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Also check out the Menger sponge, which has infinite surface area and zero volume: http://en.m.wikipedia.org/wiki/Menger_sponge
I think Sierpinski's triangle may also be of interest here, which, contrastly, approaches zero surface area: http://en.wikipedia.org/wiki/Sierpinski_triangle
Sponge has infinite area and 0 volume. Sierpinski's triangle has infinite perimiter and 0 area.
I've thought it was paradoxical that infinitely long curves could have finite integrals ever since I first took calculus. For example, the integral of 1/x is |ln x|. I wonder why it takes three dimensions before people start getting upset.
The paradox is not the convergence of infinite long curves.. Read the paradox description. To fill the horn with paint you need a finite amount. To paint the outside of the horn you need an infinite amount. That is the paradox. The surface area of the inside is equal to the outside, yet one side requires a finite amount of paint while the other side requires an infinite amount. At least that's how I understand the pa…
This reminds me of a Putnam problem from a few years back, was something along the lines of: Construct a set of discs in R^2 s.t. no infinite straight line can be drawn without intersecting at least one disc, and the sum of the areas of all the discs is finite.
(Awarded 1 point of 10 for being unable to specify the precise piecewise-linear spiral function)
I wonder what sound it would make. Can we model such a horn assuming a from, say, 1cm to some value p and simulate it?
[1] Electromagnetism, classical mechanics, etc
This reminds me of a Putnam problem from a few years back, was something along the lines of: Construct a set of discs in R^2 s.t. no infinite straight line can be drawn without intersecting at least one disc, and the sum of the areas of all the discs is finite.
I've thought it was paradoxical that infinitely long curves could have finite integrals ever since I first took calculus. For example, the integral of 1/x is |ln x|. I wonder why it takes three dimensions before people start getting upset.
On the other hand, if you take 1/x^2 as the integrand, you get the desired effect: its antiderivative is -1/x, so the integral of 1/x^2 from x=1 to infty is finite (actually, it's 1).
By the way: For good fun with convergent series, take a look at the problem of escaping a lion in a circular arena at equal maximum speeds. Here's an example link to both question and answer: http://puzzling.stackexchange.com/questions/8140/escaping-a-...
IIRC, there is an even more fascinating story associated to the history of this problem, as the first published solution was actually incorrect.
I've thought it was paradoxical that infinitely long curves could have finite integrals ever since I first took calculus. For example, the integral of 1/x is |ln x|. I wonder why it takes three dimensions before people start getting upset.
I'm not sure how you intended the example, but it's probably a bad one: If you take the integral of 1/x, say from x=1 to infty, you don't get a finite amount (because ln x goes to infinity). On the other hand, if you take 1/x^2 as the integrand, you get the desired effect: its antiderivative is -1/x, so the integral of 1/x^2 from x=1 to infty is finite (actually, it's 1). By the way: For good fun with convergent seri…
I've thought it was paradoxical that infinitely long curves could have finite integrals ever since I first took calculus. For example, the integral of 1/x is |ln x|. I wonder why it takes three dimensions before people start getting upset.
The paradox is not the convergence of infinite long curves.. Read the paradox description. To fill the horn with paint you need a finite amount. To paint the outside of the horn you need an infinite amount. That is the paradox. The surface area of the inside is equal to the outside, yet one side requires a finite amount of paint while the other side requires an infinite amount. At least that's how I understand the pa…
>Since the Horn has finite volume but infinite surface area, it seems that it could be filled with a finite quantity of paint, and yet that paint would not be sufficient to coat its INNER surface
Earlier quoted context omitted.
Or even just a convergent infinite series. Infinitely many positive numbers that sum to a finite number.
People don't tend to find that unintuitive. The Greeks had a nice geometric example in a square of side length 1: ┌┬┬─┬───┬───────┐ ├┘│3│ │ │ ├─┘-│ │ │ │ 64│ │ │ ├───┘ │ │ │ │ │ │ 3/16 │ │ │ │ │ ├───────┘ │ │ │ │ │ │ │ │ 3/4 │ │ │ │ │ │ │ └───────────────┘ 3/4 + 3/16 + 3/64 + 3/256 + ... is easy to visualize as successively filling in three quarters of an ever-smaller residual square. Intuitively, no matter how finel…