eli5:locally euclidean vs euclidean what is the difference in layman’s terms?
In: Mathematics
It’s why Flat Earther’s can think the Earth is flat. If you’re close enough to the surface, or if you zoom in enough, then the Earth is basically indistinguishable from a flat plane. You need to zoom out, or traverse large distances in order to have non-euclidean structures like curvature to begin to matter.
You may know that there is no 2d map of the Earth that preserves all the geometry — the Mercator projection, for example, ends up enlarging places like Greenland compared to Africa. To solve this, we often use atlases, a big book of local charts. Each chart is of a much smaller area, so the distortion is not as great. Also, because each chart doesn’t “wrap around” like the globe does, we can distort it to a 2d projection without doing any cutting. This is what we mean by “locally Euclidean” — the globe is not Euclidean (flat space) but each local area (chart) does look Euclidean.
In fact this is essentially the definition of a manifold in mathematics. The globe is a 2d manifold because each chart looks like 2d Euclidean space. Spacetime is a 4d manifold because, even though we have local curvature, each chart looks like 4d Euclidean space (3 space dimensions + 1 time).
The definition of “looks like” Euclidean space has a precise mathematical definition, but roughly speaking it is some invertible transformation that doesn’t involve any cutting.
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.
Locally Euclidean would just mean that the geometry behaves like a flat surface on a small enough scale. The further you “zoom out” the less the geometry behaves like a flat surface.
Imagine two people, both at the equator but separated by hundreds of miles, each get into their own car. At the same exact time they both drive North, parallel to one another. During the entire trip both drivers drive in a straight line, meaning they never once turn the wheel of the car. Once they reach the North Pole they crash into one other. They’re both very confused – they both started parallel to one another and both of them drove in a straight line, never once turning their wheel. Yet they still crashed into one another at the North Pole.
That’s because the Earth isn’t flat, it’s curved. From the * drivers perspective* they don’t notice the curvature and so the geometry *behaves* like a flat surface on a small enough scale. But as we “zoom out” and get a bigger picture we can begin to see the curvature and so the geometry behaves differently. Two straight parallel lines will converge on a curved surface, whereas they’d never do that on a flat surface. But if we zoom in close enough to any one section of the line then it *looks* and *behaves* like it’s flat.
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