Gabriel’s Wedding cake.

We now post a mathematical puzzle and ask you all to make sense of it. A little bit of Calculus is needed to do the computations, but resolving the puzzle probably only requires persistence and ingenuity.

So what is Gabriel’s wedding cake? Is it this?

cake

Well sort of. Let us begin by defining a function over the set of numbers from 1 to infinity. Call it f(x). f(x) = 1 on [1,2], f(x) = 1/2 on (2,3], f(x) = 1/3 on (3,4] and so on; f(x) = 1/n on (n, n+1]. This is a step function where the length of each step is 1 and the steps descend by smaller widths going to zero. Now rotate this function about the X-axis to generate the “cake” i.e., an infinite stack of cylinders, each of height 1 but radii 1/n as n goes to infinity. It looks something like this:

cake2

Ok, so what’s the big deal? Well from Calculus one can compute the volume and surface area of this curious, infinitely stacked cake. The volume is just an infinite sum of numbers, Pi*r^2*h, where r = 1/n and h = 1. So we are basically adding up Pi times the sum of the squares of reciprocals of natural numbers. This famous sum is the value of the zeta function at s=2 and Euler showed it is equal to Pi^2/6. So the volume of the cake is Pi^3/6 or just under 5 cubic units.

On the other hand the surface area, which includes parts of the top and all of the sides works out to include the harmonic series (the exact expression is left to you) which is the infinite sum, 1 + 1/2 + 1/3 + 1/4 + … which we know diverges to infinity. So there you have it: finite volume with infinite surface area!

In other words: a cake you can eat, but cannot frost!! How do we explain this paradox?

Spotlight: Elizabeth Pacheco, Department of Mathematics, OU.

We continue our Spotlight series where we highlight one of the member of the OU Math Department. This edition of Spotlight shines a light on one of our most accomplished graduate students, Elizabeth Pacheco

photo

Short bio (in her own words): Elizabeth is a fifth year graduate student in the math department at OU. She has a B.S. in Mathematics from the University of North Texas. She is studying (configuration spaces and Leavitt path algebras) under the guidance of Dr. Murad Ozaydin.

The blog caught up with Elizabeth to get her thoughts on mathematics, graduate study, cats and other important topics.

1. What made you decide to pursue mathematics?
I realized that the answers at the end of any math problem were far more satisfying to me than any other subject. Also, math was (and is) pretty fun – I’ve always enjoyed puzzles.
2. What are some of the rewarding things about being a mathematician?
I like how humbling mathematics is. I’ll never know all there is to know, which is both depressing and freeing.
3. What are your other interests besides mathematics? Favorite band? Snickers or M&Ms?
I like games – card games, computer games, board games. I also enjoy teasing my very grumpy cat, Schrődey. My favorite band – Queen. And definitely M&Ms – I’m allergic to peanuts.
4. Who is your favorite mathematician and why?
Although, he’s pretty popular, I have to go with Erdős. I like the problems he would come up with and/or popularize. He strikes me as a bit of a math rock star, and I like his belief in “the book.”
5. Discuss some of the challenges students face in graduate school and your suggestions to overcome them.
The struggles I notice as are feeling as if you know noting and the difficulty of learning new things. I’m still in the midst of overcoming some of them. What has helped me somewhat is taking to professors or fellow graduate students – it helps you gain clarity when you are discussing your confusions, or solidify your understanding if you are teaching a concept.
You should also attend talks – it’s sometimes surprising to realize how much you have learned by attending talks, even if you don’t understand everything of what is said.
Also, for me, playing with my cat is a big stress relief. No matter how you slice it, you are never alone in this.
6. Describe “a day in the life of a mathematician” as it applies to you.
My schedule is rather fluid, but the morning and afternoon are usually spent on classes – both those I teach and attend, and the evening is spent on reading and trying to understand papers.
7. Tell us, if possible, a little of what is your research area and what you are currently working on.
Dr. Özaydin and I used to talk about configuration spaces a lot, but right now we are discussing Leavitt path algebras. A brief explanation of this would be: take a graph, and consider the algebra of paths on this graph (elements are formal linear combinations of paths with coefficients on some field and these elements interact under addition and concatenation of paths, where two non-incident paths concatenate to be zero). You can ask questions about properties of the algebra you just created and/or you can use the construction to provide examples of rings with interesting properties, such as rings of a given basis type.

Summer 2015 REU (Research Experience for Undergraduates)

The blog is back!

We start with an announcement for students to apply to an REU opportunity at San Diego State University. Students spend a summer conducting research and are paid a stipend of $5000 for the whole summer while working with enthusiastic, like minded fellow students from around the country.

Deadline: March 6th.

More information here:

http://www.sci.sdsu.edu/math-reu/index.html