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/sci/ - Science & Math

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3379096 No.3379096 [Reply] [Original]

If you take y=1/x and rotate it around the x-axis from 1 to infinity you get a solid with infinite surface area but a volume of pi.

Post other interesting mathematical curiosities /sci/

>> No.3379104

That's not too hard to grasp.
Take a unit cube.
Surface area 6.
Volume 1.

Take a face and crease it inward to make a new face while bending another face outward in the opposite manner. This adds surface area but preserves volume. Repeat over and over.

>> No.3379116

>>3379104
How does it add surface area?

>> No.3379126
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3379126

>>3379116

>> No.3379125

>>3379116
One side has a face area of 1, a 1 by 1 square.

Crease it inward in such a way that it's two faces that are 1 x 3/4 and 1 x 3/4. New area is 3/2

>> No.3379150
File: 772 B, 729x118, cantor.png [View same] [iqdb] [saucenao] [google]
3379150

The Cantor set is as large as the set of reals

>> No.3379153

>>3379125
>>3379126
maybe im retarded but i cannot see this...

>> No.3379159

>>3379096

> lim(n->infinity) [- log(n)]

wat

>> No.3379160
File: 342 KB, 153x113, 1300833084412.gif [View same] [iqdb] [saucenao] [google]
3379160

>>3379153
>>3379153
Yeah, you're retarded.

>> No.3379161

An infiniely-dimensional sphere has volume 0.

>> No.3379170

>>3379153

+1

>> No.3379176

>>3379160

Not him, but maybe you are retarded because you're unable to explain or give a link.

>> No.3379178

>>3379150
It's not that counterintuitive when you learn that cantor set consists of fractions between 0 and 1 which in base 3 have only 0s and 2s.

>> No.3379180

http://en.wikipedia.org/wiki/Random_walk

an infinity long random walk in 1d has a 100% chance of passing through the origin
an infinity long random walk in 2d has a 100% chance of passing through the origin
an infinity long random walk in 3d has about a 34% chance of passing through the origin

explain that one /sci/

>> No.3379182

>>3379104
Nice explanation man. Thanks!

>> No.3379184

>>3379180
That's bullshit. Any point along the walk is equally likely as the origin.

>> No.3379183

>>3379161
I'm not sure waht you mean.
I know relevant fact, that as dimentiality goes to infinity, almost all ball's volume is in very small layer along its surface. The higher the dimension, the thinner the layer.

>> No.3379195

>>3379184
I should have added 'passing again through the origin ie. starting place'
heres part of the maths anyway
http://mathworld.wolfram.com/PolyasRandomWalkConstants.html
it doesnt give an explination but i imagine its got something to do with markov chains

>> No.3379301

volume under e^-r^2, r distance from origin, is pi.

1D line has same number of points as 3D space.

>>3379184
idiot?

>> No.3379342

>>3379178

This was pretty rad, i didn't know this.

Thanks.

>> No.3379354

>>3379180

ergodic hypothesis

works up to 2d and explains why they go through 0. The infinite monkey theorem is also good for explaining this (lemma of Borel and Cantelli).

In 3d that argument is not possible, i'll think about how to explain that one.

>> No.3379358

>>3379354
> vaguely invokes infinite monkeys to explain 1D and 2D
> fails to explain why analogy doesn't work in 3D

>> No.3379365

>>3379180

> Wiener process

lol'd

>> No.3379380

>>3379180
http://mathworld.wolfram.com/PolyasRandomWalkConstants.html

>> No.3379389

>>3379358

The issue is that you have 2 angles you can choose randomly in 3d.
Say we start with some random step and another one. These two steps are in a plane. Now leave the plane with the third step, keep one angle constant and only vary the second angle.

The motion now completely takes place in a plane that does not contain the origin.

The explanation for the chance to be 34% has something to do with hitting transcendent angles, not sure how exactly, but i keep thinking.

>> No.3379392

>>3379380

slowpoke.jpg

>> No.3379402

>>3379389
you wont get it
look at >>3379380, integrals and gamma functions everywhere

>> No.3379419

>>3379402

And?

I mean, those are of course valid mathematical calculations of probabilities, but they don't mean the argument is rather simple.

I could for example say that interference is due to particles taking every possible path (path integral formalism), but i'm surely not able to calculate the effect using the path integral formalism from the top of my head now.

Thanks for motivating me.

>> No.3379436

>>3379389
...not in any sense a rigorous explanation, nor even an understandable one.

>> No.3379453

>>3379436

k, better example:

If you asked me why waves (water) passing through a small slit come out as elementary waves.
My answer would be Huygen's principle. You can regard every point on a wave front as the source of an elementary wave and assume only one point passes through the slit, which then propagates as an elementary wave.

Easy enough? The math behind that sure isn't.

>> No.3379472

For anyone who understands, here is the 1d 2d 3d problem explained proved

http://www.statslab.cam.ac.uk/~james/Markov/s16.pdf

>> No.3379473

>>3379453
the fuck are you talking about?
i asked why 'infinite monkeys' explains 1D and 2D but not 3D. you posted some totally vague thing about two angles and i called bullshit.

>> No.3379481

>>3379389

That's a good explanation for why the chance is not 100% in 3D.

>> No.3379492

>>3379481
no it fucking isn't. why is it appropriate to never vary the angle again? you have zero understanding of rigour.

>> No.3379496

>>3379473

Vague? lol

The argument for 3d chance not being 100% is that you can have the entire motion in a plane that does not contain the origin.
Basically proof by counter-example.

If you don't understand that, not my fault.

The argument works for 1d and 2d because it states that (the ergodic hypothesis and the infinite monkey are equivalent here) every possible point is reached, as you cannot leave the plane containing the origin, which is especially the case in the 1d case.

>> No.3379501

>>3379492

wut

he constructed an example of an infinite motion that does not pass 0 again.

lrn2proof

>> No.3379516

>>3379501
>>3379496
2D: move forever in straight line.
2D: always increase distance from origin.
2D: move in circle around origin.

you are fucking clueless faggots.

>> No.3379528

>>3379516

wait, we were talking about open space?

I thought we were talking about bound spaces with reflection.

>> No.3379531

>>3379496
Well, in the same respect for 2D, you start on the x-axis at 0.

You could very well leave the x-axis and stay on a line parallel to it the entire time, never coming back to the origin.

>> No.3379533

>>3379528
Well yeah, any constrained finite space will by filled by an infinite random walk.

>> No.3379540

>>3379496
you cant prove by counter example in this instance

http://en.wikipedia.org/wiki/Almost_surely

>> No.3379545

ITT: scifags with no clue of how to prove things tell others to learn how to prove things.

>> No.3379607

>>3379540
>>3379545

The set of planes not containing 0 is of non-zero cardinality.

It works.

>> No.3379625

>>3379607
The set of lines not containing 0 is of non-zero cardinality.
The set of the plane with 0 removed is of non-zero cardinality.
You're a faggot.

>> No.3379692

>>3379625

> The set of the plane with 0 removed is of non-zero cardinality.

i don't even

>> No.3379700

>>3379692
it's because you're a fucking idiot.

>> No.3379736

>>3379700

y u mad son?

>> No.3379747

>>3379736

Thinking and learning is forbidden on /sci/.

Deal with it.