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Conterintuitive facts in mathematics, CS, and physics

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Re: Conterintuitive facts in mathematics, CS, and physics

#311

Post by rsj_hn »

Synaesthesia wrote: post #170
d_burfoot wrote: post #117

Surprising fact about the sun: it actually produces less heat per unit volume than a compost pile - this makes sense when you consider that fusion events are very rare. The reason the sun is so hot is that it has an enormous ratio of volume to surface area. Heat emission due to radiative cooling is proportional to the surface area, but heat creation due to fusion is proportional to the volume.

The density of the sun is very low, like much lower than the atmosphere (throughout most of it, anyway,) But yes it has an enormous volume.

The density of the sun is 1.4 g/cm^3, which is 1.4 times the density of the water.

The atmosphere is 0.0012 g/cm^3 or three orders of magnitude less dense than the sun.

Re: Conterintuitive facts in mathematics, CS, and physics

#312

Post by caf »

BeetleB wrote: post #286
caf wrote: post #130

Earlier quoted context omitted.

The way I'd heard the paint comment was along the lines that "Gabriel's Horn can hold only a finite quantity of paint, but requires an infinite quantity of paint to cover the surface" . So if you think of it as a bucket that can't hold enough paint to cover itself, that is at least a little surprising.

But that's exactly my paint. If you allow for infinitely thin paint, then a finite volume of paint can always cover an infinite surface - you don't need Gabriel's Horn to show that. If you don't allow for infinitely thin paint, then no - Gabriel's Horn surface cannot be painted even with an infinite amount of paint.

It's not supposed to be especially tricky. Yes, as soon as you realise that the horn is a bucket with a very deep section that is narrower than the assumed thickness of a coat of paint, the surprise dissipates.

I believe the point of the exercise is an introduction to curves with infinite length but finite area under them, in order to expand one's intuition about such objects, which is then transferable to other examples like space-filling curves.

Re: Conterintuitive facts in mathematics, CS, and physics

#313

Post by caf »

twic wrote: post #172
caf wrote: post #124

Earlier quoted context omitted.

Do the legs have to be even?

Not even, but the feet have to be coplanar.

And since the spacing between the feet is obviously irrelevant, that further implies that it's always possible to find 4 coplanar points on the surface arbitrarily close together. Or arbitrarily far apart.

Re: Conterintuitive facts in mathematics, CS, and physics

#314

Post by macrolocal »

niccl wrote: post #26

Missing one that still makes no sense to me: The Ramanujan Summation: 1 + 2 + 3 + ⋯ + ∞ = -1/12 https://en.wikipedia.org/wiki/1_+_2_+_3_+_4_+_%E2%8B%AF

Ok, here's my attempt at an elementary explanation. Consider the function

  f(N) = floor(N) 
       = 1 + 1 + 1 + ... + 1, whenever N's an integer
       = 1^0 + 2^0 + 3^0 + ... + N^0.
Then f(N) ~ N - 1/2 is f's simplest unbiased polynomial estimator. It's a succinct description of f and its asymptotic behavior.

Next, let's do something like integration, and increase those zeroth powers to one, ie. consider the function

  g(N) = 1^1 + 2^1 + 3^1 + ... + (N - 1)^1
       = 1 + 2 + 3 + ... + (N - 1).
In this case, the estimator ends up being

  g(N) ~ N^2/2 - N/2 - 1/12
       ~ integral(N - 1/2) - 1/12,
where I've written this estimate as the difference of a divergent term and a constant term.

Here's the point: the divergent term is really just left over (integrated up or 'induced') from our asymptotic description of the degree 0 sum. In a sense, it's degenerate and doesn't provide any sort of 'new' information. So in the same sense, the interesting semantic content of 1 + 2 + 3 + ... is just the constant term: -1/12.

Ps. This way of thinking is related to trace formula truncation.

Re: Conterintuitive facts in mathematics, CS, and physics

#316

Post by dragonwriter »

justusthane wrote: post #315

> Adding 3 feet to a tightly tied rope around the earth would allow you to raise it uniformly by almost 6 inches. There is no way my brain can accept that this is true.

> > Adding 3 feet to a tightly tied rope around the earth would allow you to raise it uniformly by almost 6 inches.

> There is no way my brain can accept that this is true.

Yeah, my brain wants the size of the Earth to matter even though I know c = πd => c1 - c0 = π(d1-d0) means only the delta matters. The Earth, a soccer ball, the Milky Way galaxy...the amount of additional diameter you get from the same added circumference is constant.

Re: Conterintuitive facts in mathematics, CS, and physics

#317

Post by 9935c101ab17a66 »

unholiness wrote: post #109
kristjansson wrote: post #40

Earlier quoted context omitted.

That one got me good, so my future at College Board is as bright as yours. However, I don't think the argument by demonstration in that video is particularly convincing. Instead, I think's its easier to note that that the _center_ of a circle of radius r travels 2 * pi * r distance over one rotation. In the problem, the center of the smaller circle has to travel further than the circumference of the bigger circle - i…

I don't think that argument holds water. Imagine rolling it on the inside of the circle – the center of the circle traces a path with only twice its radius, yet it still rolls 4 times around.

Is this correct? wouldn't the radius of the circle it's revolving around be significantly smaller if you had it follow the edge of the larger circle but from the inside? I don't know much about math or physics, so I could be wrong, but I think it would be significantly less, closer to two, right?

The reason that this problem is tricky, and has a counterintuitive solution is that Circle A is rolling around Circle B, and so the 'radius; of the circular path it is following is the radius of Circle B + the radius of Circle A.

I'm no expert, but some quick math:

Circle B has a radius of 9, circumference then is 56.55 Circle A (1.3) has a radius of 3, circumference is 18.84

56.55 / 18.84 = 3. This suggests that you could "unwind" circle A (say it was made of pipecleaner), and you would need 3 circle As to fully ensconce Circle B

BUT that wasn't the question. The question states that Circle A is rolling AROUND circle B.

So the circumference of the circle we are now trying to 'ensconce' is Circle A Radius + Circle B Radius = 12, so the new circumference is 75.39, and divided by Circle A Circumference, we end up with 4, which makes sense, and matches the demonstration from the video.

HOWEVER, if we are 'rolling' around the inside of circle B, then I think you're right, the radius of the circular path Circle A will take is 9 -3 = 6, and the circumference of said circle is 38, so we it only will take two rolls. I think this is correct, i do not think it will be 4 rolls.

Think of it this way: when Circle A is rolling around the inside of Circle B, the way you're picturing it is circle A following the outside of Circle B, which is why I think intuitively it feels like the answer will still be 4 revolutions. BUT a better way to think of it is that Circle A is actually revolving around a new Circle, Circle C, which has a circumference of 38. Circle A is not revolving around Circle B, it's revolving around this new, smaller circle within circle b. Does that make sense?

I made a quick diagram to illustrate my point. I did it in a wire-framing tool that has snap to grid, so it's definitely not perfect:

https://i.imgur.com/dospt2w.png

You can play around with what I made here:

https://whimsical.com/r-DoXHwe2kUdgvSiAu2yuGpE

Re: Conterintuitive facts in mathematics, CS, and physics

#318

Post by 9935c101ab17a66 »

kristjansson wrote: post #158
unholiness wrote: post #109

Earlier quoted context omitted.

I don't think that argument holds water. Imagine rolling it on the inside of the circle – the center of the circle traces a path with only twice its radius, yet it still rolls 4 times around.

I don’t think that’s right. It if the radius of the inner circle is one third the outer, it only rotates twice rolling around the inside, which makes sense as it’s center traces a circle that’s the the difference of the radii. Imagine the limiting case, as the inner circle approaches the size of the outer circle - the inner circle completes much less than one rotation per lap around the inside edge of the outer circl…

I think you're right!

I made a diagram that helped me think through it visually:

https://i.imgur.com/dospt2w.png

If circle A is rolling around the edge of circle B from within, it is actually revolving around a new, smaller circle C which has the Radius Circle B - Circle A.

Re: Conterintuitive facts in mathematics, CS, and physics

#319

Post by crvdgc »

barbs wrote: post #126
niccl wrote: post #26

Missing one that still makes no sense to me: The Ramanujan Summation: 1 + 2 + 3 + ⋯ + ∞ = -1/12 https://en.wikipedia.org/wiki/1_+_2_+_3_+_4_+_%E2%8B%AF

This is a good video explaining it: https://www.youtube.com/watch?v=w-I6XTVZXww Although it relies on a particular interpretation of the left side of the equation, it's used in some areas of physics.

I find a response video to it more enlightening. It's often found as the number one related video.

https://youtube.com/watch?v=YuIIjLr6vUA

Re: Conterintuitive facts in mathematics, CS, and physics

#320

Post by crvdgc »

> 6. Causation does not imply correlation: https://arxiv.org/abs/1505.03118

And in the paper, from the abstract,

> We demonstrate that the Faithfulness property that is assumed in much causal analysis is robustly violated for a large class of systems...

In preliminaries,

> The Faithfulness assumption is that no conditional correlation among the variables is zero unless it is necessarily so given the Markov property.

So probably a more accurate laymam description would be "not all causations are observable as a statistical correlation relationship".

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