[Monte314]

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[rahdam]

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strange problem.
I am imagining the projection of a plane of finite area onto the infinite area below.
It seems there is, strangely, a 1:1 correlation between each point on the initial plane and it's projection onto the infinite area below.
There being an infinite number of points in the initial, finite plane, it easily maps to the infinite number of points on the infinite area below.

[altoid]

So is Olivia related to a Mr. Torricelli?
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[Nightsun]

Solution:

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It's a fractal geometry problem.

First of all each step has a lenght of 1/k - 1/(k+1). it means that the total lenght is:

1 + 1/2 + 1/3 + 1/4 ... - 1/2 -1/3 - 1/4 ... and this series converge to 1. But the height of each step is simply 1/2, 1/3, 1/4 and this is the harmonic series.. harmonic series doesn't converge so actually also if it is contained inside (0,1) the profile has an infinite lenght (because the profile lenght is obviously greater than the sum of its height). So it has an infinite surface but doesn't have an infinite volume as said by the OP.
The fractal dimension of the profile is between 2 and 3 (that's because the surface is infinite and the volume finite).

Let's try to find it out:
the first profile is 1/2 base, 1/2 height so it's lenght (using Pitagora theorem) is: sqrt(5)/2.
the second profile is 1/6 base, 1/3 height for a total lenght of 0.687 and so on. Now going on the base became even more trascurable, so actually we can consider that the total lenght profile is a little more than 2 times the height. It means something like 2*(1/2) + 2*(1/3) + 2*(1/4)+... so the surface grow like 2*ln(k) where k is the number of step and obviously is infinite for k = infinite.

Now we can try to find the generalized dimension. Generalized dimension is defined as D=ln(L(n))/ln(L(n+1)) where L is the lenght at the step n and n+1.
At the first step we have a total lenght of 1-1/2+sqrt(5)/2 = 1.618
The second step we have 1.618-1/6+0.687 = 2.138

D = ln(2.138)/ln(1.618) = 1.579

So the surface has a dimension of 1+1.579 = 2.579

It means a dimension between 2 and 3 or said otherwise an infinite surface within a finite volume.

[Monte314]

Olivia is the jedi math dog's 16 year-old daughter. She loves to cook and bake... but she doesn't like math very much.

[thod]

I will take your word for it that the series on the bottom does not converge. Oddly enough I was thinking this exact same thing the other day. How a Mandelbrot of infinite length could enclose an area of finite area. I can't help thinking its another xeno's arrow somehow.

[Latro]

Originally Posted by rahdam To view links or images in this forum your post count must be 2 or greater. You currently have 0 posts.

( Show Spoiler ) strange problem. I am imagining the projection of a plane of finite area onto the infinite area below. It seems there is, strangely, a 1:1 correlation between each point on the initial plane and it's projection onto the infinite area below. There being an infinite number of points in the initial, finite plane, it easily maps to the infinite number of points on the infinite area below.

( Show Spoiler )

In my experience cardinality arguments don't work very well for these sorts of things. You can map the interval (0,1) to the ordered pairs of real numbers R x R in one-to-one correspondence, but that doesn't mean that the area of (0,1) is infinite.

Good to see you back, btw, Monte. This one is outside what I've fiddled with, I think, so I'll be skipping it.

[jm123]

Unedumacated answer.....

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The answer is simply that area is a two dimensional value, where as volume is a three dimensional measurement. So an infinite area in a finite space does not matter when you are filling that space with a three dimensional substance. As the space is <1 cubic foot it would require less the 1 cubic foot of substance to fill the volume, regardless of the surface area exposure.

[paleoeco]

	Originally Posted by thod To view links or images in this forum your post count must be 2 or greater. You currently have 0 posts.
	I will take your word for it that the series on the bottom does not converge. Oddly enough I was thinking this exact same thing the other day. How a Mandelbrot of infinite length could enclose an area of finite area. I can't help thinking its another xeno's arrow somehow.

Techinically, Zeno's fletcher's paradox is more about time and motion than about space. The Dichotomy paradox is more accurate, as it specifically deals with the infinite regression of division of space into a smaller point.

But, upon reading Monte's post, that was the first thing I thought of: Zeno and his damned paradox.

[DewFuel]

Are any of these solutions correct Monte? BTW, love that these problems are back

[jndiii]

Physics answer:

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It is impossible to get infinite area, because eventually the ripples are smaller than the particles of which the pan is made, never mind smaller than the particles of which the batter is made.

Math answer:

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Assuming that we're talking about an ideal situation in which the fluid that fills the pan is not made of discrete particles, it JUST DOESN'T MATTER. It is possible to have a finite volume with an infinite surface area: it is demonstrated here. The "empty space" above the pan can be a vacuum, or air (imaginary, non-particle air), or batter (imaginary, non-particle batter).

Similarly, the "finite time" isn't an issue, either: there is no inherent contradiction between a finite volume filling in a finite time, and an "infinite area" being covered in a finite time with a 3-d fluid. There's nothing magical or paradoxical or contradictory about the rate at which the area is being covered. A perfectly flat surface is covered "infinitely fast" by another perfectly flat surface.

[Monte314]

The answers given by jndiii are exactly ones I would give. It is this distinction between "true in principle" and "true in practice" that this problem is designed to highlight.