The Mathematics of Ambigrams

To celebrate Math Awareness Month (April) this year, the Mathematical Association of America has created a page featuring a different fun math topic for each day of the month. I have April 13 this year, and my topic is Ambigrams. Ambigrams are words that read in more than one way — like WOW, which is MOM upside down. Only a few words are natural ambigrams, but if you stretch the letter shapes, almost any word can become an ambigram. I’ve been creating ambigrams since 1975, and have published many on my site. Here are a few of my favorites:

Ambigrams are a playful blend of math, art, psychology and language. What is so mathematical about ambigrams? You won’t find much traditional math here — no functions, numbers or equations. Nonetheless ambigrams involve a lot of mathematical thinking. Here are the top reasons mathematicians enjoy ambigrams, and so many math teachers use ambigrams in their classroom. 

1. Symmetry. Ambigrams are a good way to introduce students to the concepts of symmetry. Symmetry is an important concept in geometry, as well as in science and art. Many mathematical proofs rely on symmetry. Take a look at the three ambigrams above. Can you name the symmetry of each design? No two have the same symmetry. Many people do not have a clear idea of the difference between rotational and mirror symmetry; here the difference is crucial.

2. Curiosity. Ambigrams do something right in front of your eyes that seems impossible, just like a magic trick. Students who see ambigrams for the first time experience a sense of wonder and curiosity that entices them to learn more. Curiosity is a great way to get kids interested in learning more about mathematics.

3. Problem solving. Creating an ambigram is an exercise in problem solving that requires both sides of your brain. Try writing the word USA so it reads the same upside down. To do this you need to think analytically about mathematical symmetry. You also have to think artistically about how far you can distort letters before they become illegible.

4. Creative exploration. Ambigrams are also a great way to show students the creative side of mathematics. Math isn’t just about solving someone else’s problems. It’s also about asking your own problems, and getting curious about questions no one has asked before. The first time I created an ambigram I didn’t know whether it would be possible, or how to do it. The moment I created my first ambigram a whole world of creative exploration opened up to me, and I’ve been continuing to push its boundaries ever since. Every time I create a new ambigram I have a chance to try something I’ve never tried before. The thrill of exploring uncharted territory is what hooks mathematicians on doing original mathematics.