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

Volume 10, Number 4 opens with an interview with Nalini Joshi, Professor of Mathematics in the School of Mathematics and Statistics at the University of Sydney. Prof. Joshi received her doctoral degree from Princeton University. She studied differential equations. The cover represents the iterates of an integrable third-order difference equation that arose out of joint work between her and Dr. C.-M. Viallet of CNRS and Sorbonne Universités. Dr. Viallet created the cover image.

In the 10th installment of *In Search of Nice Triangles*, Emily and Jasmine apply the knowledge they learned from Prof. Alison Miller from Part 9 and succeed in finding the minimum polynomials of the cosines of “nice” angles, a big step toward their goal of classifying all triangles that have 3 “nice” angles and 2 sides of integer length.

In *Anna’s Math Journal*, Anna gives in to the temptation of the generating function and uses it to prove her conjecture about the number of “special” tilings of a 1 by rectangle.

Next up, Milena Harned and Miriam Rittenberg explain different ways of counting the number of ways a game of NIM can be carried out. In this first half, they find explicit formulas when the game begins with two piles, one of which has a small number of counters. In the final half, which will appear in June, they will examine the asymptotics of the general 2-pile case.

Recently at the Girls’ Angle club, some members have been studying perspective drawing and practicing the theory by making drawings of geometric objects such as a checkerboard, like the one in the floor of Vermeer’s *Art of Painting*. This issue’s *Math In Your World* shows how the harmonic mean lurks within this topic. In fact, there’s a lot of beautiful mathematics in perspective drawing.

Milena Harned decided to generalize an idea she found on the 2002 American Invitational Mathematics Exam I. Along the way, she found some polynomials that relate to the famous way of obtaining the Fibonacci numbers from Pascal’s triangle shown at left.

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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After you make your intuitive guess, compute to find out the truth!

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Volume 10, Number 3 opens with an interview with Sommer Gentry, professor of mathematics at the US Naval Academy. Prof. Gentry invented a system for optimizing kidney transplantation that positively. This interview was conducted by Girls’ Angle progam assistant Long Nguyen. As has been our practice, we truncate the interview in the electronic version. For the full version, please subscribe!

Next comes Villanova assistant professor of mathematics Beth Malmskog‘s concluding part of *Quilt-Doku!* Here she shows that a 5 by 5 row complete Latin square is impossible. Shown at right is a row complete Latin square of order 15. It remains an unsolved problem to determine if there are any row complete Latin squares of odd prime order. In fact, it isn’t even known that there isn’t one of order 13. Can you prove it?

Test your area computation skills in *Area Area*. Can you solve these area problems in your head?

Anna B. continues to search for a combinatorial proof that the conjectured formula she found for the number of special tilings of a 1 by rectangle is, in fact, valid. She remains stumped. Can you help her prove that the close-form formula she found is correct?

Emily and Jasmine continue their pursuit of triangles with 3 nice angles but only 2 sides of integer length at Cake Country where they run into Alison Miller, a Benamin Peirce and NSF Postdoctoral Fellow at Harvard University. Prof. Miller gives them a lot to chew on for next time! She also mentions Gauss’s Lemma, but doesn’t have time to prove it, so we provide an installment of *Learn by Doing* where you can prove the lemma for yourself.

The cover pertains to Addie Summer’s follow-up to her article on the quadratic formula. In this issue, she creates a graphical representation of monic quadratics and interprets properties of them geometrically. On the cover, the surface represents monic cubics with multiple roots. Specifically, it is the surface of points (*b*, *c*, *d*) in *bcd*-coordinate space such that the cubic has multiple real roots. (Note that if a cubic with real coefficients has multiple real roots, then all of its roots are real.) This graph was created using MATLAB, a powerful suite of math software created by MathWorks. MathWorks has been a valuable sponsor of the Girls’ Angle Bulletin for several years.

For fun, we offer a self-referential true/false quiz that was inspired by a self-referential multiple choice test created by Jim Propp.

Finally, we conclude with Notes from the Club, which contains a description of one of our more versatile and popular games: Describe that Drawing.

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 10, Number 2 opens with an interview with Doris Schattschneider, professor emerita of mathematics at Moravian College. Among many other things, Professor Schattschneider was the first woman to serve as editor of the MAA’s *Mathematics Magazine*. This interview was conducted by Girls’ Angle summer intern Sandy Pelkowsky. As has been our practice, we truncate the interview in the electronic version. For the full version, please subscribe!

The cover is a picture of the function , modulo the prime number 503. For more examples and an explanation of how the image was made, check out this issue’s *Mathematical Buffet*.

We feature a contribution from Villanova assistant professor of mathematics Beth Malmskog, *Quilt-Doku!* Prof. Malmskog shows how she went from a quilting friend’s request to unsolved problems in mathematics. She blogs for the American Mathematical Society – check it out!

We’d also like to take special note of a member contribution in *Member’s Thoughts*. Here, ** π** takes us along on her derivation of the volume of a regular

Emily and Jasmine continue their pursuit of triangles with 3 nice angles but only 2 sides of integer length. While they make some progress in their investigation of the minimal polynomials of cosines of nice angles, the journey begins to appear rather daunting!

Last issue, Anna B. found a recursive formula for the number of special tilings of a 1 by rectangle. In this issue, she manages to guess the closed-form formula, but is unable to prove it. Can you help her prove that the close-form formula she found is correct?

Addie Summer explains how she discovered the quadratic formula. In the process, she shows that the algebraic technique known as “completing the square” corresponds to a natural geometric idea: shifting the parabola sideways so that it is symmetric about the *y*-axis.

In Notes from the Club, we mention a few of the things that have been happening at the Girls’ Angle club and give a sampling of problems from our traditional end-of-session Math Collaboration.

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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So we did.

Before giving our “proof”, please note that for the statement “potatoes are made of wood” to be a contradiction, it must also be true that “potatoes are not made of wood”. That potatoes are *not* made of wood, was the standing assumption that allowed for the statement “potatoes are made of wood” to be regarded as a contradiction.

Now to our proof.

Let *W* be the set of food items that are made of wood. Let *N* be the complement of *W*, that is, the set of all food items that are not in *W*.

Let *s* be the size of the intersection of *W* and *N*.

By definition of *W* and *N*, no food item is in both *W* and *N*. Therefore, *s* = 0.

Potatoes are not made of wood, and therefore, potatoes are not in *W*, and, hence, are in *N*. On the other hand, we are assuming that potatoes are in *W*. Therefore, potatoes are in both *W* and *N* and therefore the intersection of *W* and *N* contains at least one food item (namely, the potato). Therefore, *s* > 0.

However, we also know that *s* = 0.

Since *s* > 0, we can divide the equation *s* = 0 by *s* on both sides to arrive at the equation

1 = 0.

It is well-known that potatoes are not made of ducks. In particular, potatoes contain 0 ducks. However, 1 = 0, so if potatoes contain 0 ducks, then potatoes contain 1 duck.

Furthermore, if potatoes contain 1 of anything else, then, since 1 = 0, potatoes contain nothing of anything else.

We conclude that potatoes contain 1 duck and nothing else. Therefore, potatoes are made of ducks.

Click here for a printer friendly version of this “proof.”

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Volume 10, Number 1 opens with an interview with Dr. Brandy Wiegers, an assistant professor of mathematics at Central Washington University and associate director for the National Association of Math Circles. This interview was conducted by Girls’ Angle summer intern Sandy Pelkowsky. As has been our practice, we truncate the interview in the electronic version. For the full version, please subscribe!

The cover features an image of an opened Quintetra assembly by Jane and John Kostick. Jane visited Girls’ Angle last month and led the girls in an exploration of tetrahedra, cubes, and rhombic dodecahedrons. For more on the Quintetra assembly, please see Volume 7, Number 4 of the Girls’ Angle Bulletin.

After the interview, we offer 3 theorems to meditate upon in *Meditate to the Math*. All 3 theorems involve the construction of dropping a perpendicular from a point to a line, and, although it might be a challenge, all 3 theorems can reasonably be proven without use of scratch paper – just sitting comfortably, observing, thinking, meditating.

Next up is the concluding half of Prof. Helen Wilson‘s article on chocolate flow. This half is considerably more mathematically involved than the first half. In it, Prof. Wilson gets into the details of **scaling analysis** by leading us through an example involving liquid chocolate. It’ll help if you’re familiar with calculus. Just don’t be intimidated by the big partial differential equations that govern fluid flow! The whole point of the article is to show how to tame such a massive equation.

Emily and Jasmine return to their pursuit of triangles with 3 nice angles but only 2 sides of integer length. They decide to embark on an investigation of the minimal polynomials of cosines of nice angles.

Anna B. follows through on a question that sprouted up in the last issue: how many special *N*-tilings are there of a 1 by rectangle?

Addie Summer goes crazy (well, in a way) and counts some familiar sets in a complicated way, and is rewarded with beautiful identities involving binomial coefficients.

We conclude with Notes from the Club, featuring just a few of the things that have been happening at the Girls’ Angle club so far this semester.

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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Do you pay attention to the dimensions of objects? If so, you’ll have a nice advantage with this crossword.

Scan in or take a digital photo of your completed grid and send it to girlsanglepuzzler “at” gmail.com. We will randomly draw from the correct answers received by midnight on **October 16, 2016** to select a “winner” and send the winner a small prize. Girls’ Angle members will be put into a separate pool for a different prize. The prize will probably involve chocolate.

Good luck!

Special thanks to Becky J. for test solving!

(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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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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