I had a large collection of Legos as a kid, which my father saved and eventually shipped to me so our own kids could use them. One day, a math colleague of mine said, "Do your kids play with Legos? Because mine used to, but doesn't anymore, and we'd like to give the Legos to someone who'll play with them. We have two boxes." I said sure, we'll take'em.
Old saying: Mathematicians only know three numbers: Zero, one, and infinitely many.
Mathematicians tend to be literal minded about numbers (it goes with the territory). I neglected to ask "How big are the boxes?" As it turns out, they were two 45-gallon totes filled with Legos. We spent two weeks cleaning and sorting. Conservatively, our Lego collection quadrupled in size, overnight.
Now, I'm "old school" Lego. Back in my day, if you couldn't build it with bricks and flats, you weren't trying. Since the bricks and flats were all right angles, it was a serious challenge to make anything with a slope or a curve---but you could do it, if you tried. I'm convinced that playing with Legos helped me become a mathematician, since (if nothing else) I had to think about how things were put together from bricks and flats.
I was playing around with the pieces one day and thought "Hey, there are some interesting things you can do with Legos if you know math..." The pegs (or posts) of Legos occur in particular positions; consequently, every distance that can be spanned by a Lego brick must correspond to a whole number. In other words, Lego space is a Pythagorean paradise! No mathematician can resist that.
Since they solve some of the problems facing classic Lego builders (like making slopes or curves), and illustrate some really nice mathematical theorems, I thought it might be worthwhile to post them. I'm sure others have already figured out the solutions, but the alternative to putting this page together is doing some real work, so here goes.
Oblique Lines
Curves
Making a circle