Playing a game of ping-pong by yourself in the past typically involved hitting a ball against a wall or playing against a computer-generated opponent in a video game. Now, thanks to the Mathematical Computing Laboratory at the University of Illinois at Chicago, you can play against the perfect opponent: yourself.

Students and professors working in the lab created a virtual reality (VR) game in which you hit a ping pong ball with a paddle in a three-torus. This space is created by “gluing” opposite sides of a cube together: top to bottom, left to right, front to back. So, if you move through one face of the cube, you will reappear through the opposite side. Because of this, if you hit the ball forward through the front face, it will come back at you from behind with the same speed and trajectory at which the ball was hit.

To help understand what a three-torus is, let us first look at the two-torus. Instead of gluing opposite ends of a cube together, a two-torus is created by gluing opposite sides of a square together, creating a 3-dimensional object. A common representation of this is a donut. A three-torus, like a two-torus, exists one dimension up from its base shape — a cube — in a four-dimensional space. Thus, providing an accurate visualization of one is physically impossible. However, we can still express a three-torus in three dimensions, which looks something like the image found below. If the space is small enough, you will look forward and see the back of your body. Likewise, if you look to the side or up or down you will see yourself through the opposite face of the cube. This will create seemingly infinite versions of yourself, but it is important to note that the three-torus is a finite space. Regardless, playing ping-pong in a three-torus can be quite distracting, as you may struggle to figure out which ball to go for.

The original article did not mention if or when this new game would be marketed to the public for use. Regardless, this cool new innovation in VR technology allows us to play games like ping-pong by ourselves in a way not previously possible. In this new form of ping-pong, the only person stopping you from victory is yourself. The only question is: would you be up to the challenge?

**Top:** A three-dimensional visualization of a two-torus. **Bottom:** A three-dimensional visualization of a three-torus.

*An article by Joe Gyorda*

View the original article and a video of the ping-pong VR program in action:

https://blogs.scientificamerican.com/roots-of-unity/ping-pong-for-introverts/

More information about the three-torus and source for the image:

https://blogs.scientificamerican.com/roots-of-unity/a-few-of-my-favorite-spaces-the-three-torus/