Comments by "Muizz" (@muizzy) on "3Blue1Brown" channel.

@Huntracony: Can confirm. A sphere (any dimension) is simply all points in equidistant from a center point. Doesn't matter in what metric (way to measure distance) you are or in what dimension. So for the unit sphere in the real numbers it's the points -1 and 1.
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@ Joey Ruler Try Khan Academy. It's really quite good in its explanations and is designed to educate anyone anywhere so starts from counting sheep to multivariable calculus. @ Basile Saint-Martin The dimensions of the universe are not estimated to be between 7 and 14. That's just some weird thing coming out of the mouths of string theorists who have a very different way of thinking about dimensions than anyone else. @ Mike Jones No it could not. Mathematics doesn't deal with the "real world" but instead assumes a few principles (called axioms) and says: IF these are true THEN this is ALSO true. Example: If we postulate "When it rains, it rains cows." as an axiom, a mathematician will have no problem using this to say "Your car is outside, it will get damaged when it rains." and doesn't care about whether that's the truth. Only about the logic that follows.
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@ I am curious: Distance is still defined, just not very useful. @ Ganondorfchampin: It doesn't work in 0D because there exist no spheres in 0D. You see, any sphere needs to - per definition - have a finite radius r. That is to say 0 > r > infty. (This radius can explicitly not be 0 or infty.) Then since a sphere is defined as all points x such that d(0,x) = r, and d(x,x) = 0 for all x (and in specific d(0,0) = 0), no point lies on the sphere and thus the sphere does not exist.
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Right. The best arguments in math are so diabolically clever they make you love and hate the problem at the same time :)
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@ Samuel An interval is not a circle, even in 1 dimension.
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Which is why it's not an interval. An (open) interval (a,b) is a collection of points x such that a<x<b. S^0 just has 2 points, -1, and 1. Not the points in between. So he's not right as it is not an interval but the set {-1,1} (pick x = 0, then x does not lie in S^0 but x does lie in [-1,1]) The line segment you seem to have in mind would be the half open line segment [0, 2pi) (or (0, 2pi] ) which in topology may be glued together under an equivalence relation that equates in the following way: x and y are equivalent if and only if x mod 2pi = y mod 2pi. This would result in a space that is topologically equivalent to a circle. That said, it is not actually a circle, but just something that can be made to look like it. (You need distance for a sphere which is not a priori present in a topological space.)
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ganondorfchampin Any introductory text book on analysis, or geometry will give you those definitions. The proof is my own. Keep in mind that we're doing maths here, not wishing upon a star. We're not going to be changing the definitions because some other concept doesn't work. This formula is fine for all strictly positive numbers, so that's part of the conjecture.
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xoppa09 the points - 1 and 1.
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Ah, but you didn't read carefully here. I never said make definitions, I said change them. Interchanging definitions requires equivalence which in this case is not true. If you want to fight the definitions, be my guest: Give me a formal definition of a sphere and I'll probably be able to disprove it when it calls upon the 0-D case. Note that any useful definition is general: It should be defined in n-dimensions, whether n=0 or n=10. If you need exceptions you haven't got a definition.
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