And now, an exclusive behind the scenes interview with Diana Davis about how she put this incredible project together without any prior experience:

Diana Davis: “One challenge was that we recorded all the pentagon dancing once, using the Math Department video camera, but the resolution was not good enough to make a great video, so we had to do the whole thing again. So then the question is, what video camera can we get that has good resolution? It turns out that an iPad was the best option. But then, how do you use an iPad to record video from directly above? So we filmed in a building at Brown that used to be a church, and put the camera in the organ loft pointing straight down. To do this, I bought an 8-foot plank from Home Depot, which I tied to a chair using an Ethernet cord, and taped the iPad to the end of the plank, and then leaned out over the railing to press Record and Stop. We taped the pentagons to the floor directly below the iPad.

The second challenge was making the magic happen. Sometimes people ask who did the video editing. It was ME! And I had never done any video editing before. I learned, by putting in hundreds of hours and failing and trying again, how to use these two powerful video editing programs, Adobe After Effects and FinalCut Pro. Adobe After Effects is what makes the magic happen, where Libby dances across one edge and reappears on the other side — basically you can make part of the screen show one video and another part of the screen show another video, so I made the two parts of the screen show two parts of the same video. Making this work perfectly took forever. If you pause the video at any point, you will find that it looks perfect, with about 105% of a Libby on the screen at any one time. I had to blend in the shadows so that they didn’t disappear all of a sudden. I think that Libby dancing on the double pentagon surface is a work of art! And then I used FinalCut Pro to put the pentagon videos together with the Math Hatters and the text.

The best place to start is this FAQ video: https://vimeo.com/47273811 because as you know, it’s better to *show* you how we made it than to *tell* you how we made it.”

Suppose you need to walk through a wet parking lot. The lot is covered with puddles and you would like to keep your shoes as dry as possible. If you know the depth of the puddles at every point, how do you choose the path that minimizes the maximum depth of the puddles you cross? A hiker might want to solve a similar problem if they want to avoid fatigue by seeking low elevations. How do you traverse a mountainous area while remaining as low as possible?

To make the problem more formal, say the area of interest is and the elevation of the ground (or puddle depth) at each point is , with continuous. Choose areas that are acceptable start and finish points and call them and . If you want to walk from the north end of the square to the south, then you could choose and . The problem can be stated as the minimax

over continuous paths such that and .

Initially, you might suppose that this is a fundamentally continuous problem since there is a continuum of possible routes . However, we can make the problem discrete by considering sublevel sets of .

The -sublevel set of is the set of points where is below elevation , that is

.

The minimax problem is solved by finding a that lies in the lowest sublevel set connecting and . That lowest elevation is

.

Watch the animated figure below and notice the first time that the white sublevel set connects the top and bottom. A path between top and bottom must first appear when the boundaries pass through a saddle point of because saddle points are where connected components of sublevel sets join each other. This is a consequence of the “mountain pass theorem”: if is optimal and is maximized by , then is a stationary point of . That stationary point is a normally a saddle point (though it could be part of a flat-topped area).

Now we can write a discrete process for finding an optimal path:

- Locate all saddle points of .
- Order the saddle points by elevation so that .
- For each saddle point , determine whether its sublevel set has a connected component intersecting and .
- If the lowest such saddle point is , then is the minimum value for the minimax problem, and the optimal path is any contained in that connects and .

For something else to think about, let’s consider a connection to graph theory. If you watch the above animation a few times, you’ll notice that connected components of the white sublevel sets are created at local minima. They spread out and meet other components at saddle points. You might imagine a graph (or network) being created between the local minima: when two components touch, an edge is created between two minima (one in each component).

Is there a natural way to choose a representative from each component to connect? If a hole forms so that the component is no longer simply connected, should an edge be added to make a cycle? Also, a graph on the maxima could be created in the same way. Should the graph on the maxima be the dual of the graph on the minima?

Post any thoughts or questions in the comments below!

]]>“Goodbye, mission control. Thanks for trying.” ~aiken_~

to the lighter

“I leave. Dog panics. Furniture shopping.” ~Reed~

That said, there seem to be very few–I found two or three–flash fiction stories pertaining to math. With this in mind I want to propose a challenge:

**Write your own six-word story/stories capturing the life, experiences, or work of mathematicians and graduate students, and post them in the comments below. The best ones will be highlighted in my post next month!**

To give some sense of what I mean, and maybe to help get your creative thoughts rolling, here are two six-word stories I wrote about common experiences in the lives of math graduate students.

Working hard, checked arXiv. Start again.

From: Grad Admissions. “We are sorry…

]]>*Originally published by Scientific American *

From the profound revelations of the shape of space to the furthest explorations reachable by imagination and logic, the history of mathematics has always been seen as a masculine endeavor. Names like Gauss, Euler, Riemann, Poincare, Erdös, and the more modern Wiles, Tao, Perelman, and Zhang, all of them associated with the most beautiful mathematics discovered since the dawn of humanity, are all men. The book Men of Mathematics, written by E.T. Bell in 1937, is just one example of how this “fact” has been reinforced in in the public consciousness.

Even today, it is no secret that male mathematicians still dominate the field. But this should not distract us from the revolutionary contributions women have made. We have notable women to thank for modern computation, revelations on the geometry of space, cornerstones of abstract algebra, and major advances in decision theory, number theory, and celestial mechanics that continue to provide crucial breakthroughs in applied areas like cryptography, computer science, and physics.

The works of geniuses like Julia Robinson on Hilbert’s Tenth Problem in number theory, Emmy Noether in abstract algebra and physics, and Ada Lovelace in computer science, are just three examples of women whose contributions have been absolutely essential.

**Julia Robinson (1919-1985)**

At the turn of the twentieth century the famed German mathematician David Hilbert published a set of twenty-three tantalizing problems that had evaded the most brilliant of mathematical minds. Among them was his tenth problem, which asked if a general algorithm could be constructed to determine the solvability of any Diophantine equation (those polynomial equations with only integer coefficients and integer solutions). Imagine, for any Diophantine equation of the infinite set of such equations a machine that can tell whether it can be solved. Mathematicians often deal with infinite questions of this nature that exist far beyond resolution by simple extensive observations. This particular problem drew the attention of a Berkeley mathematician named Julia Robinson. Over several decades, Robinson collaborated with colleagues including Martin Davis and Hillary Putnam that resulted in formulating a condition that would answer Hilbert’s question in the negative.

In 1970 a young Russian mathematician named Yuri Matiyasevich solved the problem using the insight provided by Robinson, Davis, and Putnam. With her brilliant contributions in number theory, Robinson was a remarkable mathematician who paved the way to answering one of the greatest pure math questions ever proposed. In a Mathematical Association of America article, “The Autobiography of Julia Robinson”, her sister and biographer Constance Read wrote, “She herself, in the normal course of events, would never have considered recounting the story of her own life. As far as she was concerned, what she had done mathematically was all that was significant.”

**Emmy Noether (1882-1935)**

Sitting in an abstract math course for any length of time, one is bound to hear the name Emmy Noether. Her notable work spans subjects from physics to modern algebra, making Noether one of the most important figures in mathematical history. Her 1913 result on the calculus of variations, leading to Noether’s Theorem is considered one of the most important theorems in mathematics—and one that shaped modern physics. Noether’s theory of ideals and commutative rings forms a foundation for any researcher in the field of higher algebra.

The influence of her work continues to shine as a beacon of intuition for those who grapple with understanding physical reality more abstractly. Mathematicians and physicists alike admire her epoch contributions that provide deep insights within their respective disciplines. In 1935, Albert Einstein wrote in a letter to the New York Times, “In the judgment of the most competent living mathematicians, Fräulein Noether was the most significant creative mathematical genius thus far produced since the higher education of women began.”

**Ada Lovelace (1815-1852)**

In 1842, Cambridge mathematics professor Charles Babbage gave a lecture at the University of Turin on the design of his Analytical Engine (the first computer). Mathematician Luigi Menabrea later transcribed the notes of that lecture to French. The young Countess Ada Lovelace was commissioned by Charles Wheatstone (a friend of Babbage) to translate the notes of Menabrea into English. She is known as the “world’s first programmer” due to her insightful augmentation of that transcript. Published in 1843, Lovelace added her own notes including Section G, which outlined an algorithm to calculate Bernoulli numbers. In essence, she took Babbage’s theoretical engine and made it a computational reality. Lovelace provided a path for others to shed light on the mysteries of computation that continues to impact technology.

Despite their profound contributions, the discoveries made by these three women are often overshadowed by the contributions of their male counterparts. According to a 2015 United Nations estimate, the number of men and women in the world is almost equal (101.8 men for every 100 women). One could heuristically argue, therefore that we should see roughly the same number of women as men working in the field of mathematics.

One large reason that we don’t is due to our failure to recognize the historical accomplishments of female mathematicians. Given the crucial role of science and technology in the modern world, however, it is imperative as a civilization to promote and encourage more women to pursue careers in mathematics.

*Originally published by Scientific American *

Let’s suppose your students have been multiplying matrices for a couple days and have just started to get the hang of it. Try presenting them with this matrix:

and this stick figure position on the coordinate plane as shown:

Remind students that an ordered pair, such as the location of the stick figure’s right foot, can be represented as a column vector. Have students individually or in a group multiply various key points on the stick figure by R to see what happens to the drawing.

Obviously the figure was stretched, and it should be obvious that the direction and magnitude of the stretch correspond to the diagonal entries in R. Point out as well that the angle of the figure’s left leg has changed from about 45^{o} to about 30^{o}, so in effect, those points have been rotated around the origin. Ask students to identify which points were not rotated at all, merely pulled outward from the origin.

Now repeat the process with this matrix:

The points on the x-axis will be stretched as before, but the figure’s left hand will leave the y-axis and move rightward as the entire figure appears to stretch diagonally and right rather than upward and right like before. Once again, ask students to identify which points experience no rotation. This time, it should be points on the x-axis and points on the line y=x (the figure’s crotch in this case). I like to compare matrix multiplication to stretching out the fabric of the x-y plane like a stretchy bed sheet. I imagine a pair of people standing on the left and right sides of bed sheet pulling in opposite directions, and another pair at oppose corners, pulling slightly less hard. As the bed sheet is stretched, the designs on the fabric get distorted in a particular way. Challenge students to identify in what two directions the x-y plane is being stretched and by what magnitude.

At the point, the students will have basically invented eigenvectors on their own—all you need to do is provide the name “eigen” and then let the students formulate a definition themselves. Here’s one possibility:

__Def:__ An **eigenvector** is the direction in which a matrix stretches vectors through multiplication. The factor by which they are stretched in that direction is called an **eigenvalue.**

It’s fine for now if the definition is cast in geometric terms; it will give the students something concrete they can picture in their mind’s eye. Later in the course when you circle back to eigenvalues, students will have a clear foundation for what exactly they are and why they are important.

This might also be a great time to build on your students’ natural curiosity and encourage them to pose conjectures or ask questions. For instance, do all matrices have eigenvectors? How many? Does one of them always have to lie along the x- or y-axis? How can we find eigenvectors without having to make drawings? Have students jot down these questions on the inside cover of their notebooks or hang them up somewhere in the classroom where students can continue to ponder them as the semester advances.

At its most basic level, spiral teaching is about on circling back to the same concept in more depth once it’s had a few weeks or months to sink in. You introduce eigenvalues with a concrete example, give students some practice, and then return to them later on to develop theorems. But sparking students’ curiosity can make this technique even more successful. If, in the intervening weeks, students find themselves pondering these questions every time they open their notebooks or gaze absentmindedly at the wall above the chalkboard, they’ll remember the concepts better when you return to them. Some students may already have foreseen some the theorems you plan to introduce thanks to their conjectures, while others will be itching to know the answer to the questions.

]]>However, after introducing limits, the students still encountered difficulties understanding the subtleties of one-sided versus two-sided limits. When you explain how to handle *some *types of functions, students will not pay attention to “some” because they only look at the outside definition without thinking about the internal meaning. In the beginning, it helped to give them instructions on how to approach very specific common types of functions such as roots, absolute value functions, piece-wise defined functions, and .

Another way I try to decrease the confusion for my students is to represent the function as a box and tell them that this box is a control system, which means that we have input and output.Therefore, if we have a number or variable as input, this number/variable needs to be plugged into any variable that appears in the box (I tell them that our function is inside the box). Then, after plugging in whatever we have as input, we will have the result which is the output. It also helps my students when I write math problems on the board in colors and sometimes with shapes because it is very helpful for students to focus on the functions and the changes we do to them.

Students are familiar with the usage of the variable in functions, say , but if we change that to , some students may not be able to plug in, say , to find . To avoid this confusion, I give them at least four examples: one with , one with , one with a Greek letter, and one with . This method can make students familiar with the usage of other variables in functions.

Similarly, when the students worked in groups for a class activity on functions, I noticed some other common mistakes. For example, suppose they are given and are asked to find . The first time I gave them a problem like that, they did not know where to plug in ; often they would plug it in as . I guided them to the step , but we ended up with another problem because they did not know how to simplify . They thought that . I always try to help students by writing handouts to address the common errors and how they can avoid making them on exams. (You can view the handout that I created on this topic here.) When I see something is very difficult for them to understand using the traditional method, I create my own method using interactive techniques. (See my earlier blog post: *How to change the Traditional Mathematics Teaching from the Memorization-Based Method to Interactive-Based Method.)*

In conclusion, based on my experience, handling functions as abstract objects is one of the most difficult mathematical concepts for non-engineering and non-math students. I believe that a successful math teacher must teach students how to use advanced mathematical thinking in order to help them become more comfortable with abstract mathematical objects.

To see more teaching handouts, please see my course webpage. Best of luck and feel free to reach out if you have questions!

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Yesterday I received a disheartening 44/50 on a homework assignment. *Okay okay*, I know. 88% isn’t *bad*, but I had turned in my solutions with so much confidence that admittedly, my heart dropped a little (okay, a lot!) when I received the grade. But I quickly had to remind myself, *Hey!* G*rades don’t matter.*

The six points were deducted from two problems. (Okay, fine. It was three. But in the third I simply made an air-brained mistake.) In the first, apparently my answer wasn’t explicit enough. *How stingy! *I thought. *D**oesn’t our professor know that this is a standard example from the book? I could solve it in my sleep*! But after the prof went over his solution in class, I realized that in all my smugness I never actually understood the nuances of the problem.*Oops*. You bet I’ll be reviewing his solution again. *Lesson learned.*

In the second, I had written down my solution in the days before and had checked with a classmate and (yes) the internet to see if I was correct. Unfortunately, the odds were against me two-to-one as both sources agreed with each other but not with me. But I just couldn’t see how I could *possibly* be wrong! Confident that my errors were truths, I submitted my solution anyway, hoping there would be no consequences. But alas, points were taken off.

Honestly though, is a lower grade such a bad thing? I think not. In both cases, I learned exactly where my understanding of the material went awry. And that’s great! It means that my comprehension of the math is clearer now than it was before (a*nd* that the chances of passing my third qualifying exam have just increased. Woo!) And *that’s* precisely why I’m (still, heh…) in school.

So yes, contrary to what the comic above says, grades *do* exist in grad school, but – and this is what I think the comic is hinting at – they don’t matter. Your thesis committee members aren’t going to say, “Look, your defense was great, but we can’t grant you your PhD. Remember that one homework/midterm/final grade from three years ago?” (They may not use the word “great” either, but that’s another matter.) Of course, we students should still work hard and put in maximum effort! **But the emphasis should not be on how well we perform, but rather how much we learn. **

So to all my future imperfect homework scores out there: *bring it on.*

*This post originally appeared on the Math3ma.com blog on March 9th, 2016.*

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These conversations are always short, and I have to remind myself that I didn’t fall in love with math because someone told me that I could use it to build bridges or maximize profit. I love math, because I’ve spent time seeing what math can do and doing it myself.

Now, I finally have a better answer. Last August, Khan Academy and Pixar teamed up to release Pixar in a Box!

Pixar in a Box is a compilation of math lessons all about the different parts of animation that go into making a Pixar film. Talk about seeing math in action! Each section lists an appropriate grade level so you can see what level of math is required to understand the material, and there are sections for everyone from “all ages” through high school. As a grad student, even I’m having fun! You can use code to create a character, learn the math behind animating curves, or use combinatorics to create a crowd of robots! This is an awesome way for students to see math being used to create something rad, and they get to try it on their own. NOTE: All Khan Academy content is available for free at www.khanacademy.org.

Some food for thought: I like math because I find it intrinsically interesting, not because it’s used in a wide variety of professions. Will Pixar in a Box help foster an interest in mathematics for the sake of mathematics? I don’t know the answer, but I would argue that more positive exposure to the subject can’t hurt.

Next time I’m asked “who uses math in real life?” I’m going to direct them straight to Pixar in a Box to experience firsthand how math is used in animation. Maybe this doesn’t answer the question, “When will I ever need the definition of a derivative?” but I think that by seeing how animators use elementary, middle, and high school level math, students may start to be convinced that all sorts of professions benefit from mathematics.

Do you have other resources for students to experience mathematics? I’d love to know what they are, so leave a comment below!

]]>COMPUTING PI VIA BILLIARDS

Special thanks to Y. Fan, who communicated the following problem.

Let’s consider two billiard balls of equal mass constrained to move along the real axis. Suppose that Ball A is initially stationary and is located at the point *x* = 1 on the real axis, while Ball B is traveling towards it from the right in with some fixed velocity. Suppose moreover that there is a wall at *x* = 0 off of which the billiard balls reflect. We will of course assume that there is no friction, and that the billiard balls are point masses which are perfectly elastic.

In this simple situation, it is evident that there will be a total of three collisions amongst the two billiard balls and the wall. First, B strikes A, resulting in A moving to the left (with the same velocity as B originally had), and B becoming stationary. Then A reflects off the wall, reversing its velocity. Finally, A strikes B from the left, resulting in A becoming stationary and B moving off to the right. In particular, note that the number of collisions is independent of the masses of the billiard balls (given that they are equal), and the incoming velocity of B.

Now suppose that the balls are *not* of equal mass, but rather that B is significantly more massive than A, say 100 times more massive. Then after B initially strikes A, A will move rapidly to the left, while B will still be moving to the left with a slightly diminished velocity. As before, A then reflects off of the wall and strikes B, which slows down B slightly. After enough cycles of A reflecting off the wall and striking B, B will start to move to the right, and it is easily seen that eventually B will be moving to the right with sufficient speed so that no more collisions of any kind occur. In order that we do not have to worry about B itself getting too close to the wall, let us pretend that every so often we pick up the wall and move it to the left. (This does not affect the velocities of the balls.)

We can now ask the question: in total, how many collisions occur amongst the two billiard balls and the wall? It turns out that if B is 100 times as massive as A, there are (perhaps up to plus or minus one extra collision), 31 total collisions (regardless of the incoming velocity of B and the numerical values of the masses of A and B). If B is 10000 times as massive, it turns out that there are approximately 314 collisions! We thus have the following:

Prove that if B is 10^{2n} times as massive as A, then the total number of collisions within our system of billiard balls is approximately π times 10^{n}.

No advanced physics is needed to solve this problem – you can get by with only the conservation of energy and the conservation of momentum!

HINTS FOR THE BILLIARDS PROBLEM

Breaking with tradition a bit, I thought I would post a hint for the billiards problem, since it’s a bit more computational than the previous problems I have posted. There aren’t any other riddles in this post, so if you don’t want to read the hint, you can turn off your computer now and go frolic in the sun. Even given the hint, there’s still some work, but it may be helpful for getting and overall idea of the solution:

Let’s represent the velocity of A (*v _{A}*) and the velocity of B (

Almost anyone in the math world will know that today is π Day. Every year on March 14, the date reads 3/14, the first 3 digits of our favorite constant. It may be an irrational holiday, but it is about the best that mathematicians get, and it seems to get more popular each year. Some of my Calculus II students discovered my favorite new way to write π on their final exam today. I found this by chance as I was trying to come up with a suitably difficult problem for them to solve. Here it is:

The function inside this integral has a wonderful graph shown below, and the integral above represents the area under the graph on the right side, stretching all the way to infinity– an extremely long and thin slice of π. (If you want to solve this integral without any hints, read no further!)

As Calculus II problems go, this one is a monster. First, it is improper at both limits of the integral (it is undefined at 0, and ∞ is always improper), so technically you should to split it into two integrals and take a limit of the bounds at either end. To make things worse, you have to integrate by parts, which may not be obvious at first. I’ll let you work it out on your own, but you will end up with exactly π, thanks to an arctangent function showing up along the way. I doubt I’m the first to find this integral, but I don’t yet see it on Wolfram’s list of integral representations of π (feel free to send me a check when you update this, Wolfram).

While you are working on this integral, I hope you are enjoying some pie as well (which you might find being sold for $3.14 today).

Happy π Day!

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