Euclidean and non-Euclidean spaces?



Like I kinda know they have to do with actual space but what?

In: Mathematics

Euclidean space is flat; non-Euclidean space is curved.

For example, on a flat piece of paper, the angles inside a triangle always add up to 180°…but for a triangle across the surface of a sphere, the angles add up to more than 180° because the sphere is non-Euclidean, so it has different math.

Euclid made a few assumptions of space when we developed geometry. Four of these five have turned out to be pretty accurate and applies in every situation we can think of. However the fifth one does not always apply. Specifically it is the assumption that you can draw a line that is parallel with any given line. Most of the geometry you have been taught in school assumes this and is therefore classified as Euclidean. However it does not take much to get a non-Euclidean space. For example the surface of a sphere is non-Euclidean. And most of the geometry rules no longer applies. For example a sphere can have a triangle where the sum of the angles is more then 180 degrees.

Euclid described the mathematics of geometry using axioms, rules that are true by definition – because these are the rules that are what describe geometry. One of those axioms is the 5th axiom describes what are parallel and perpendicular lines.

There are subtle and interesting consequences to this rule. One consequence is that the sum of the angles of a triangle are equal to two right triangles (not 180 degrees, geometry doesn’t use numbers). But that has a “hidden assumption”, namely that you’re drawing triangles on a flat plane, like a sheet of paper.

What if you drew a triangle on the inside, or outside of a sphere? We draw on the outside of a sphere to navigate the Earth, we draw on the inside of a sphere to map the stars.

What happens then? The sum of the angles of a triangle are then more or less than two right angles. This initial assumption about the sum of a triangle wasn’t noticed for hundreds of years. And once it was realized, we started looking at what it meant to draw a triangle on all sorts of surfaces – cylinders, cones, wackadoodle blobs… All of Euclid’s geometry held for all these surfaces, and that was unexpected and profound. Enter the world of non-euclidian geometry, geometry that isn’t on a flat surface.

And from that we got abstract geometry, which we can use to map the geometry of one space to another space. And this is important because there are all sorts of mathematics that have underlying geometric principles – cryptography, random number generation, statistics, and physics are some maths that have geometric properties. For example, I was reading about some random number generation the other day that showed a weakness in one algorithm we used to use in cryptography – the damn thing produced numbers only on the surface of a 15-dimensional surface (don’t sweat it, don’t try to imagine what 15 dimensions looks like, mathematicians largely gave up on trying to visualize math after Riemann surfaces were described in the middle 1800s), not in that volume or above that volume. That means the production of random numbers was predictable, exactly what you don’t want in cryptography. It’s well above my pay grade that someone figured that out.

Maybe the easiest way to view it are parallel lines. If you have two parallel lines, is the distance between them always the same no matter where exactly you measure them? If so, then you have Euclidean space, otherwise you don’t.

Sometimes you don’t even have parallel lines at all, like on the surface of a sphere. Assuming the Earth to be a sphere, the equator is a straight line on its surface (it never goes left or right). All meridians are similarly straight lines. The latitudes other the equator are not straight lines – if you walk on the Earth so that you follow one of these latitudes, you’ll have to slightly curve towards the left or right to do so.