# The Mattress Group

Dr. Strogatz

Steven Strogatz is a professor of applied math at Cornell University.  In addition to doing lots of interesting math, he writes a column for the New York Times about mathematics.

His most recent article is about group theory in the bedroom (which is a lot less racy than it sounds!).  As you know, you are supposed to periodically rotate and/or flip your mattress that it doesn’t get squished flat from people sleeping in the same spot each night.  Dr. Strogatz discusses the fact that if you have a standard rectangular mattress, then there are a total of four possible positions that the mattress can be in (either upside-up or upside-down and either your head on one end or your head on the other end).  But, as Dr. Strogatz discusses, what is actually even more interesting is the actual operations which move the mattress from position to position.  The basic ones are to either rotate the mattress or to flip the mattress, and all the rest are some combination of these two.

Since if you do one of these operations and then another one, you get a third such operation, it turns out that the collection of all four of these forms a group.  Just by checking the various possibilities, it turns out that there are two groups with four elements:

$\mathbb{Z}_{4} \text{ and } \mathbb{Z}_2 \times \mathbb{Z}_2$

The first is the integers with addition modulo 4 (you might have learned it in Discrete Math as modular arithmetic) and the second one is the Klein Four Group.  So which one is the Mattress Group?  Well, in $\mathbb{Z}_4$ there is an element (the number $\bar{1}$) which you can do with itself (ie. add to itself) once, twice, thrice, or four times and each time get a different answer.  But it’s not too hard to see that in the Mattress Group, if you do any operation twice, the mattress is back to where you started!

via the NYTimes

So, by process of elimination, the Mattress Group is the Klein Four Group! (P.S.  Technically, it’s isomorphic to the Klein Four Group.)

You should definitely go to the NYTimes and read the whole article and Dr. Strogatz’s other interesting essay!  For example, check out this one on the Pythagorean Theorem which is closely related to the Primitive Integral Triangles (PITs) that Dr. Miller talked about last month in Math Club.

Thanks to Dr. Ravi Shankar for pointing out the article by Dr. Strogatz.  Dr. Shankar also pointed out that if you have a square mattress or a pillow top, then you get other interesting Mattress Groups.  Which makes us think that Dr. Shankar has a nicer bed than we do!