eli5:locally euclidean vs euclidean what is the difference in layman’s terms?
In: Mathematics
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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