This term seems to come up in science fiction every so often, and the layman description is that it’s just 2d geometry on a curved surface?
In: Mathematics
Euclidean geometry makes the following constructive assumptions.
* We can draw a line segment between two points
* We can extend a line segment as far as we want
* We can draw a circle if we are given a center point and a radius length
These amount to the things we can do with a compass and a straight edge. Note that we didn’t define the concept of a ‘line’ – this will be important. It also makes the following assumptions about the space in which you’re drawing.
* All right angles are equal to one another.
* **All triangles have internal angles that add up to 180 degrees**
These essentially define our geometry as happening on a flat plane. These imply that our ‘lines’ must be what we intuitively think of – a flat line on a plane.
Let’s do something wild and re-define what we mean by line. Instead of being a line on a flat plane, I will arbitrarily say lines are now great circles on a sphere – great circles being the largest circle you can draw. The equator is an example.
Do our constructive assumptions still work? If we draw two points, we can trace the great circle between them, and we can extend it. Our straightedge now has to be a hoop going around the sphere, so we can even construct it physically. If we start at a point we can draw a circle around it. Our compass needs an extending drawing tip, but that’s fine as well.
What about our assumptions about the space? A right angle is when two great circles intersect so that when you zoom in they form a right angle – these are definitely the same everywhere. However, we can draw [this bizarre triangle](https://qph.fs.quoracdn.net/main-qimg-65dd7fa64e7cd74cea6e7263708e2a97.webp). All the lines are straight, since they’re segments of great circles, so it’s a valid triangle – but the internal angles add up to 270 degrees! The bold axiom above does not apply to these kinds of lines.
This is an example of *a* non-Euclidean geometry – what we get by dropping the triangle postulate. You’ll also see it called the parallel postulate, because there’s a way of rephrasing it in terms of parallel lines. Intuitively, when we drop that postulate our space becomes curved. For any curved space, we can define ‘line’, ‘point’, ‘circle’, etc. in a way that makes the other postulates true, but breaks the triangle postulate.
There are of course other modifications to Euclidean geometry. There are geometries where not all right angles are equal to each other, though they must have more distortions than this kind of smooth curvature. There are a few more assumptions which I didn’t put in because they’re borderline common sense, but you can break those as well. These are all technically non-Euclidean, but they’re less common so you would normally specify that you’re breaking even more postulates than normal.
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