*So far, all I've written is a description of the different kinds of unusual geometry.
I plan to write about the rendering some time in the future.*
## What is "Non-Euclidean"?
The term "Non-Euclidean" has several meanings. Here are the definitions I will use.
### Euclidean Manifold
A space is a *Euclidean manifold* if it can be covered by patches that act exactly like Euclidean geometry.
Video game examples:
- [*Portal*](https://store.steampowered.com/app/400/Portal/)
- [*Antichamber*](https://store.steampowered.com/app/219890/Antichamber/)
- [*Manifold Garden*](https://manifold.garden/)
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.
### Locally Euclidean
A space is *locally Euclidean* if as you look at smaller and smaller patches, they act more and more like 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.
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've thought a lot about how to render a four-dimensional world.
To a lesser extent, I've thought about how to render non-Euclidean worlds.
I plan to write about what I've learned.
Soon, I will have to split these into their own books. But for now, they're here.

TODO