Happy Deep Fried Torus Day

Last Friday was National Doughnut Day and I hope everyone celebrated accordingly. Aside from being delicious with a cup of coffee, these sweet treats are a great way to think about math. From simple brain teasers to advances in quantum teleportation, this shapely pastry has a lot to offer.

How many doughnuts are in this picture?  What sort of things would you need to know to answer that question?

How many doughnuts are in this picture? What sort of things would you need to know to answer that question?

Let’s start out with something fun. The photo above was popping all over the internet this weekend, and it’s a great way for students to work on their understanding of spatial reasoning and geometry. The math teacher focused site YummyMath guides students through a doughnut inspired estimation exercise, and BedtimeMath gives a series of doughnut questions for all ages. Between doughnut flavors, different toppings, and doughnut holes, there are plenty of questions to be asked!

Of course doughnuts as math objects reached their cultural apex around 2006 with the solution to Poincare’s Conjecture, inspiring the popular joke,

A topologist is a mathematician who can’t tell a coffee cup from a doughnut.

Or its lesser known modification

How many topologists does it take to change a light bulb?
Just one. But what’ll you do with the doughnut?

courtesy of the blog MathJokes4MathyFolks. Even Stephen Colbert got in on the action, smooshing a doughnut on the Colbert Report.

But beyond the fun questions and smooshing of baked goods, the doughnut — or torus as we call it in the biz — plays a very important role in advanced mathematics and physics. The special shape of the torus means that it has a relatively large amount of surface area relative to its size. Compare the torus, for example, to a sphere or a cube. Tori are also special because when you rotate them around a central axis, every point moves. But wait, doesn’t that work on any 3-dimensional shape? Not so fast, think about the sphere-like planet earth, it rotates around the axis through the north and south pole, but this means that points exactly at the north and south pole don’t actually move. This torus is special in this way. If you rotate it around an axis through the “doughnut hole,” then every point moves.

This observation was critical in a recent development of new techniques in quantum teleportation. If you imagine sending encoded pieces of data from one torus to another, the large surface area means there are many more points on which to place the data bits, and the nice axis of rotation makes these pieces easier to encode. The math blog and community forum Mathesia gives a nice down-to-earth explanation in donuts, math, and superdense teleportation of quantum information.

Ok, now I want a doughnut.

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