Alice and Bob live on the modular curve \(X_0(1) = \mathbf{H}/\mathrm{PSL}_2(\mathbb{Z})\). What does the world look like to them, assuming that they view the world in hyperbolic perspective?
To those who are not used to hyperbolic geometry, there may be a few mild surprises. Suppose that Alice is at the point x=i and Bob is at y = 10i. Let us also imagine that Alice is looking in the direction of the cusp along the projection of the geodesic given by the y-axis. What does she see? Take a moment to think about it if you like; we will give the answer in the next paragraph.
Lifting Bob to the universal cover, there are infinitely many Bobs spaced equally along the horosphere (10i + t). A naive guess is that all of these Bobs would fill out Alice’s field of vision. But this can’t be true; since geodesics in \(\mathbf{H}\) are given by semi-circles perpendicular to the \(x\)-axis, most geodesics through x=i don’t cross Bob’s horosphere. In fact, Bob only takes up about \(10^{\circ}\) of Alice’s vision, and those Bobs who are at (10i + n) for large integers n appear almost to be directly in front of Alice (although a long way away). Of course, Alice also sees copies of herself receding similarly into the distance directly in front of her.
All this and more can be seen in the 80’s inspired video game of my undergraduate summer students Jasmine Powell and Justin Ahn (funded by the NSF!). The basic setup is as follows: you are a cube wondering around on \(X_0(1)\) and you need to shoot the monsters, which are in the shape of a pill. Occasionally, some bonus feature will appear (extra shields, freeze, extra life, etc.) which you can collect. Some mathematics that is hiding in the background but is only partially relevant for game play: the monsters travel along closed geodesics, and the goodies appear at CM points. The game was also partly inspired by the video not knot. Here’s a link to a video capture from the game:
(The transition to video has made it look a little wonky.) If you notice carefully, you will see that at one point in the video you crash into yourself by passing through the cone point \(i\), losing a life.
The alpha-release of the game itself can also be downloaded here (sorry, macintosh only). Please play around with it and offer suggestions and improvements! Various possibilities include upgrading to a 3-manifold (probably a Bianchi manifold), and also the ability to pass to congruence covers \(X_0(p)\) of \(X_0(1)\).