Euclidean and Non Euclidean spaces and geometry



What is the differences and what are the uses of non Euclidean geometry in the real world if there are any?

In: Mathematics

Spherical geometry: the earth is a globe(ish) so geodesics (shortest paths) come in handy when sailing and flying.
Hyperbolic geometry: curved space time was discovered to follow this geometry.
Taxicab geometry: getting around a grid city

Just to name a few.

Euclidean geometry has a couple of axioms that look different in non-Euclidean space. One of the key ones is about parallel lines – in Euclidean space, parallel lines remain at constant distance to infinity. In hyperbolic geometry, they diverge and the distance grows. The easiest example of non-Euclidian space is the surface of a sphere – imagine two lines that are parallel to each other in one point of the sphere. Extend them one the surface of the sphere…they will start coming closer until they eventually meet.

Euclidean geometry is the geometry that you learn in grade school: the rules are the same regardless of where you move/rotate yourself in a Euclidean space. Distances between points are simply the length of the line segment connecting those points, and so on.

Non-Euclidean geometry is anything that breaks those rules. (Portal is a good example, if you’ve ever played it: the distance between two points depends on whether you go through a portal or walk, and it’s possible to get back to where you started even if you never turned around by creating infinite loops with portals, etc.)

In the real world, the surface of Earth is actually non-Euclidean: if you start at the south pole, go 100 feet north, 100 feet east, and 100 feet south, you’ll end up back where you started (Euclidean geometry demands that you end up 100 feet east of where you started). I live in Colorado, and even though it’s a “rectangle,” the northern border is about 26 miles shorter than the southern, due to the curvature of Earth. Higher physics also involves non-Euclidean geometry, in the form of wormholes and such.

All math is based on a few rules, that are combined to make new rules.
If you don’t use all of those rules, you’re doing a different kind of math.

Euclid set 5 rules.

1. You can draw a line from any point to any point.
2. You can make a line longer as far as you like
3. You can draw a circle.
4. All right angles are the same.
5. That, if a straight line falling on two straight lines make the interior angles on the same side less than two right angles, the two straight lines, if produced indefinitely, meet on that side on which the angles are less than two right angles. IE parallel lines never meet and they never get further apart either.

Number 5 was much more complicated than the other 4, and this really bugged Euclid and math guys for thousands of years. They kept thinking it could be derived from rules 1-4 if put in the right order.
Turns out they couldn’t.

Most of non-Euclidean geometry decides to mess with rule 5.

So say instead of parallel lines never meeting either they meet eventually, or they split apart.

So how is this useful?
We’ll say you’re at a spot on Earth, and head straight north and just keep going, you’ll eventually cross the north pole.
Say another person at a different spot also heads north and just keeps going, he’ll eventually cross the north pole as well.
So the two of you had parallel paths, and yet your paths met, because the Earth is a globe, and not flat.

And that’s where non-euclidean geometry works. Euclidean geometry describes what geometry is on a flat surface. Non-euclidean geometry describes what geometry is on a not flat surface, for example: the globe we live on.

Think of a globe.

Put a pen on the North Pole of that globe and draw a line along the Prime Meridian down to the equator.From that point, drag the pen east along the equator 1/4 of the way around the globe. Now you’ve got a big “L” on the globe, a 90-degree angle back where the meridian and equator intersect in the Atlantic Ocean. You’ve come 90 degrees east into the Indian Ocean. Now drag the pen back up to the North Pole.

Now you have a triangle.

But you have a triangle with three right angles.


This doesn’t work in Euclidean geometry because Euclidean geometry begins with the premise that we’re restricted to the surface of a flat plane, and builds up rules from there.

Spherical geometry begins with the premise that we’re restricted to the surface of a sphere, and builds up rules from there. If you’re restricted to moving across the surface of a sphere, then the shortest path between two points is an arc on the surface of that sphere, not a straight line. A ray on a *plane* has a starting point and stretches off to eternity. The spherical equivalent of a ray would start at a point and come back around to where it started. I suppose there’s no meaningful difference between a line and a ray on a sphere since stretching infinitely far across the surface of a sphere in one *or both* directions will wrap around the sphere and make a circle.

But we can use either set of rules, planar or spherical, to build up a set of internally consistent rules that let us describe the universe.