Snakes on a Projective Plane

Snakes on a Projective Plane - A game

This is a game that I made for fun and maybe to help people understand projective spaces, and identification spaces.

The idea is to take the classic game of snake which (you may know) takes place on a torus, and add a twist (literally) so that you can play on a Klein bottle or on a projective plane. When you go off the top, you won't just end up on the bottom, but you will end up on the other side.

Math Behind the Game

Torus

If one wanted to make Snake without walls, an obvious way would be to connect the top to the bottom and the right to the left. This is the way the original Pac-Man is played. This however is how a topologist likes to think of a donut, or torus.

To see how this would make a donut, take a piece of paper and try to stretch it to make this identification space.

Higher Dimensions

This type of construction is called an identification space. It gives us an easy way to make new shapes that cannot be fully realized in only 3 dimensions. When going off the left side, if instead of coming onto the right side in the usual way, we twisted the sheet of paper first, we would come onto the right side but now going the other direction. (i.e. NW would become SW from the point of view of the identification space).

These various spaces are easily constructed from changing our identifications. They produce the Klein Bottle and projective space.

Something strange happens with these new spaces that did not happen before. Direction changes, left is now right and right is now left, but why? The answer has to do with one-sidedness. The snake has moved the "back" of the identification space as we view it. If we imagine ourselves standing where the snake is, what we view as left and right would be opposite of what somebody staring at the screen would view as left and right.

How did the snake get here? Unlike the snake on the torus, who only ever stayed on the "top," the snakes on these new spaces don't have a "top" and a "bottom;" they are one-sided. From the point of view of the snake, there is no difference between the two sides of the identification space. When we look at it, we must keep in mind that we are looking at what is just a convenient representation of the actual surfaces. There is no actual concept of back.

Analogs

The snakes in this game sit on a surface. To the snake, these surfaces all just look like planes. It's not until the snakes get really long and wrap into themselves that they should suspect that they're not in a plane, but on some funny surface. Mankind made a similar discovery when we realized the world is round.

Similarly as humans in a vast space, we shouldn't suspect that the universe extends forever in all directions without wrapping. In fact, modern physics would suggest the exact opposite.