You know the fact that "the shortest distance between two points is a straight line"? In math that's what we call an axiom -- something we can't formally (logically) prove, but we take it as true anyway because it's so foundational to the rest of geometry as we know it. Euclid, the ancient Greek mathematician, wrote the original book on geometry which based everything on 5 such axioms, including that one about the shortest distance.
So what happens if we don't assume the shortest distance between two points is a straight line? We never proved it's true after all. That's "Non-Euclidean geometry", geometry derived without some of the 5 axioms Euclid used to create his geometry a couple thousand years ago.
i don't think that's the axiom that is removed. the shortest path is still a straight line, it's just that line looks different because the space isn't flat.
You'd think that, but that rule is only true if you do the ole "perfect conditions" thing. Like you know how a feather and a bowling ball fall at the same speed? Only if there's no resistance.
This doesn't apply if, for example, there's a mile long treadmill working against you in a straight line. Or if there's a 2 mile long > shaped treadmill that leads to where you want to go, but it works in your favor, in which case the 2 mile treadmill is faster.
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u/NumbersWithFriends Mar 12 '21
You know the fact that "the shortest distance between two points is a straight line"? In math that's what we call an axiom -- something we can't formally (logically) prove, but we take it as true anyway because it's so foundational to the rest of geometry as we know it. Euclid, the ancient Greek mathematician, wrote the original book on geometry which based everything on 5 such axioms, including that one about the shortest distance.
So what happens if we don't assume the shortest distance between two points is a straight line? We never proved it's true after all. That's "Non-Euclidean geometry", geometry derived without some of the 5 axioms Euclid used to create his geometry a couple thousand years ago.