[Rendello]

One of my favourite games of all time is called Knossu and is also non-Euclidian (edit: this is wrong, see comment below), although it takes a very different approach to confusing the player. The author, Jonathan Whiting, is probably best known for his article "Why I Write Games In C" around here

It seems like the best games throw you in with no context, and let you explore their consistent but mysterious mechanics and worlds

1. https://jonathanwhiting.com/games/knossu/

2. https://jonathanwhiting.com/writing/blog/games_in_c/

[zenorogue]

Knossu is cool but not non-Euclidean... It changes topology but its geometry remains Euclidean. Weird connections and non-Euclidean geometry are two very different things.

[nine_k]

I wold say that being Euclidean requires very plain topology, an equivalent of a plane. Else you can't get Euclidian distance, for example.

By this token, classic Doom is already non-Euclidean because of teleports.

What people often want when they ask for non-Euclidean is a curved space, like, well, the surface of a globe. Or maybe a torus. Or space being infinitely repeated while looking flat (also much like a torus). Some kind of hyperbolic geometry could be fun but likely too mind-boggling.

(The ultimate in non-Euclidean worlds I've seen so far in games is the paradoxical space of Monument Valley.)

[zenorogue]

Geometry is the curvature of the fabric the space is made of.

Topology is how you sew/glue this fabric.

You can make a cylinder out of flat paper, so it is an Euclidean manifold.

You cannot make a sphere out of flat paper, so it is a non-Euclidean manifold.

Non-Euclidean geometry refers specifically to the geometry, not to the topology. It is not "anything where the distance is not the Euclidean distance" or "anything other than the Euclidean space" or "anything not related to Euclid's proof that there are infinitely many primes". Such a concept would not be useful, because it would be so broad that nothing interesting could be said about it (as you said, it would include Doom levels because of the teleports).

So a cylinder or a flat torus or a space being infinitely repeated are not non-Euclidean geometry. Monument Valley also has no relation to non-Euclidean geometry.

[nine_k]

I agree that you can glue flat paper in a way that connects disparate points without making it spherical and thus breaking the Euclidean properties: parallel lines, sum of angles in a triangle, distance, except for the teleport points. You can draw two lines parallel to a given line through a given non-teleport point: one passing through a teleport, another not passing.

Since the very notion of a line or a distance in the Mountain Valley world is ill-defined, I would not call it Euclidean either, even though each static configuration of it may be Euclidean, minus doors (which are teleports again).

[zenorogue]

Usually "non-Euclidean" is used for Riemannian manifolds, so if something is not a Riemannian manifold, it is not called Euclidean or non-Euclidean.

Just like "irrational number", which is not any number that is not rational -- we still assume that it is a real number, so i or Aleph-Null are not considered irrational.

If we include teleport points in our metric, we no longer have a manifold (if teleports are one-way, not even a metric space). Similarly for Mountain Valley, or affine/projective manifolds that can be found in some games, it is wrong to call them Euclidean or non-Euclidean.

Anyway, originally "non-Euclidean" was just the hyperbolic plane (the only geometric structure which satisfies all the Euclid's axioms except the fifth), and then (depending on whom you ask) it was extended to also mean other situations which are different but similar in nature -- so spherical geometry (which is just the opposite), Nil/Solv geometry (which also play with parallel lines), but not cylinders, affine manifolds, teleports, taxicab metric, and so on (they don't really do anything interesting to parallel lines).

[nine_k]

Thanks, that's informative!

Games on "spherical", or rather, "polar" surfaces are well-known: space flight around a gravitating body, along circular geodesics.

A shooter in a hyperbolic space, and with gravity, could be hilarious.

[Rendello]

That makes sense, I'd only seen the term in reference to similar games before, I guess I'm not the only one online that was mistaken!

[zenorogue]

Indeed, it seems that Antichamber called itself "non-Euclidean" and people did not know what it means and called all games which did something weird to their spaces got called non-Euclidean. We have lots of cool truly non-Euclidean games in development today, so hopefully they will understand it better :)