eli5:locally euclidean vs euclidean what is the difference in layman’s terms?
In: Mathematics
Consider the geometry fact you were taught in grade-school of *”The angles in a triangle add up to 180-degrees.”*.
That fact is *actually* only fully true for Euclidean spaces like the xy-plane (or xyz-coordinates, etc.); it doesn’t necessarily apply to non-Euclidean spaces.
The most straightforward (counter-) example is drawing a big “triangle” on a globe: start at the North Pole and head south until hitting the Equator, turn 90-degrees and travel straight along the Equator for 1/4 of the way around the globe, then turn 90-degrees and travel back to the North Pole, when you get there you’ll see that there is a 90-degree angle between your original path south and your latest path north… but that’s 90+90+90=270-degrees! The three local angle measurements were done properly, but the “Triangle Sum Theorem” just clearly doesn’t apply globally. But what about “locally”? If you “zoom in” to where you’re standing on the surface of the Earth, you could totally draw triangles on the imperceptibly-curved pavement with some chalk where the angles would add up properly to 180-degrees.
So, the surface-of-a-sphere (like Earth) is non-Euclidean because it doesn’t technically preserve rules like “Triangle Sum Theorem” at the full global scale, but at the same time – if you zoom in enough – things locally start to behave properly and Euclidean-like again.
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