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Conterintuitive facts in mathematics, CS, and physics

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Re: Conterintuitive facts in mathematics, CS, and physics

#121

Post by robocat »

brrrrrm wrote: post #15

a couple more: - at any time while stirring a cup of coffee, there will be a point on the top that is right where it started. (if we pretend coffee stirring is 2-dimensional, Brouwer's Fixed Point theorem) - a drunk man will eventually make it home, unless he can fly. in which case he only has a 34% chance. (if we assume the man is walking/flying on a grid, Pólya's recurrence theorem)

- a chair with four even legs placed on any undulating continuous surface can always be rotated such that all four legs touch the ground at once.

Re: Conterintuitive facts in mathematics, CS, and physics

#122

Post by caf »

hinkley wrote: post #59
lapetitejort wrote: post #25

My favorite paradox in physics: what happens when you spin a disk at relativistic speed? The circumference should contract, since it's parallel to the direction of motion, but the radius is perpendicular, and thus should not contract. https://en.wikipedia.org/wiki/Ehrenfest_paradox

I believe Veritasium covered this: spinning at relativistic speeds generates centripetal forces that overcome the nuclear force. There is no material you can use to test this paradox, and there can be no such material because your device would atomize - or worse - in the attempt. Effectively there are g-forces so high that you end up with subatomic particles.

Yes, this is similar to transmitting a message faster than the speed of light by moving a very long perfectly rigid rod; a perfectly rigid rod is not realisable and neither is a perfectly rigid disk.

Re: Conterintuitive facts in mathematics, CS, and physics

#123

Post by caf »

Ansil849 wrote: post #82
guerrilla wrote: post #79

Earlier quoted context omitted.

The area enclosed by a circle is πr^2 while 12 and 18 are the diameters, right? The radius of a disc is half the diameter, so 6 and 9, respectively. 2π6^2 < 1π9^2 ~~ 226.2 < 254.5. In other words the area is proportional to the square of the radius, not linearly proportional to the diameter in any way.

> In other words the area is proportional to the square of the radius, not linearly proportional to the diameter in any way. Awesome, thanks for that geometry refresher, that makes sense now of course :).

Similarly, if the toppings are distributed evenly across the pizza, the average distance of a piece of topping from the edge is one third of the radius: most of the pizza is near the edge.

Re: Conterintuitive facts in mathematics, CS, and physics

#124

Post by caf »

robocat wrote: post #121
brrrrrm wrote: post #15

a couple more: - at any time while stirring a cup of coffee, there will be a point on the top that is right where it started. (if we pretend coffee stirring is 2-dimensional, Brouwer's Fixed Point theorem) - a drunk man will eventually make it home, unless he can fly. in which case he only has a 34% chance. (if we assume the man is walking/flying on a grid, Pólya's recurrence theorem)

- a chair with four even legs placed on any undulating continuous surface can always be rotated such that all four legs touch the ground at once.

Do the legs have to be even?

Re: Conterintuitive facts in mathematics, CS, and physics

#126

Post by barbs »

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.

Re: Conterintuitive facts in mathematics, CS, and physics

#127

Post by marcosdumay »

d_burfoot wrote: post #114

Here is one of my favorites. Take a light source and shine it at a filter that is polarized up/down that is in front of a filter that is polarized left/right. None of the light will get through: the first filter removes all of the L/R components of the light, and the second filter removes all the remaining light. Now add a third filter, between the two, which is polarized at 45 degrees. Now some of the light goes thr…

Not surprisingly, this is related to the observer effect of Quantum Mechanics.

Re: Conterintuitive facts in mathematics, CS, and physics

#128

Post by lifeplusplus »

duxup wrote: post #103
6gvONxR4sf7o wrote: post #11

> 16. If you let a 100g strawberry that is 99% water by mass dehydrate such that the water now accounts for 98% of the total mass then its new mass is 50g: https://en.wikipedia.org/wiki/Potato_paradox I really like this one. It's a perfect combo of intuitive from one perspective and mind bending from another. > 18. A one-in-billion event will happen 8 times a month: https://gwern.net/Littlewood This one, on the other…

16 is like the money hall problem. I understand the answer, the answer makes sense to me. And yet but when I think of it how I think of it initially ... it still makes no sense.

For me the mistake was 1:1 relation between percentage and weight which didn't remain true after weight loss but I thought it did

Re: Conterintuitive facts in mathematics, CS, and physics

#129

Post by Kranar »

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

I love that one, what this taught me is that infinite sums are fundamentally different from finite sums, and that when operations performed on the finite are generalized to work with the infinite, there are often ambiguities and subtleties about how to perform that generalization.

We wish to use the same familiar notation with the infinite that we do with the finite, but we must keep in mind that while they do share similarities, they are not the same operation. The way certain ambiguities are resolved in order to extend the finite to the infinite can lead to counterintuitive results and absurdities that may not even be apparent at first.

For me, seeing how one approach to generalizing infinite sums yields the -1/12 result, and how that result actually has some relevance in physics is quite profound and insightful and I am happy that you reminded me of it.

Re: Conterintuitive facts in mathematics, CS, and physics

#130

Post by caf »

BeetleB wrote: post #85
Causality1 wrote: post #16

Considering spacetime, matter, and energy are all quantized, why is something like Gabriel's Horn significant? I don't see how it has any more relation to reality than phrases like "negative surface area" would. Also, it's patently absurd someone would include Fitch's Paradox, a piece of philosophy, on a list of "counterintuitive facts."

Gabriel's Horn was cool till someone pointed out to me that you can have a line of infinite length within a square (trivially). When comparing something of a certain dimension with something of a higher dimension, it's not at all surprising that the lower one can be infinite and the higher one finite. Usually it's phrased as "a finite amount of paint can paint an infinite area." But why do I need the Horn to realize…

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.

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