The term "Non-Euclidean" has several meanings. Here are the definitions I will use.

A non-Euclidean space is a space that does not have Euclidean geometry.
But this can have several meanings, depending how strictly we interpret "Euclidean Geometry".

### Euclidean Manifold

1. We could interpret "Euclidean geometry" to mean "literally the Euclidean plane or Euclidean 3-space".
    Under this definition, games with portals could be considered non-Euclidean. But I think this definition is too strict.

A space is a *Euclidean manifold* if it can be covered by patches that act exactly like Euclidean geometry.

    Non-euclidean, under definition 1:
    - [*Portal*](https://en.wikipedia.org/wiki/Portal_(video_game))
    - [*Antichamber*](https://en.wikipedia.org/wiki/Antichamber)
    - [*Manifold Garden*](https://en.wikipedia.org/wiki/Manifold_Garden)

Video game examples:
- [*Portal*](https://store.steampowered.com/app/400/Portal/)
- [*Antichamber*](https://store.steampowered.com/app/219890/Antichamber/)
- [*Manifold Garden*](https://manifold.garden/)

2. A looser definition is that a space is Euclidean if for almost every point in the space,
    you can find a small region around it which acts exactly like a piece of Euclidean 3-space.
    This is the definition I use.

Sure, the areas in these games may connect to themselves in strange ways.
But if you look at a region of space that isn't large enough to self-connect, it just has ordinary Euclidean geometry.

    Under this definition, games with portals still count as Euclidean.
    Similarly for games like [*Asteroids*](https://en.wikipedia.org/wiki/Asteroids_(video_game)),
    where you can wrap around th edges of the screen.

### Locally Euclidean

    Non-euclidean, under definition 2:
    - [*HyperRogue*](https://www.roguetemple.com/z/hyper/)
    - [*Hyperbolica*](https://store.steampowered.com/app/1256230/Hyperbolica/)

A space is *locally Euclidean* if as you look at smaller and smaller patches, they act more and more like Euclidean space.

3. An even looser definition is that a space is Euclidean if as you look at smaller and smaller regions,
    they get closer and closer to acting like pieces of Euclidean space.

Consider the surface of a sphere. A piece of a sphere, no matter how small, is at least slightly curved.
But as you look at smaller and smaller pieces, they get less and less curved.
So the surface of a sphere is not Euclidean, but it is *locally Euclidean*.
Video game examples:
- [*HyperRogue*](https://www.roguetemple.com/z/hyper/)
- [*Hyperbolica*](https://store.steampowered.com/app/1256230/Hyperbolica/)
### Not even locally Euclidean
I don't have any examples of video games that aren't locally Euclidean. How about a story?
- [*Dichronauts*, by Greg Egan](https://www.gregegan.net/DICHRONAUTS/DICHRONAUTS.html)
This is a fiction story, set in a world with *Minkowski geometry*.
Since it isn't locally Euclidean, things work *very* differently.
Even something as simple as *turning around* isn't possible in this world!
The book can be confusing, precisely because the geometry is so unfamiliar. But it's the best example I have.
## Homogeneity and Isotropy
TODO
## A Digression on Spacetime
General relativity brings together all of these ideas.
Spacetime is non-homogeneous, non-isotropic, curved, and locally Minkowski.
And on top of that, to actually use general relativity, it's not enough to say that spacetime is curved.
You need to be able to describe precisely *how* spacetime is curved. No wonder GR is hard.

    This definition is too loose, so we usually give it its own terminology; we call a space with this property "locally Euclidean".

But *special* relativity is a different story.
You still need to understand Minkowski space. But it's homogeneous, isotropic, *flat* Minkowski space.
Once you really get your head around how Minkowski space behaves, special relativity is actually pretty easy.

    I don't have any examples of video games that aren't locally Euclidean. How about a story?
    - [*Dichronauts*, by Greg Egan](https://www.gregegan.net/DICHRONAUTS/DICHRONAUTS.html)

        - [The Easy Case](./non_euclid_raster.md)
            - [Projective Geometry](./projective.md)