Seems rather paradoxical in my opinion.
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Imagine you have a pie with a radius of 1 unit of length. Conveniently, the area of the pie is pi units of area. The total amount of edge of the pie is just its circumference (2 pi units of length).
What if you cut the pie into eight slices? Its area is still only pi units of area. But now the amount of edge is 2 pi + 16 units of length, because each of the eight slices now has a curved bit of crust, and two straight edges towards the center.
What if you cut the pie into eight million slices? Its area is still only pi units of area, but now the amount of edge is 2 pi + 16000000 units of length.
This shows the principle that a fixed area can have a wildly varying perimeter depending on how it is *shaped*. A circle is compact; an infinite number of slices with the same area is not.
Gabriel’s Horn is the same principle, in one higher dimension. You could have a certain volume shaped like a sphere to be as compact as possible. Or you could start stretching that volume or slicing it to get a shape that has the same volume, but is much less compact. After stretching the Horn infinitely in one dimension but squishing inward in two other dimensions, the volume stays the same, but you’ve made it infinitely less compact.
Imagine you have a pie with a radius of 1 unit of length. Conveniently, the area of the pie is pi units of area. The total amount of edge of the pie is just its circumference (2 pi units of length).
What if you cut the pie into eight slices? Its area is still only pi units of area. But now the amount of edge is 2 pi + 16 units of length, because each of the eight slices now has a curved bit of crust, and two straight edges towards the center.
What if you cut the pie into eight million slices? Its area is still only pi units of area, but now the amount of edge is 2 pi + 16000000 units of length.
This shows the principle that a fixed area can have a wildly varying perimeter depending on how it is *shaped*. A circle is compact; an infinite number of slices with the same area is not.
Gabriel’s Horn is the same principle, in one higher dimension. You could have a certain volume shaped like a sphere to be as compact as possible. Or you could start stretching that volume or slicing it to get a shape that has the same volume, but is much less compact. After stretching the Horn infinitely in one dimension but squishing inward in two other dimensions, the volume stays the same, but you’ve made it infinitely less compact.
As it turns out, dealing with infinite are both hard and non-intuitive.
First, lets simplify the problem a bit. The horn is a rotational integral of a curve like 1/x, so lets just work with 1/x instead. It has all of the same properties that matter for this paradox, so it will work.
What does 1/x do as it goes to infinity? It *converges* toward 0. While it gets extremely close to 0 and limit of 1/x as x approaches infinity is 0, it never actually reaches zero. This is a case where the difference actually matters.
Since 1/x is approaching 0, the distance between the x axis and the curve become infinitely small. This means that when you take the integral and add up all of those distances, the values near infinite don’t actually change the total very much which means it also converges towards a specific number as well. This means that we can bound the value of the integral between a pair of two other finite values, so it must be finite as well.
However, because 1/x never actually reaches 0, it must have infinite length (or surface areas as a 3d curve). By convention, a curve is “closed” and ends when it crosses the x axis. This means as the curve goes to infinity, that small distance between the curve and zero prevents it from closing and means that the length of the curve keeps increasing and doesn’t have an upper bond. This means that it must be infinite as well.
As it turns out, dealing with infinite are both hard and non-intuitive.
First, lets simplify the problem a bit. The horn is a rotational integral of a curve like 1/x, so lets just work with 1/x instead. It has all of the same properties that matter for this paradox, so it will work.
What does 1/x do as it goes to infinity? It *converges* toward 0. While it gets extremely close to 0 and limit of 1/x as x approaches infinity is 0, it never actually reaches zero. This is a case where the difference actually matters.
Since 1/x is approaching 0, the distance between the x axis and the curve become infinitely small. This means that when you take the integral and add up all of those distances, the values near infinite don’t actually change the total very much which means it also converges towards a specific number as well. This means that we can bound the value of the integral between a pair of two other finite values, so it must be finite as well.
However, because 1/x never actually reaches 0, it must have infinite length (or surface areas as a 3d curve). By convention, a curve is “closed” and ends when it crosses the x axis. This means as the curve goes to infinity, that small distance between the curve and zero prevents it from closing and means that the length of the curve keeps increasing and doesn’t have an upper bond. This means that it must be infinite as well.
As it turns out, dealing with infinite are both hard and non-intuitive.
First, lets simplify the problem a bit. The horn is a rotational integral of a curve like 1/x, so lets just work with 1/x instead. It has all of the same properties that matter for this paradox, so it will work.
What does 1/x do as it goes to infinity? It *converges* toward 0. While it gets extremely close to 0 and limit of 1/x as x approaches infinity is 0, it never actually reaches zero. This is a case where the difference actually matters.
Since 1/x is approaching 0, the distance between the x axis and the curve become infinitely small. This means that when you take the integral and add up all of those distances, the values near infinite don’t actually change the total very much which means it also converges towards a specific number as well. This means that we can bound the value of the integral between a pair of two other finite values, so it must be finite as well.
However, because 1/x never actually reaches 0, it must have infinite length (or surface areas as a 3d curve). By convention, a curve is “closed” and ends when it crosses the x axis. This means as the curve goes to infinity, that small distance between the curve and zero prevents it from closing and means that the length of the curve keeps increasing and doesn’t have an upper bond. This means that it must be infinite as well.
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