Euclidea: Winning Geometry

In the future, everything will be gamified. That’s the premise of my favorite dystopian video, Hyper-Reality. At times it may seem like we’re headed down that path, but in practice, a lot of life is pretty resistant to gamification. Life kind of sucks that way.


It’s not like it’s going to be possible to turn your math homework into a fun game. Or is it? I was intrigued this weekend to see that an app called Euclidea was a top puzzle game in Apple’s App Store. Euclidea is based on proving propositions in Euclidean geometry. The fact that they turned Euclid’s Elements into a highly rated iPad game reeled me in. I downloaded it. Sure enough, it’s pretty great. Can it deliver on the notion of being fun while helping you actually get better at geometry? I think it can.

For each level, you’re given a proposition to demonstrate (“Construct the tangent line through a point on a circle”). Your job is not only to provide the necessary demonstration, but to do it in optimal ways. This can be tricky. I was pleased with myself for solving one of the problems easily enough: inscribe a square inside a circle. But then they had the temerity to assert that my solution was bloated. In fact, you could find the solution with exactly seven elementary steps: either using a compass or a straightedge, but nothing else. Or so they said. I convinced myself that this was impossible.

Fortunately, I know how to play (cheat at) games in the modern age. YouTube will always tell you what you need to know, and after I beat my head against the problem for a while, I gave in. This video did the trick. Sometimes you cheat and you feel bad. I could have figured that out if I wasn’t so lazy. But sometimes you cheat and you realize that you’ve been schooled in something altogether new, something you were never going to discover on your own. This was one of those situations. I was impressed. Spoiler: Here’s what my screen looked like after I cheated.

Seven magic steps I never would have found without YouTube

This raises a question: if it’s so easy to cheat, if YouTube is always one click away, then who will bother to learn? This is exactly why gamification becomes so important. It gives you the story, the motivation to pay attention to the cheat video when it finally comes. On my own, I didn’t find the optimal solution to the inscribed square problem, but I watched carefully and was amazed when I finally saw it demonstrated. No other pedagogical approach would have glued me in place quite so thoroughly. I call this cheating your way to mastery. It’s a real thing, and it works. More than that, it’s a prominent feature of our age. Remember, the problem isn’t cheating. The problem is being sufficiently motivated to learn.

Incidentally, the idea of using modern graphics and UI to explore planar geometry has been around for a long time. One good example is the Geometer’s Sketchpad. Sadly, it was acquired by a textbook company, so it was promoted as an instructional tool rather than an explorational game. It never achieved the widespread audience it deserved. I hope Euclidea will go farther.