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Mode RP²
Controls: Arrow keys / WASD · Pause: Space · Restart: R

What is the Real Projective Plane (RP²)?

This game takes place on the real projective plane. You can build it by taking a square and gluing opposite edges together with a twist. When you do this with just one pair of opposite edges, you get a mobius band, a shape with just one side and one edge.

Square whose sides are identified with reflection, forming a mobius band Möbius_Strip

If you take the mobius strip and glue its one edge to itself (without adding a twist) you get a Klein bottle, a non-orientable surface with no edges, whose inside is the same as its outside, and has zero volume. This cannot exist in 3 dimensions without intersecting itself, but can be pictured as below.

Square with Klein bottle edge identifications Klein bottle image

And if you connect both top and bottom, and left and right, with a twist, you get the Projective Plane, which goes even crazier!

Square with all opposite edges are identified with reflection, forming the real projective plane Square with all opposite edges are identified with reflection, forming the real projective plane Square with all opposite edges are identified with reflection, forming the real projective plane

If you go through an edge in this game, you come out reflected in the opposite location, so that top right becomes bottom left and bottom right becomes top left. This is difficult to get used to, but that's the fun! You can also play in "Head-centred View", where you see the world from the snake's point of view, including seeing the reflected world through the portals, and the camera itself being reflected when you pass through portals. It is like stepping into a mirror world...