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 is based on a set of assumptions. Non-euclidean geometry is anything that doesn’t use those assumptions.
Most of the assumptions are about triangles. So think about drawing triangles on a paper. There are certain things that are true like the sum of angles and intersections and so forth. Most people take a whole semister in high school about triangles on paper. This is within Euclidean geometry.
Now imagine drawing triangles on an orange. You could have a normal triangle with three 90 degree corners so some of the rules change. There would be another set of rules for this space. Now imagine drawing triangles on a toilet paper roll. There would be yet another set of rules different than the first two cases here.
None of the three are wrong or incorrect, just different rules based on different initial assumptions. And you don’t have to use “curved space”, but curving space is an easy way to break the Euclidean assumptions.
Scifi usually points to non-Euclidean geometry as a way to work outside of normal logic. Similar to multiverse or some quantum phenomena. And by saying ‘non-euclidean’ they don’t ever have to actually specify. It’s a nice way out of the many paradoxes that scifi gets itself into.
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.
Not an expert, so feel free to correct me if I’m wrong, but one example could be railroad tracks. In Euclidean geometry, parallel lines never meet, but if you stand on the tracks and look down them (eg/ perspective (non-euclidean) geometry), they appear to converge despite being parallel.
Euclidean geometry: geometry on a flat thing.
Non-euclidean geometry: geometry on a curved thing. (At least the non-euclidean geometry that the most mathematicians find interesting)
We just so happen to live on a globe, not on a flat thing.
Notice how on flight maps the course between places isn’t a straight line, it’s a curve?
That’s because it’s a straight line on a curved surface, and someone just did some non-euclidean geometry to figure out where to go.
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