Apparently today, 4/4/16, is Square Root Day. (I supposed we could also have celebrated 4/2 to have a long Square Root Weekend.) How should a math enthusiast celebrate this holiday, which won’t come again until May 2025?

Of course, one option is to be a curmudgeon. As a curmudgeon myself, I heartily support you in this endeavor, and Michael Lemonick of Scientific American does as well. If, however, you are not as cold and dead inside as he and I are, I’ve got some lovely square facts and activities for you.

I’ve got to lead off with one of the coolest things I learned last month, courtesy of Matt Baker.

Let n be a positive integer. It is easy to see that a square can be dissected into n triangles of equal area if n is even (Exercise). What if n is odd? If you play with the question for a bit, you probably won’t be surprised to learn that in this case it’s impossible. But you may be surprised to learn that this result was not proved until 1970, that the proof involved p-adic numbers, and that no proof is known which does not make use of p-adic numbers!

I find it shocking and delightful that a fairly simple question about plane geometry requires p-adics to solve. If, like me, you’re a bit uncomfortable with p-adics, cut-the-knot math has a p-adic page where you can learn about these strange completions of the rationals.

A more traditional way of celebrating might be to ponder the wonderful Pythagorean theorem. Cut-the-knot has more proofs of it than you can shake a stick at, and I’m still enchanted by Albert Einstein’s elegant proof, which Steven Strogatz wrote about last November.

Arts and crafts have many opportunities for square-making. One of my favorite square-centric designs is a curve of pursuit: out of tilted squares, curves seem to appear. The excellent mathematical knitting site Woolly Thoughts has some information about how to knit curves of pursuit. A few years ago, my grandparents celebrated their 8^{2}-th anniversary, so I made them a tablecloth with 8 squares arranged in a curve of pursuit. You can see it in the picture at the top of this post or read about it here.

Last year, Math Munch shared a project called SquareRoots by John Sims. Inspired by the quilts of Gee’s Bend, he made mathematical quilts based on the base 3 digits of pi. And for more mathematical quilts, check out the gorgeous ones on Elaine Ellison’s website.

If you want to take your square explorations up a dimension or two (because who has time to wait until 4/3/64 or 4/4/(2)256?), Numberphile has a lovely video about higher-dimensional Platonic solids, including the hypercube, and Mike Lawler has been making hypercubes with his kids.

It’s baseball opening day today as well (√ √ √ for the home team!), and Patrick Vennebush of Math Jokes for Mathy Folks has combined the two topics for a guess-the-graph game.

How will you be square today?