# Random Walks Mural

I’ve been meaning to give the back wall of my classroom a makeover for a while. This summer I finally found some time to tackle the big project. I took down all the decorations and posters. I fixed up the wall and painted it a nice tan color. Then, I let loose the randomness!

I struggled with what the new mural would be–I’ve thought about it over the last few years. I considered doing some kind of fractal like the Mandelbrot Set. But it should have been obvious, given the name of my blog!! What you see in the picture above is three two-dimensional random walks in green, blue, and red. In the limiting case, one gets Brownian motion:

Brownian motion of a yellow particle in a gas. (CCL)

I honestly didn’t know what it was going to look like until I did it. I generated it as I went, rolling a die to determine the direction I would go each time. I weighted the left and right directions because of the shape of the wall (1,2=right; 3,4=left; 5=up; 6=down). For more details about the process of making it, here’s a documentary-style youtube video that explains all:

Actually, I lied–it doesn’t tell “all.” If you really want to know more of my thought process and some of the math behind what I did, watch the Extended Edition video which has way more mathematical commentary from me. I’ve also posted the time lapse footage of the individual green, blue, and red. Just for fun, here’s an animated random walk with 25,000 iterations:

A two-dimensional random walk with 25,000 iterations. Click the image for an animated version! (CCL)

I think the mural turned out pretty well! It was scary to be permanently marking my walls, not knowing where each path would take me, or how it would end up looking. At first I thought I would only do ONE random walk. However, the first random walk (in blue) went off the ceiling so I stopped. And then I decided to add two more random walks.

In retrospect, it actually makes complete sense. I teach three different courses (Algebra 2, Precalculus, and Calculus) and I’ve always associated with each of theses courses a “class color”–green, blue, and red, respectively. I use the class color to label their bins, to write their objective and homework on the board, and many other things.

The phrase “Where will mathematics take you?” was also a last-minute addition, if you can believe it. There just happened to be a big space between the blue and red random walks and it was begging for attention.

What a good question for our students. The random walks provide an interesting analogy for the classroom. I’d like to say I’m always organized in my teaching. But some of the richest conversations come from a “random walk” into unexpected territory when interesting questions are raised.

Speaking of interesting questions that are raised, here are a few:

• Can you figure out how many iterations occurred after looking at a “finished” random walk? Or perhaps a better question: What’s the probability that there were more than n iterations if we see m line segments in the random walk?
• Given probabilities $p_1, p_2, p_3, p_4$ of going in the four cardinal directions, can we predict how wide and how high the random walk will grow after n iterations? Can we provide confidence intervals? (might be nice to share this info with the mural creator!)
• After looking at a few random walks, can we detect any bias in a die? How many random walks would want to see in order to confidently claim that a die is biased in favor of “up” or “left”…etc?

Some of the questions are easy, some are hard. If you love this stuff, you might be interested in taking a few courses in Stochastic Processes. Any other questions you can think of?

Where will math take you this coming academic year? Welcome back everyone!

# The Mathematics of Juggling and more from George Hart

[Dr. Chase guest blogging again]

You’re probably familiar with Vi Hart’s math videos. Less well-known are her father’s math videos. Although I was aware of his mathematical sculpture, I was not aware until today that since August 2012, he has been producing a mathematical video series called mathematical impressions for the Simons Foundation. The 10th in the series is The Mathematics of Juggling. Check it out!

# Fearless Symmetry

I come to you today with a recommendation for the book Fearless Symmetry by Avner Ash and Robert Gross. I started it this summer and finally had a chance to finish it over the Christmas break. I didn’t understand the last half-dozen chapters, but my dad did warn me that would happen. I wouldn’t even attempt reading it unless you’ve already been exposed to some undergraduate mathematics. But if you have, or if it’s been a while and you need a refresher, I highly recommend the book.

In the book, Ash and Gross attempt to explain some of the math underlying Wiles’ proof of Fermat’s Last Theorem. So you can understand why the math gets a bit hard at the end.

Along the way, you’ll get a very conversational, well-written, fun-loving introduction to the Absolute Galois Group of the Algebraic numbers. This is a group that is so complicated and messy and theoretical that we can only explicitly write down two elements of the group. In order to talk about it, we need representations, which the authors also introduce in a gentle way. In particular, we need linear representations.

Elliptic curves become very important too. I have studied elliptic curves in two of my classes before, but I really liked the way they introduced them here: We know everything about linear equations (highest exponent 1), and everything about conics (highest exponent 2 on x and y), but suddenly things become very interesting when we allow just ONE of the exponents (on x) to jump to 3. These are elliptic curves. Amazingly, you can define an arithmetic on the points of an elliptic curve that yield both a GROUP and an algebraic VARIETY. Incredible. Of course, the authors introduce what a variety is too.

After reading this, I also gained a much bigger view of abstract algebra–a course I’ve taken, but I found myself guilty of seeing the trees but not the forest. I loved the way Ash and Gross introduce the group SO3 and relate it to A4 with the rotations of a sphere inside a shell. Very nice visualization!

I could go on, but just know that there are lots of little mathematical gems scattered throughout this book. It’s a refreshing jaunt through higher-level mathematics that will demystify some of the smart-sounding words you’ve been afraid to ask about :-).

Go check it out!

# For Reals

A few people have pointed me to this mathy web comic:

Thanks to smbc-comics for a great mathy web comic! (http://www.smbc-comics.com/comics/20120517.gif)

I’m not sure how often discrete mathematics uses the phrase “for reals”….I would think “for natural numbers” would be more appropriate, don’t you?