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.
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