Ping-Pong is great, but there’s a catch: you have to play with someone else. What is a committed introvert to do?
You could always hit the ball against some fixed object, but as Mitch Hedberg observed about tennis (easily generalized to other hit-a-ball games), you’ll never be as good as a wall. Demoralizing! You could play a Ping-Pong video game, but what if your virtual opponent is too good or not good enough to have a good game?
Topology to the rescue! A crackerjack team of mathematicians has perfected Ping-Pong for introverts and misanthropes by letting you play against a perfectly-matched opponent: yourself. Students and professors at the Mathematical Computing Laboratory at the University of Illinois at Chicago created a VR program that lets you hit a ball with a racket in a three-torus.
The three-torus is the space created from a cube by “gluing” opposite sides together. The top is attached to the bottom, so if you climb up through the ceiling, your head will pop up from the floor. Likewise, if you walk forward through the front wall, you will emerge behind where you were, and if you sidle to your right, your right arm will start emerging from the left wall. If you play Ping-Pong in a three-torus, you can serve to yourself, but you have to keep your head on a swivel. The ball will come back to you from behind.
But explaining it that way obscures one of the space’s more disconcerting properties: if the space is small enough, you will look forward and see the back of your own head. In fact, you’ll see many copies of yourself in every direction. A dream for narcissists, perhaps, but possibly an unwelcome distraction, especially if you’re trying to figure out which ball to go for when you’re playing Ping-Pong against yourself. (You know how you always end up looking at the thumbnail of yourself on a video call? A three-torus would dial that up to 11.)
Watch a demo of Ping-Pong in the three-torus below.
I learned about VR three-torus ping-pong a week ago at the beginning of a semester devoted to mathematical illustration and visualization at the Institute for Computational and Experimental Research in Mathematics at Brown University, which provided funding for me to attend through a Sloan Foundation grant. Expect more stories about mathematics you can look at over the course of the next few months.