In an excellent Numberphile video, Katie Steckles (previously appearing on this blog here and here) tells us about the fold and cut theorem: any region with straight sides can be created by folding a piece of paper and cutting it out with a single straight cut. I'll let her explain.

To me, the fold and cut theorem is a perfect illustration of the peculiar laziness of mathematicians. We’ll work for hours to find a way around a twenty-minute computation, and in this case, we’ll fold paper for hours to avoid opening and closing the scissors a few extra times.

A single-cut square within a square, or the letter O in my fold-and-cut alphabet.

Towards the end of the video, in a virtuoso fold-and-cut demonstration, Steckles folds 26 pieces of paper from memory and produces the whole alphabet in 26 cuts. I spent some time this afternoon in my office folding a few letters of my own. There are some resources out there for fold-and-cut guidance, but I followed Steckles’s advice and tried to figure them out myself. I had some early successes, some pretty weird failures, and a lot more fun than if I had looked up the answers. Although my initial goal was to cut out the letters of my and my office mate’s names to put on our door, I ended up settling on just my initials for now. 

Making one-cut letters is a math activity anyone old enough to use scissors can enjoy. Mike Lawler and his two kids have been folding and cutting shapes during the past week, so if you’re a parent or teacher who wants to try it with kids you might get some ideas from them. I think I'll be incorporating one-cut letters into sewing projects for math friends, and I can only hope someone will create a font based on a fold-and-cut alphabet. 

Take that, four color theorem! One cut suffices.