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