What is Non-Euclidean Geometry?
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An example of non-Euclidean geometry is the geometry of 2d objects on the surface of a globe.
We are introduced to geometry (nearly always) by assuming that the 2d objects exist on a flat plane. In this plane, internal angles of triangles add up to 180 degrees and parallel lines never meet. (The parallel lines thing is Euclid’s fifth postulate – ELI5) From here we develop things like cartesian coordinates. Distance can be measured using Pythagoras.
Non-Euclidean geometry abandons the parallel postulate and imagines geometry (can be 2D, 3D etc) in curved spaces. It introduces the concept of curvature (which is a measure of non-flatness)
It’s essentially geometry on a curved surface, where, for example, parallel lines can meet. It started when mathematicians attempted to prove Euclidean axioms by assuming they weren’t true and figuring out what that would mean for geometry.
But it became relevant in the real world when Albert Einstein proved that over astronomical distances space itself is curved. Suddenly over astronomical distances space was non-Euclidean. Parallel lines could meet, etc.
Starting at the North Pole, walk due south. Once you hit the equator, turn 90 degrees to the right and walk west for as long as you want. Then turn 90 degrees to the right again and walk north.
In Euclidean gemoetry you will never reach your starting point. In non-Euclidean geometry, youll end up back at the North Pole.
This is also how we know the Earth is not flat.
You know how a black hole distorts light? It doesn’t actually. Light goes in a straight path, black holes distort space itself meaning the path doesn’t look straight anymore.
That is non euclidian geometry. Space that isn’t straight.
Euclidean geometry is based on 5 unprovable truths called Postulates. In basic modern English, they are:
1. You can draw one straight line between any two given points.
2. You can infinitely extend any given line segment in a straight line beyond either end.
3. You can draw a circle given a center point and a given radius.
4. All right angles are equal to each other.
5. If two lines cross a third, the two lines, if extended, will eventually cross each other on the side of the third line where those two lines make angles smaller than right angles. (Or, two lines that cross a third at right angles are infinitely parallel.)
Non-Euclidean geometry discards or alters at least one of these 5 postulates. Usually the 5th.
Elliptical geometry, like that on the surface of the Earth, allows for parallel lines to cross. You can see this by looking at a globe. Any two lines of longitude are at right angles to the equator, but cross at the poles.
I feel like these comments aren’t really ELI5 so here’s my take:
Take a piece of paper and draw two straight lines that cross in a 90° angle.
even If you had a really really big piece of paper those lines will never meet twice, right?
now take a balloon and pick a point on it.
From that point draw two small lines at a 90° angle, then carefully extend them around the balloon.
But now the lines meet for a second time at the other side of the balloon! How can two straight lines meet at two different points??
It turns out that it depends on the surface on which you are drawing, and if the surface isn’t flat (or euclidean) the rules can be different than if it was.
Essentially it’s the illusion of straight lines on surfaces that aren’t actually flat.
You go South a 1000 miles from the northpole, west 1000 miles and north 1000 miles again. That’s two 90 degree angles and a far far smaller angle.
It doesn’t add up to the 180 of a triangle despite having three sides.
In reality if you drew your route on a map, the lines are curved.
This behaviour gets way more complex when you look at shapes that are more complex than spheres.
Geometry on a curve, so basically all of the manmafe stuff that you see. Lovecraft was using it as a curse because he was shit at maths
Euclidean geometry is based on several foundational principles. One of these principles is that corresponding points on two parallel lines will always be the same distance from each other, regardless of how far up or down the line you go.
Non-Euclidean geometry emerged from people being dissatisfied with how awkwardly this was worded and wanting to see how math would work if you changed this rule.
Just in case this is in response to the colloquial “non-euclidian” that gets used in some writing to describe impossible and mind bending spaces with paranormal effects.
It was popularized by HP Lovecraft in his cosmic horror stories to describe maddening chaos and extra-dimensional horrors.
But HP was just bad at math and didn’t understand how to use the term. Now it get’s used pretty commonly in that same fashion thanks to him.
Really it’s just geometry in 3-dimensions and it accurately describes the real and non elder god manipulated reality we all live in.
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