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Anonymous Wed 24 Nov 22:21:00 2021 No.13887545 View Reply Original Report

Quoted By: >>13887587 >>13887609 >>13887650

So in 3 dimension we have something like gabriels horn: infinite surface, finite volume.

And in 2 dimensions you have whatever fractal with an infinitly long edge, but finite area.

Do these type of objects exist in lower and/or higher dimensions as well?
Like a 4d object with infinite volume, but finite whatever4thdimensioniscalled

Anonymous

Anonymous Wed 24 Nov 2021 22:36:34 No.13887587 Report

Quoted By: >>13887650

>>13887545
Very good question. It seems like such objects should exist.

I think it relates to the summability of an infinite non-negative sequence: you have an infinite collection of objects but they may still have a finite sum. Or similarly, you can integrate a non-zero value over an infinite vector field, and still obtain a finite value.

I'm taking a grad course on algebraic topology at the moment and I feel like I don't understand any of it, but I think it might be at least loosely related to the question you're asking. It's all about boundaries, and spaces, at any rate.

Anonymous

Anonymous Wed 24 Nov 2021 22:51:09 No.13887607 Report

Quoted By: >>13887784

the horn is interesting because it isn't a fractal

Anonymous

Anonymous Wed 24 Nov 2021 22:51:43 No.13887609 Report

Quoted By:

>>13887545
OP that sounds like a black hole

Anonymous

Anonymous Wed 24 Nov 2021 23:11:57 No.13887650 Report

Quoted By: >>13887673

>>13887545
I don't think a fractal is what you want in 2d since it has no well-defined length.
For the general case you could probably consider a strictly positive smooth function $f: \mathbb{R}^n \rightarrow \mathbb{R}$ for which the integral $\int f$ is finite (this is related to the summability thing >>13887587
mentioned). The graph of this function is a subset of $\mathbb{R}^n \times \mathbb{R}$ and the region "above" the $\mathbb{R}^n \times {0}$ hyperplane and "below" the graph of this function would be connected, have finite (n + 1)-volume and its boundary would have infinite n-volume. An example of such a function would be $f(x) = e^{-r^2}$ where r equals the distance from the origin to x. If the use of the $\mathbb{R}^n \times {0}$ seems artificial, you can also consider the region between the graphs of f and -f, which would still have a boundary with infinite n-volume and an interior with twice the original finite (n + 1)-volume.

Anonymous

Anonymous Wed 24 Nov 2021 23:18:12 No.13887673 Report

Quoted By:

>>13887650
I should add that the boundary of this thing wouldn't be connected so it might not be what you want

Anonymous

Anonymous Wed 24 Nov 2021 23:57:58 No.13887784 Report

Quoted By:

>>13887607
the horn is a delusion

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