Consider a hollow cylinder. The outside surface loses area faster than the inside because dA/dr=kr. Now twist it to make a Mobius strip: there are no longer two different surfaces. The only problem left is its finite width, which I don't see how to get around. Besides, a wide, thin Mobius strip would make a weird cake of soap. How about a Klein bottle? Would a four-dimensional person need a shower? Steve Gray ----- Original Message ----- From: "David Wolfe" <wolfe@gustavus.edu> To: "math-fun" <math-fun@mailman.xmission.com> Sent: Wednesday, April 26, 2006 9:26 AM Subject: [math-fun] A problem for the shower
A question struck me in the shower this morning:
Suppose you would like a piece of soap which dissolves at a uniform rate. I.e., as it gets smaller, the soap bar's surface area remains constant until it vanishes.
I have convinced myself that one should be able to construct such a bar of soap with arbitrarily small holes. As the soap gets smaller, the holes are exposed to increase the surface area.
Need it have holes?
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