How can Gabriel’s Horn have an infinite surface area, but a finite volume?

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Seems rather paradoxical in my opinion.

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15 Answers

Anonymous 0 Comments

…and hence the name *The Paradox of Gabriel’s Horn*.

The paradox comes from the intuition that if the volume is finite, approaching π units, then we could fill the inside of the horn. But if we can fill, in effect paint, the inside of the horn, then wouldn’t we have enough to paint the outside of the horn?

Anonymous 0 Comments

…and hence the name *The Paradox of Gabriel’s Horn*.

The paradox comes from the intuition that if the volume is finite, approaching π units, then we could fill the inside of the horn. But if we can fill, in effect paint, the inside of the horn, then wouldn’t we have enough to paint the outside of the horn?

Anonymous 0 Comments

…and hence the name *The Paradox of Gabriel’s Horn*.

The paradox comes from the intuition that if the volume is finite, approaching π units, then we could fill the inside of the horn. But if we can fill, in effect paint, the inside of the horn, then wouldn’t we have enough to paint the outside of the horn?

Anonymous 0 Comments

How much math background do you have? That’ll kinda be a basis for what should be used to explain this. Below a certain point there’s not a great way to show exactly how it works.

Anonymous 0 Comments

You can divide any number in infinite number of parts.
0,9+0,09+0,009+… is still less than 1, though the sequence goes forever.
You can paint with 1 liter paint, infinite number of diminishing surfaces within 1 liter volume, because you need *less* paint with each iteration.

Anonymous 0 Comments

How much math background do you have? That’ll kinda be a basis for what should be used to explain this. Below a certain point there’s not a great way to show exactly how it works.

Anonymous 0 Comments

How much math background do you have? That’ll kinda be a basis for what should be used to explain this. Below a certain point there’s not a great way to show exactly how it works.

Anonymous 0 Comments

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.

Anonymous 0 Comments

You can divide any number in infinite number of parts.
0,9+0,09+0,009+… is still less than 1, though the sequence goes forever.
You can paint with 1 liter paint, infinite number of diminishing surfaces within 1 liter volume, because you need *less* paint with each iteration.

Anonymous 0 Comments

You can divide any number in infinite number of parts.
0,9+0,09+0,009+… is still less than 1, though the sequence goes forever.
You can paint with 1 liter paint, infinite number of diminishing surfaces within 1 liter volume, because you need *less* paint with each iteration.