[HN Gopher] The most counterintuitive facts in all of mathematic...
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The most counterintuitive facts in all of mathematics, CS, and
physics
Author : raviparikh
Score : 116 points
Date : 2021-10-05 19:28 UTC (3 hours ago)
(HTM) web link (axisofordinary.substack.com)
(TXT) w3m dump (axisofordinary.substack.com)
| tunesmith wrote:
| Someone asked about #4 then deleted after I typed the response,
| so here it is.. :)
|
| It's queuing theory in general, related to "utilization". The
| utilization curve is always shaped the same; 50% utilization
| always doubles waiting time, and the curve is practically
| vertical when you get to 99% utilization. That's what explains
| the difference - 5.8 customers per hour, for a throughput of 6
| per hour, shoves the efficiency to the almost-vertical part of
| the curve, which impacts waiting time.
| whiterock wrote:
| very cool collection - truly something to pick from at a party.
| should have much more upvotes imho.
| anonymousDan wrote:
| Agreed, some real mindbenders there!
| antognini wrote:
| Here is one of my favorites: The specific heat of a star is
| negative. (This also applies to a galaxy or any other
| gravitationally bound object.)
|
| As a star loses energy, it heats up. If you inject it with
| energy, it cools down. It's a trivial corollary from the virial
| theorem, but it leads to counterintuitive behavior (like the
| gravothermal catastrophe).
| bo1024 wrote:
| 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
| zero! The sphere is tangent to every one of the 2n faces, yet
| takes up almost none of the space inside the cube.
|
| Note that the distance from the middle of any face of the cube to
| the opposite face is 1, yet the length of a diameter of the cube
| (corner to opposite corner) is sqrt(n).
| 6gvONxR4sf7o wrote:
| > 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 hand, I don't like. Depending on a whole
| bunch of subjective definitions, a one-in-billion event can
| happen a million times a second or practically never or whatever
| else you choose.
| lotsofpulp wrote:
| >https://en.wikipedia.org/wiki/Potato_paradox
|
| The Wikipedia link above says:
|
| > Fred brings home 100 kg of potatoes, which (being purely
| mathematical potatoes) consist of 99% water. He then leaves
| them outside overnight so that they consist of 98% water. What
| is their new weight? The surprising answer is 50 kg.
|
| It annoys me when mass is used interchangeably with force
| (weight), so I went to the Wikipedia source, and the source is
| accurate in using units of force throughout.
|
| https://web.archive.org/web/20140202214723/http://www.davidd...
|
| Wonder why the person that wrote the Wikipedia article changed
| it up when it is supposed to be a direct quote.
| mint2 wrote:
| I really don't like that example because it makes no sense.
| In no logical circumstance could the potatoes dehydrate so
| quickly when left out over a single night.
| jgtrosh wrote:
| In this situation, the mass and weight are proportional and
| irrelevant to the problem. Other than proper respect of
| units, why would it really matter? I would agree that using
| mass+kg would remain correct and be less unusual, but it
| doesn't matter a lot.
| lotsofpulp wrote:
| It does not matter, it is just a pet peeve of mine. Might
| be due psychological trauma from when I was a kid and
| arguing with an older cousin about how pounds and kilograms
| are not units of the same thing, and the older cousin
| "winning" the argument in the eyes of the elders because
| the cousin was quite a few years older than me and
| considered to be smart in school.
| dwohnitmok wrote:
| 29 is not correct as stated and falls prey to logical errors.
| Hamkins presents a formal take on it here:
| https://mathoverflow.net/questions/44102/is-the-analysis-as-...
|
| His conclusion (which I agree with) is
|
| > The claims made in both in your question and the Wikipedia page
| on the existence of non-definable numbers and objects, are simply
| unwarranted. For all you know, our set-theoretic universe is
| pointwise definable, and every object is uniquely specified by a
| property.
|
| Despite arguments about countability, which ignore how difficult
| it is to pin down "what is definable," it is possible (although
| not necessary) for all real numbers to be describable/definable
| in ZFC.
| hodgesrm wrote:
| > 0% selected the right answer on this SAT question: Circle A has
| 1/3 the radius of circle B, and circle A rolls one trip around
| circle B. How many times will circle A revolve in total?
|
| That's fun. I of course immediately selected 3 which means I
| could have a bright career in test preparation ahead of me.
| clon wrote:
| It was implied that the correct answer is 4.
| brrrrrm wrote:
| 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, Polya's recurrence theorem)
| [deleted]
| Causality1 wrote:
| 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."
| tunesmith wrote:
| Remedial question about zero-knowledge proof. Isn't "proof" a
| misnomer since the concept is about just making it incredibly
| _likely_ to be true?
| eruleman wrote:
| No -- the word 'proof' is accurate, it establishes with
| certainty that you know the value of x.
| jancsika wrote:
| > Knowing just slightly more about the value of your car than a
| potential buyer can make it impossible to sell it:
| https://en.wikipedia.org/wiki/The_Market_for_Lemons
|
| From that Wikipedia page:
|
| > This means that the owner of a carefully maintained, never-
| abused, good used car will be unable to get a high enough price
| to make selling that car worthwhile.
|
| This is bullshit-- the word "enough" was sneaked in there without
| any rationale provided. The most we can say is that a seller--
| _if they decide to sell_ -- won't get as high a price as they
| would if the information assymmetry didn't exist. But that's just
| a truism.
|
| I'd be willing to agree for the sake of argument that we are
| representing humans here as some commonly known set of JSON
| values. But before we go anywhere else from that argument, I'd
| need to know that the speaker will at some point halt the
| simulation and come back to Earth with insights into the real
| world.
|
| Does that happen in this paper? If not, then how does the paper
| have relevance for the economic transactions among the set of
| bona fide human beings?
| rumpelstilz18 wrote:
| > Knowing just slightly more about the value of your car than a
| potential buyer can make it impossible to sell it
|
| > This means that the owner of a carefully maintained, never-
| abused, good used car will be unable to get a high enough price
| to make selling that car worthwhile.
|
| Both statements are a wrong understanding of the phenomenon.
| The market collapses because the transactions happen outside of
| the market. Something similar has in my opinion happened on the
| job market. Most good jobs are outside the regular job boards.
| Most good applicants could not be bothered to apply for a job
| opening but are asked by recruiters or friends.
| MatteoFrigo wrote:
| The list is missing one of the most astonishing discoveries of
| all time: if you reflect the universe in a mirror, you can tell
| whether you are in our universe or in the mirror because the laws
| of physics are different in the mirror. See
| https://en.wikipedia.org/wiki/Wu_experiment
| pishpash wrote:
| The most counterintuitive fact, er... fib is this nugget:
|
| "0% selected the right answer on this SAT question: Circle A has
| 1/3 the radius of circle B, and circle A rolls one trip around
| circle B. How many times will circle A revolve in total?"
|
| You know how hard it is to get 100% of the people to do
| something? Don't insult our intelligence like that, or of SAT
| test takers in general.
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(page generated 2021-10-05 23:00 UTC)