight on September 30, 2015**. For example, you might send the string “NSEWNSEWNSEWNSEWN” if you think those are the directions they went in that order, left to right. On October 1, 2015, two random “winners” will be drawn from correct entries, one from among Girls’ Angle members and the other from among the general public. The two “winners” will receive a modest prize. (The member prize is bigger.)

Exhibits are marked by flag poles. Determine the path that goes from flag pole to flag pole. They start at the museum entrance and end at one of the exhibits.

Good luck!

(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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