10 ms·
Gabriel's Horn
- dubin 11y agoAlso check out the Menger sponge, which has infinite surface area and zero volume: http://en.m.wikipedia.org/wiki/Menger_sponge http://en.m.wikipedia.org/wiki/Menger_sponge
- wz1000 11y agoNon-mobile: https://en.wikipedia.org/wiki/Menger_sponge https://en.wikipedia.org/wiki/Menger_sponge
- darkmighty 11y agoOr equivalently any object with a fractal dimension between 2 and 3.
- mturmon 11y agoYou can do the first few levels with business cards: http://theiff.org/oexhibits/menger02.html http://theiff.org/oexhibits/menger02.html As you go to higher levels, that "infinite surface area" thing starts to become difficult to manage. ;-)
- relicscattergun 11y agoI think Sierpinski's triangle may also be of interest here, which, contrastly, approaches zero surface area: http://en.wikipedia.org/wiki/Sierpinski_triangle http://en.wikipedia.org/wiki/Sierpinski_triangle
- gohrt 11y agonot constrast; it's analogous, in lower dimension. Sponge has infinite area and 0 volume. Sierpinski's triangle has infinite perimiter and 0 area.
- deleted 11y ago[deleted]
- TeMPOraL 11y agoI wonder what sound it would make. Can we model such a horn assuming a from, say, 1cm to some value p and simulate it?
- currentoor 11y agoDepends on how many of the laws of physics you want to ignore. The horn has infinite length and sound waves have a finite speed (assuming sound is defined as the usual compression wave through a medium). So a sound wave starting at the narrow end would take an infinite amount of time to reach the other end. But the sound it self shouldn't be anything special.
- daniel-levin 11y agoOne of the reasons that these sort of paradoxes can exist is because real numbers can get arbitrarily small. In principle, you could build a horn that very closely matches Gabriel's horn, but it would necessarily be an approximation of the mathematical object because you do not have available particles of arbitrary size with which to build. This last notion hints at an important property of the real numbers: they can be divided into pieces smaller than any size you can specify. They are useful abstractions upon which to build extremely successful models of reality [1], because at those scales, real numbers work very well. However, at extremely small scales, the correspondence between real numbers and reality breaks down. What does it mean to model a horn made of atoms with numbers that are arbitrarily small fractions of the width of an atom? This is not a 'real numbers / axiom of choice bad' rant, but rather I'd like to point out that real numbers exist in our minds, and using our physical intuition about to reason about them at very small scales is bound to lead to paradoxes. [1] Electromagnetism, classical mechanics, etc
- jveld 11y agoI'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.
- parallel 11y agoOr even just a convergent infinite series. Infinitely many positive numbers that sum to a finite number.
- reagency 11y agoOr in general, any infinite size n-dimensional structure can be embedded in some finite size n+1 dimensional structure.
- wz1000 11y agoI don't think that's paradoxical. We deal with convergent infinite series all the time in everyday life. 10/3(3.333...) is 3 + 3/10 + 3/100/ + 3/1000 + 3/10000... π(3.1415...) is 3 + 1/10 + 4/100 + 1/1000 + 5/10000... Its quite easy to see that both of these series will be finite, and many numbers, lets say 4 or 3 + 5/10 will be greater than either of them.
- thaumasiotes 11y agoPeople 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 finely you detail it, you're never going to stop fitting inside the original area-1 square. edit: better text art
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- mgob 11y agoI always describe this to "non-mathy" people when they ask what could possibly be fascinating/beautiful/etc about math. I'd like to think I've changed at least a mind or two.
- davidrusu 11y agoThis 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.
- gohrt 11y ago(approximate) logarithmic spiral of discs whose radii are the harmonic sequence? (Awarded 1 point of 10 for being unable to specify the precise piecewise-linear spiral function)
- cousin_it 11y agoTake a set of discs with radii 1/n. Their total area is finite but the sum of radii is infinite, so we can just use them to cover the X and Y axes. Yeah, this is pretty similar to Gabriel's horn.