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This is something that's always bothered me. I am well aware that you can easily see why this is the case with math. I mean, even in the 2-D case, take a square with side length $1$, and it has a perimeter of $4$ and an area of $1$. Now take a rectangle with side lengths $0.5$ and $1.5$, then the perimeter is $4$, but now the area is $0.75$.

But where did all the extra area go? We have the same amount of material there. Intuitively I'd like to say that the same area is covered, but it's just distributed differently, but this clearly isn't the case. I know that different shapes with the same surface area have different volumes, but I can't picture why. I can take a sheet of (ideal) paper, turn it to a sphere, and then turn it into a cube, and they'll have different volumes, despite using the same sheet of paper.

To extend on this question a little further, what makes one shape "better" with volume than another?

I hope this isn't a remarkably trivial issue wherein I'm missing something obvious.

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    $\begingroup$ Intuitively: if you deflate a football. The area stays the same (maybe changes a little due to stress on the fabric but lets ignore this) but the volume is significantly less. $\endgroup$ – Kaladin Feb 27 '15 at 23:37
  • $\begingroup$ "what makes one shape "better" with volume than another?" youtube.com/watch?v=DS68v2zyZS4 math.berkeley.edu/~hutching/pub/bubbles.html $\endgroup$ – leonbloy Feb 28 '15 at 0:06
  • $\begingroup$ maa.org/sites/default/files/pdf/upload_library/22/Ford/… $\endgroup$ – Aretino May 27 '16 at 20:01
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    $\begingroup$ This will probably come across more harshly than I intend, but the question is (to me) like asking "Why, intuitively, do buildings that measure the same from east to west have different measurements from north to south?" They simply measure different things, and I don't see any reason for intuition to suggest otherwise. $\endgroup$ – Greg Martin May 27 '16 at 20:28
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Tie the two ends of a thread several inches long together. Fill your kitchen sink with water. Let the waves settle down. Let the loop of thread float on the placid surface, but place it so that the thread meanders erratically --- it's not a circle but a meandering curve returning to its starting point. Then drop a tiny drop of liquid dish detergent into the part of the surface surrounded by the thread. The detergent is a surfactant: it decreases the surface tension. But it does so in the region surrounded by the thread and not outside it. The result: surface tension on the outside pulls the thread outward and it suddenly assumes a circular shape!

Conclusion: the circle surrounds a larger area than does any other closed curve of the same length.

So think about that.

That's the "physicists' solution" of the isoperimetric problem. Dym & McKean's book on Fourier transforms solves the same problem by using Fourier series.

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The problem is wth your intuition. You have the feeling that a loop of length $L$ [cm] should enclose an area of $\alpha L^2$ [cm$^2$], where $\alpha$ is some universal constant, independently of how the loop is laid out.

But this is not how area is measured. After person A has laid out the loop in some way person B comes with a bag full of little $1$ cm$\times 1$ cm stickers and counts how many of these stickers (s)he can place (aligned, and without overlapping) in the finite domain determined by the loop. Person B does not know the size of $L$, nor will (s)he measure this quantity to accomplish his (her) task.

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Here's another thought experiment: Picture a zip-loc bag. Now, picture the zip-loc bag with different amounts of air or water sealed inside. The volume of the sealed bag depends on how much fluid is sealed inside, but the area of the bag (assuming you don't pressurize it to the point where the plastic stretches) remains constant.

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Imagine taking a circle(Let's call it C1) and cutting a pie slice out of it. You have decreased the SA but increased the perimeter let's call this shape A. So now, imagine a circle with the same perimeter as shape A, let's call this shape B. Evidently shape B has a bigger SA than shape A, but the same perimeter. You can extend this to the third dimentions with a similar idea but with spheres.

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