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Make Your Own Font, 1 Cut at a Time

Move over, four-color theorem! It's time for the one-cut theorem: You can cut out any shape you want with one cut if you're willing to do some clever folding

Single-cut initials. Like Katie Steckles, I find the "negatives" delightful.

This article was published in Scientific American’s former blog network and reflects the views of the author, not necessarily those of Scientific American


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.


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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.