Analytic geometry is a fantastic area of mathematics which is populated by all kinds of crazy objects. Being that it is Christmastime (and therefore Christmas math time) please check out the paradoxical object shown on the left. This object is sometimes known as the Infinite Gift. A related object is known as Gabriel's Wedding Cake [7].
In the Infinite Gift the length of the side of the nth box is 1/√n, so the area of one side of the nth box equals (1/√n)² = 1/n. Since a box has 6 sides the surface area of the nth box is 6·(1/n). Then what you find is that in the limit as n → ∞ that the Infinite Gift has infinite surface area but finite volume!
Here's an interesting aside: In the limit the area of the Infinite Gift equals 6 times the harmonic series (which we know diverges).
The continuous version of this object is known as Gabriel’s Horn (aka Torricelli’s Trumpet) and is shown in the figure on the right [1,2]. Gabriel's Horn is the surface of revolution of the function y = 1/x about the x-axis for x ≥ 1 [1,2]. As we can see in image, in the limit Gabriel’s Horn has volume = π and area = ∞.
These properties lead to an interesting situation known as the Painter’s Paradox [3,4].
This is the Painter's Paradox: Somehow even though you can fill Gabriel’s Horn with paint (its volume is finite), you still won’t have enough paint to cover its inside surface (its area is infinite)!
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