Conterintuitive facts in mathematics, CS, and physics • Hacker News Board

Re: Conterintuitive facts in mathematics, CS, and physics

#331

Post by intuitionist » Thu, Oct 07, 2021, 3:37 PM UTC

0xmarcin wrote: post #154

ncmncm wrote: post #150

Number 19 is a clinker. Banach-Tarski applies only to objects in real-number space, but there are no such objects. For that to work, objects have to be infinitely divisible, but all of our objects are made out of atoms. Real-numbered space is a good enough approximation to our experience that we hardly ever encounter a model failure like this one.

Also this only works when the highly contested Axiom of Choice is used.

The paradoxical decomposition of the free group doesn’t rely on choice though, and that’s arguably the “heart” of BT.

Re: Conterintuitive facts in mathematics, CS, and physics

#332

Post by rsj_hn » Thu, Oct 07, 2021, 7:04 PM UTC

kmm wrote: post #328

rsj_hn wrote: post #311

Earlier quoted context omitted.

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.

The density of the Sun isn't uniform of course, but according to this[0] StackExchange answer, the density of matter in the Sun already reaches that of air at about 95% of its radius. https://astronomy.stackexchange.com/questions/32729/at-what-...

Obviously by "density" we mean weight divided by volume. Here is a collection of four textbooks that all agree on the 1.4 figure.

https://hypertextbook.com/facts/1999/MayKo.shtml

Re: Conterintuitive facts in mathematics, CS, and physics

#333

Post by spott » Thu, Oct 07, 2021, 8:56 PM UTC

bo1024 wrote: post #10

Great list! > 33. "...if you flip fair coins to generate n-dimensional vectors (heads => 1, tails => -1) then the probability they're linearly independent is at least 1-(1/2 + o(n))^n. I.e., they're very very likely independent! Counterintuitive facts about high dimensional geometry could get their own list. A side-1 cube in n dimensions has volume 1 of course, but a diameter-1 sphere inside it has volume approaching…

as an aside: what the hell is the "o(n)" here? Some term that is proportional to n?

Re: Conterintuitive facts in mathematics, CS, and physics

#334

Post by makeset » Fri, Oct 08, 2021, 5:38 AM UTC

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.

*Monty Hall

Re: Conterintuitive facts in mathematics, CS, and physics

#335

Post by geogra4 » Wed, Oct 13, 2021, 3:35 PM UTC

dartharva wrote: post #300

TMWNN wrote: post #297

Earlier quoted context omitted.

How do you know, unless you do the calculation?

Don't know about you, but I have never seen any place that charges more for one large pizza than for two medium pizzas.

For example the place by me charges 11 for a 12 inch pizza and 25 for a 24 inch. That's 4x the pizza for a little over twice the price!