This animation illustrates the connection between roots of the Chebyshev polynomials (of the first kind) and the vertices of a regular polygon inscribed in the unit circle centered at the origin. For a still image, see the cover of the Volume 8, Number 5 of the Girls’ Angle Bulletin.

Recently at the Girls’ Angle Club, we mentioned that from a contradiction, one can prove anything. One of our members, Barry Allen (pseudonym!), challenged us to prove that potatoes are made of ducks given that potatoes are made of wood.

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!

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!

(Added October 17, 2016: This raffle is now closed. Thank you to all who entered. Congratulations to Muriel and Tim D. for winning the Measure Up crossword puzzle raffle!)

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.

(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.)

Virtual Reality excites me because I believe it is the ideal home for a math museum.

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.

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!

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!