[HN Gopher] p in Other Universes
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p in Other Universes
Author : azeemba
Score : 320 points
Date : 2023-10-29 22:35 UTC (1 days ago)
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| IanCal wrote:
| > Mathematics can be seen as a logic game. You start with a set
| of assumptions and you come up with all the logical conclusions
| you can from that. Then, if someone else finds a situation that
| fits those assumptions, they can benefit from the pre-discovered
| logical conclusions. This means that if some conclusions require
| fewer assumptions, then those conclusions are more generally
| applicable
|
| This is a really, _really_ nice expression of something my mind
| 's been hovering around for a while.
| greydius wrote:
| The same principle applies to programming. Functions that know
| less about their arguments are more generally applicable.
| lachlan_gray wrote:
| It blows my mind to think of mathematics/logic almost like a
| huge cellular automaton. "axioms" don't necessarily correspond
| to "truth", to me they're arbitrary constraints that can give
| rise to complexity. And sometimes the resulting systems can be
| useful
| nylonstrung wrote:
| I love this idea.
|
| You might enjoy Stephen Wolfram's writing- it's exactly what
| you're talking about
| gumby wrote:
| The whole of the puzzles of cosmology actually all might be
| obvious if we had a few different fundamental theorems. But
| because we hit on some that _almost_ work, and then build
| upon them a hole edifice of mathematics that is internally
| consistent and _almost_ fits the universe we keep beating on
| it, not realizing that backing up a little and then driving
| forward again at a slightly different angle might yield a
| simpler, and even more consistent and explanatory system.
| moring wrote:
| This has happened, and is happening all the time. Many
| groundbraking theories in physics can be framed this way.
| The problem is that "slightly different angle" is a huge
| space, so scientists throw a lot of theories at it and see
| what sticks.
| chongli wrote:
| Most mathematics has no application to science whatsoever.
| It's a huge parts bin which scientists delve into when they
| build their models. And then much of the work is in trying
| to shoehorn the mathematics into being tractable.
|
| Mathematics is also not provably internally consistent.
| This was famously shown by Godel [1].
|
| [1] https://en.wikipedia.org/wiki/Godel%27s_incompleteness_
| theor...
| tsimionescu wrote:
| Most mathematics originates from trying to solve physical
| or engineering problems. Typically physicists have been
| on the forefront of mathematical research - this has only
| really changed significantly in the last few decades.
|
| Also, mathematics as practiced is internally consistent.
| It is incomplete, though. That is how it stays afloat of
| Godel's result. Basically Godel's results showed that no
| matter how much we strive, there will always be
| propositions which might be true, but which we will not
| be able to prove are true. Unless of course we start
| using methods that sometimes prove false propositions,
| which we have not done.
| gumby wrote:
| > Most mathematics originates from trying to solve
| physical or engineering problems.
|
| Has this been true since the early 20th century? I have
| no feel for what constitutes "most" in the vast corpus of
| pure mathematics, so am not challenging your claim but
| rather am curious.
| tsimionescu wrote:
| You're right, that might actually be wrong.
|
| However, the claim I was actually thinking of, which is
| right I think, is that the maths used in the physical
| revolutions of the turn of the century (SR, QM, GR, and
| probably QFT, QED, and QCD as well) was invented by
| physicists or by mathematicians working with physicists
| for the express purpose of developing this theories, not
| the other way around.
|
| Also, the basis of mathematics and the first few thousand
| years were indeed motivated by these kinds of concerns.
| defrost wrote:
| I wouldn't agree - consider the hyperbolic transforms
| used to describe space time "bending" wrt relativity:
|
| https://en.wikipedia.org/wiki/History_of_Lorentz_transfor
| mat... In mathematics, transformations
| equivalent to what was later known as Lorentz
| transformations in various dimensions were discussed in
| the 19th century in relation to the theory of quadratic
| forms, hyperbolic geometry, Mobius geometry, and sphere
| geometry, which is connected to the fact that the group
| of motions in hyperbolic space, the Mobius group or
| projective special linear group, and the Laguerre group
| are isomorphic to the Lorentz group.
|
| Mathematicians were following up on "what happens when
| you discard one of Eucilids Axioms" and discovering there
| was an entire world of consistent hyperbolic geometry and
| more.
|
| Some time later: In physics, Lorentz
| transformations became known at the beginning of the 20th
| century, when it was discovered that they exhibit the
| symmetry of Maxwell's equations. Subsequently, they
| became fundamental to all of physics, because they formed
| the basis of special relativity in which they exhibit the
| symmetry of Minkowski spacetime, making the speed of
| light invariant between different inertial frames.
|
| If you read mathematics histories it's a common complaint
| that it's nigh on impossible to discover something new
| and esoteric that doesn't soon end up with a military
| application; the ongoing search for interesting but
| useless mathematics is akin to the search for the
| fountain of youth.
|
| It is the case (IIRC) that quaterions arose directly from
| Hamilton's search for a better way to describe mechanical
| motions in three dimension spaces - ie created to be
| useful from the outset.
| gumby wrote:
| Yes, I agree to some extent with the restricted claim,
| which only (slowly) started to break down in the 17th
| century in the west.
|
| A lot of Indian mathematics was rather abstract going
| back to Vedic times, but since they didn't develop the
| concept of proof, it sadly had little impact on other
| mathematics practice (except as inspiration to Persian
| and Arab scholars) other than the the famous cases of
| zero and positional notation. The mathematical documents
| I've seen from that practice have been in the form of
| essays.
|
| I know little of Chinese or Mesoamerican mathematics and
| wonder where they were on this axis. It seems pretty
| likely that maths started in support of
| astronomy/planting predictions in the cultures I know of
| so likely also for East Asia and the Americas, but
| whither thence did it go?
| WorldMaker wrote:
| I think there's a lot of fascinating mathematical
| "dualism" in how many of those were developed at the same
| time together by both "practical" mathematicians (such as
| physicists) and "theoretical" mathematicians. You feel it
| is easy to argue that because the practical
| mathematicians had an easily defined "need"
| (hypothesis/experiment) they were the "leaders" and the
| arrow flowed from them to the theoretical mathematicians
| working with them, but there's just as much evidence in
| some of those cases that those theoretical mathematicians
| were already doing the theory building on their own and
| had a "need" to find practical use cases/outlets. In some
| cases we know the theoretical mathematician sought out
| the physicist to try to find ways to test a theory and
| were really the ones building the hypotheses. In some of
| the cases we know that though both are generally credited
| for "deep" collaboration after the fact, because they
| never really worked together and did all of their work in
| parallel and it is likely both would have completed just
| about the same work even if they never crossed paths.
| Newton and Leibniz famously never corresponded until
| after both published their own takes on the fundamental
| principles of The Calculus. Alonso Church had already
| developed the Lambda Calculus before corresponding with
| Alan Turing on the fundamentals of Computing and Alan
| Turing couldn't even share most of his practical work
| because it was still state secrets (and there was an
| ocean's distance in their correspondence anyway).
|
| I think as often as not the "arrows" in the diagram point
| both directions _at the same time_ : the practical needed
| the theorist to explain the patterns they were seeing and
| the theorist needed the practical to take the simple
| beautiful thing they were working on and make it
| practical and find the edge cases and complications.
|
| That sort of "dualism" seems an interesting pattern in
| math.
| thriftwy wrote:
| Axioms are not wrong if you can derive some math from them.
|
| They may not correspond to anything in our world, and then we
| usually discover something that does.
| tsimionescu wrote:
| The point wasn't that they're wrong, but instead that they
| are arbitrary. You could create a mathematical system with
| entirely different axioms than what we explore typically,
| and it would only be different in how usefully it maps onto
| real world concepts.
| tetha wrote:
| This is also a part of why I am somewhat fascinated by the idea
| and the state of Lean4 and mathlib in Lean4. People put more
| and more formally verified proofs into mathlib, which in turn
| makes formally proving further theorems in mathlib easier.
|
| If you start with nothing (like in the numbers game), simple
| proofs are a lot of ... just effort, because you have specify a
| lot of rewrites and overall work. In mathlib, however, systems
| like simp (the simplification system) or linarith ("There is a
| solution by linear arithmetic") seem to do a lot of heavy,
| repetitive lifting by now.
|
| It's a really interesting snowball effect. Sadly, everything I
| understand is most likely already in there, so I doubt I could
| contribute meaningfully, haha.
| IanCal wrote:
| That's very interesting, I'm no mathematician but I should
| have a play around with it.
|
| > Sadly, everything I understand is most likely already in
| there, so I doubt I could contribute meaningfully, haha.
|
| I wouldn't be so sure - and even if so then remember there's
| enormous benefit to improving tooling around a system. If you
| want to be involved somehow, better devx, tutorials, output,
| packaging, error messages all make a big difference to end
| users.
|
| Edit -
|
| As another thought, is there benefit in going through papers
| and translating that work into lean4? I'm not really familiar
| enough with it but if so that may
|
| 1. Find issues in current work, like Tao did in his own work
|
| 2. Add to a reusable body of work
| lmpdev wrote:
| Former computational mathematics major
|
| You absolutely can contribute meaningfully
|
| The maths world is incomprehensibly broad and deep, even if
| you just take the Erdos approach and go for interesting but
| shallow problems
| random_ wrote:
| A good example of that is how the Axiom of choice impacts the
| measure/probability theory.
|
| It imply the existence of some sets that cannot be Lebesgue
| measured (which is an generalization of width, volume, etc for
| arbitrary sets, also generalization of probability for
| arbitrary sets)... but it's not possible to present a single
| example of those non measurable sets, only prove that they
| exist.
|
| And it's possible to construct an alternative theory with the
| axiom of determinacy, then any subset of R is measurable.
|
| * https://en.wikipedia.org/wiki/Axiom_of_choice *
| https://en.wikipedia.org/wiki/Axiom_of_determinacy *
| https://en.wikipedia.org/wiki/Lebesgue_measure
| jerf wrote:
| Mathematics is humanity's longest running, largest-scoped, most
| complicated game.
|
| It also happens to be useful, and you can dive into a lot of
| philosophy about that which is all very interesting. The
| utility itself is a large thing on its own. But I think of that
| utility as something separate from the game itself. The game is
| just a game. You can do whatever you want with it. If you want
| to convert your cookbook to hexadecimal just for fun, you can.
| The fact that it is (broadly speaking) useless, that it will
| produce no new knowledge, and if anything negative utility in
| general, doesn't mean you can't do it.
|
| That's the game.
|
| You can also try to play the game to prove the Twin Prime
| conjecture. That's a much harder level.
|
| This game is scalable to all ages and skill levels, has the
| best level variety, and can be done with anything from just
| your personal noggin, to a pencil & paper, to the largest
| computing cluster in the world. Technically all other games you
| play are a subset of this game; that may not always be a useful
| way to think of it, but it is technically true. And while there
| are a few rules, generally, nobody can tell you how to play it.
| You want to color pretty pictures? The game has lots of ways of
| doing that. You want to smash atoms together? The game can help
| with that. You want to simply count to the highest number you
| possibly can? Go for it. It's a very popular play with the
| younger players, but anyone can do it.
| defrost wrote:
| _IF_ I 've correctly assesed the zeitgeist of HN postings
|
| _THEN_ it follows Terence Tao 's Introduction to Measure Theory
| must be a bullet.
|
| https://news.ycombinator.com/item?id=38064211
|
| But seriously, who's going to read|skim a free 260+ tract on
| measure theory?
|
| https://en.wikipedia.org/wiki/Measure_(mathematics)
| thomasahle wrote:
| You don't just read/skim Tao's lecture notes. I used them to
| teach myself measure theory to skip some prerequisites at
| university, and they were _hard_. Every other page is a list of
| exercises. I doubt you would learn much if you didn't take time
| to solve them. But they are hard exercises.
| ajkjk wrote:
| > who's going to read|skim a free 260+ tract on measure theory?
|
| Why is that so hard to believe? People read 260 page books all
| the time.
|
| I'm not going to read this one, but only because it's not my
| area of interest. I'm busy reading 100+ page books on other
| subjects.
| defrost wrote:
| Take that as a tongue in cheek comment - I've read such
| things with close attention, I was studying measure theory
| back in the 1980s when I first met the author of this work
| here in Australia.
|
| There is a subset of people on HN that do read and enjoy
| mathematical texts, they _appear_ outnumbered by a larger
| group that seem to post and comment on anything Terence Tao
| without seeming to be that deep in the actual math, which is
| fine, but it has struck me as a HN trend of late.
| tedunangst wrote:
| Not sure about your universe, but here on earth, pi is 2. The
| length of the equator is 4 times the distance from the pole.
| (Approx.)
| simonblack wrote:
| Sorry, but you're wrong.
|
| Pi is 3. (more accurately, 3.2).
| https://cs.uwaterloo.ca/~alopez-o/math-faq/mathtext/node18.h...
| cvoss wrote:
| Why stop there? Take the circle with its center at one pole,
| its radius running an entire meridian, and its perimeter making
| an infinitesimally tight loop around the other pole. That
| exhibits a pi that's zero.
| quickthrower2 wrote:
| You can have any Pi_Earth you like where 0 < Pi_Earth <
| Pi_Euclidian
| quickthrower2 wrote:
| A flat earth would have pi at about 3.14159 though.
| tragomaskhalos wrote:
| Based on the attitude of its advocates, a flat earth would
| lack science and mathematics entirely, so pi would be
| undefined?
| xigoi wrote:
| You're making the assumption that most flat earth advocates
| aren't trolls who actually know science very well.
| dclowd9901 wrote:
| I must be missing something. In the example of using a sailboat
| with constant wind and distance, wouldn't sailing against the
| wind (let's call it any constant oppositional force), cause us to
| get a circle, just shifted from the origin? Not an ellipsis?
| SamBam wrote:
| I think you're right. Just apply a transform equal to the speed
| and direction of the wind.
| mannykannot wrote:
| There are a number of complications. Two of them are that
| firstly, when when the wind direction is within about 45
| degrees against the direction you want to go, you have to tack,
| and secondly, a reasonably efficient sailboat is fastest when
| it is on a reach, with the wind coming from the side.
|
| https://physics.stackexchange.com/questions/186515/why-is-a-...
| TheOtherHobbes wrote:
| All of these assume your background metric is Euclidean.
|
| If your background 2D metric is a projection of a warped 3D
| space, you can make p as big as you want by tugging on the centre
| of the circle.
| azeemba wrote:
| There is no concept of the "background metric" here. Both the
| radius and the circumference are measured in the defined metric
| itself.
|
| Any metric that "pulls on the origin" compared to Euclidean
| distance will have to do the mapping in a continuous way. This
| will basically result in both the radius and circumference
| being expanded in that metric.
|
| Matter of fact, I linked an article that proves that for _all_
| metrics, the value of p is always between 3 and 4 (inclusive).
| Unfortunately the article might have gotten the hug of death so
| here is an alternative link:
| https://www.researchgate.net/publication/353330827_Extremal_...
| charlieyu1 wrote:
| How is circumference defined?
|
| And I can think of a counterexample on a sphere, just using
| Euclidean distance on the surface. Consider a circle with
| centre at North Pole and radius being the distance from the
| North Pole to a point on the equator. For this circle it is
| easy to find out that pi=2
| Filligree wrote:
| Hmm. And if you keep increasing the radius, pi will shrink
| all the way to 0.
| jimmySixDOF wrote:
| relatively completely off topic but everything I understand
| about pi has come from 3D gif models I never saw in school they
| should be a core part of the learning curve much further to the
| start of it than 3B1B
| seanw444 wrote:
| Those GIFs really do make it super simple. I learned it the
| same. The unit circle made absolutely no sense to me, and
| appeared as yet another dogmatic arbitrary "rule" shoved down
| my throat in school. Had they made an attempt to make it
| intuitive by showing one single GIF, it'd have all come
| together for me much quicker.
|
| Math is far more elegant than public school allows it to
| appear.
|
| https://raypatrick.xyz/blog/2023/10/27/were-you-
| mathematical...
| Miiko wrote:
| It's not the background metric but the space geometry is
| assumed Euclidean - in non-Euclidean geometry the ratio of of
| the circumference of any circle to the diameter of that is not
| a constant, it depends on such diameter (so you simply cannot
| define 'pi' in that case)
| cjfd wrote:
| Well.... One still obtains pi for the ratio of circumference
| vs diameter in the limit that the diameter goes to zero.
| quickthrower2 wrote:
| A p-norm of 3 make quit a funky "retro" rounded corner style for
| avatars. It also shows what the "opposite" of rounded corners
| might look like. In CSS you can only go to a circle.
| tzs wrote:
| Note that even if another universe has a different p when it
| comes to geometry they are still going to also have an important
| constant that has the same value as our p.
|
| E.g., the zeros of the function defined by the series x - x^3/3!
| + x^5/5! - x^7/7! + ... are np where n is an integer and p is our
| p. Another place our pi will come up is in the exponential
| function. It's periodic with period 2pi.
| passion__desire wrote:
| would their p re-emerge if unit distance i.e distance between 2
| and 3, 5 and 6 is defined by their metric. sort of like change
| in base in number systems.
| svat wrote:
| Right. Also (just a few more concrete examples):
|
| * the sum of the series 4(1 - 1/3 + 1/5 - 1/7 + ...) will still
| be our p:
| https://en.wikipedia.org/wiki/Leibniz_formula_for_%CF%80
|
| * the sum of the series (1 + 1/4 + 1/9 + 1/16 + 1/25 + ...)
| will still be p2/6: https://en.wikipedia.org/wiki/Basel_problem
|
| * (therefore) the probability that two numbers chosen uniformly
| at random from [1...N] are relatively prime will still approach
| 6/p2 as N grows large
|
| * the product 2(4/3)(16/15)(36/35)(64/63)(100/99)... will still
| be our p: https://en.wikipedia.org/wiki/Wallis_product
|
| * the value of (n!/([?]n (n/e)^n))2/2 as n grows large will
| still (very slowly) approach p:
| https://en.wikipedia.org/wiki/Stirling%27s_approximation (e.g.
| https://www.wolframalpha.com/input?i2d=true&i=N%5C%2891%29Di...
| )
|
| and so on, for most of the non-geometry results listed:
| https://en.wikipedia.org/w/index.php?title=List_of_formulae_...
| NeoTar wrote:
| In a change from the normal refrain of 'there's an XKCD about
| that' - in this case there is an Saturday Morning Breakfast
| Cereal (SMBC) about it: https://www.smbc-
| comics.com/comic/pi-2?ref=refind
|
| For those unwilling to click-through, it essentially posits
| an alternate history where infinite series were explored by
| mathematicians before geometry, so rather than being
| surprised that the 'circle constant' is found in many
| infinite series, we would instead be surprised that the
| 'infinite series constant' is found in the geometry of a
| circle.
| thechao wrote:
| Pi is the scaling factor of the diameter of a circle to its
| circumference; there's an infinite set of such scaling
| factors: one for each ellipse (the circle is a special
| case). I wonder which/what sort of infinite series arise
| from/for the generalized elliptic scaling factors?
| NeoTar wrote:
| I rather suspect that the generalised elliptic scaling
| factor is a continuous function, so the answer may be a
| bit boring. For any infinite series with a finite sum I
| would be able to give you an ellipse (indeed, probably an
| infinite number of ellipses) whose scaling factor is a
| rational multiple of the sum.
| nlitened wrote:
| > Another place our pi will come up is in the exponential
| function. It's periodic with period 2pi.
|
| Isn't it the opposite? As I understand, we (European
| civilization humans) historically _define_ our complex
| exponential function to have a period of 2pi to match the
| period of our previously defined sin and cos functions.
|
| We could have defined it to have another period -- for example,
| if we define "360deg angle" to be equal to 1 instead of 2*Pi,
| and define sin0=0, sin0.25=1, sin0.5=0, sin0.75=-1, sin1=0,
| we'd also define periodicity of e^ix to be 1.
|
| UPD: Same idea as for why we use base-ten numbers. The only
| reason is that we have ten fingers on two hands, and
| historically we've been using base-ten numbers for the past few
| hundred years. But there's no reason to expect that "aliens"
| would be having ten digits also.
| moring wrote:
| Where in the definition of the complex exponential function
| is p used? IIRC its: the exponential function is defined to
| be its own derivative (note 1), i is the square root of -1,
| and exp(ix) is observed to have a period of 2p. There isn't
| any arbitrary choices in there that could be said to be
| defined in such a way that p results.
|
| note 1: exp(x) can alternatively be defined by the
| exponential series, but that series does contain arbitrary
| numbers that could be said to be selected in such a way that
| p results.
| nlitened wrote:
| As I understand, "complex exponential" function f(x) = e^ix
| must satisfy only two equalities: f(0) =
| 1 f'(x) = i f(x)
|
| So any function that satisfies these equalities can work as
| a "complex exponential" function which we denote as e^ix.
|
| So we can define a function with period of 1, and use it
| everywhere -- then "2p" vanishes from most equations, and
| the complex math still works and all equalities hold.*
| messe wrote:
| The _only_ function that satisfies those two equalities
| is e^ix which has period 2p.
| nlitened wrote:
| Hey, you're actually right! My bad
| jojobas wrote:
| Consider simultaneous functional equations:
|
| f(x) = dg(x)/dx
|
| g(x) = df(x)/dx
|
| Its only linearly independent solutions are sin(x+c) and
| cos(x+c) with, x being in radians, periodic by 2pi.
| nlitened wrote:
| Yep, you're right, my bad.
| xigoi wrote:
| You forgot a minus before one of the equations, otherwise
| you get sinh and cosh.
| jojobas wrote:
| Yep, silly me.
| jlokier wrote:
| _> As I understand, we (European civilization humans)
| historically _define_ our complex exponential function to
| have a period of 2pi to match the period of our previously
| defined sin and cos functions. We could have defined it to
| have another period -- for example, if we define "360deg
| angle" to be equal to 1 instead of 2_Pi, and define sin0=0,
| sin0.25=1, sin0.5=0, sin0.75=-1, sin1=0, we'd also define
| periodicity of e^ix to be 1.*
|
| No, it doesn't work in degrees.
|
| The definition of e isn't that arbitrary.
|
| 2p is the unique period which satisfies the definition of e
| using derivatives and the extension of real number algebraic
| laws to complex numbers. This shows up as a real world
| physical measurement, which I describe below.
|
| The (natural) exponential function ex is defined as the
| unique function which _equals its own derivative_ and
| satisfies e0 = 1 (like other exponentials). The value of e
| comes from this.
|
| Combine that with the definition i2 = -1 and using basic
| rules of algebra which are observed on real numbers with
| exponentials and derivatives (such as (xa) = xa) and you find
| the function ex must be periodic with period 2pi.
|
| This comes from sin(x) and cos(x) and their derivatives. The
| derivative of sin(x) is cos(x), and of cos(x) it is -sin(x),
| but _only_ if sin(x) and cos(x) are defined in the usual math
| way with period 2p.
|
| Those sin/cos derivatives and that little negative sign are
| enough to make them components of the unique solution to the
| derivative definition of ex applied to a complex argument,
| and thereby fix its period in the complex plane and prove
| Euler's famous identity (without needing the Taylor
| expansion).
|
| That in turn has.a more physical basis. Asin(x+B) with
| constants A, B are the family of functions whose second
| derivative equal themselves negated.
|
| Physically, it means an object whose acceleration is
| proportional to its displacement from a fixed position and in
| the opposite direction will oscillate with a period of
| exactly 2p seconds, if the acceleration is -1m/s2 per 1m
| displacement.
|
| This setup is called a harmonic oscillator.
|
| In this way, 2p arises (and is measurable!) from physical
| properties of time, force and inertia, of things moving in
| straight lines.
|
| No circles required.
| nlitened wrote:
| I agree with you that _e_ is not arbitrary. I say that the
| period of 2p for e^ix is arbitrary, because we 've
| arbitrarily defined periods of _sin_ and _cos_ as 2p.
|
| If we defined a function _sin_ to take not an angle in
| radians, but in degrees (with a period of 360.0), and used
| that definition of _sin_ in our math, then our complex e^ix
| would have a period of exactly 360, and the entire complex
| math would still work -- for example, Euler 's formula
| below would still hold: e^ix = cos x + i
| sin x
|
| And people in comments would rave about how magic number
| 360 is, and its magic properties were discovered by Romans
| two thousand years ago.
| codeflo wrote:
| You seem to think that the 2pi is injected into the
| definition of e^ix somewhere, but actually it's the other
| way round, 2pi comes out as a theorem. I'll give the
| rough outline.
|
| exp(x) for complex x is simply defined to be the infinite
| sum from k = 0 to infinity of x^k/k!. That is, exp(x) = 1
| + x + x^2/2 + x^3/6 + x^4/24 + x^5/120 ...
|
| (BTW, the motivation for this definition is that exp'(x)
| = exp(x), which shouldn't be too hard to see because it's
| already a Tailor series.)
|
| Purely from this you can prove that exp(ix) with real x
| is periodic with period 6.28...
|
| It just so happens that this number is also the
| circumference of the unit circle.
| quickthrower2 wrote:
| I had never spotted that before, each term of the series
| is the integral of the previous. That is pleasing!
|
| The Pi thing feels now less of a coincidence than the
| fact that exp is a power. That probably falls out of
| expanding the polynomials but it so ingrained as taken
| for granted that it is wonderous when you think about it.
| nlitened wrote:
| Hey, you're right! So Pi _is_ special :)
| quickthrower2 wrote:
| I guess radians are "magic" in that sin and cos can be
| defined by infinite series that look nice (and feel
| canonical). You have to manipulate those series to get
| 360 or even just revolutions.
| nlitened wrote:
| As codeflo showed in a sibling comment, actually I am
| wrong, and Pi's special magic also becomes from Taylor's
| series expansion. So turns out that Pi is the real magic
| number rather than just our arbitrary choice!
| zoomablemind wrote:
| Fact that pi is irrational may point at something fundamental
| missing in our knowledge system.
|
| It appears that we cannot precisely measure circle
| length/area in units of radius and vice versa. Basically, the
| unity as such does not exist in our knowledge, nor can we
| truly comprehend infinity.
|
| Perhaps, unity and infinity are just our abstractions for
| something else.
| xigoi wrote:
| The fact that p is irrational has absolutely nothing to do
| with physically measuring circles or with infinity. We know
| the value of p exactly.
| zoomablemind wrote:
| > ...We know the value of p exactly.
|
| Perhaps you could share that exact value with the rest of
| humanity. And I mean the number value, not the nominal
| value.
| gus_massa wrote:
| I think it's better to say that p is the same number everywhere
| 3.14... , but in other universe you don't use p in the formula
| of the length of a circle.
|
| * Manhattan (L_1): C = 8 R
|
| ...
|
| * Euclidean (L_2): C = 2p R
|
| ...
|
| * Maximal Distance (L_infinity): C = 8 R
| pr337h4m wrote:
| This is the sort of thing that makes me want to learn VR
| development.
| lfnoise wrote:
| This person is not a sailor. Sailing orthogonal to the wind, a
| "beam reach", is the fastest point of sail due to the lift of the
| sail.
| jacquesm wrote:
| What's interesting is that if you manage to exceed the hull
| speed doing that you'll end up surfing on your own bow wave!
| Zanni wrote:
| I knew someone would make this comment. I love HN for this kind
| of pedantry when it's specific, accurate and doesn't dismiss
| the entire article for one inaccurate analogy.
| jws wrote:
| Also, "broad reach" would like a word with lfnoise. (It's
| complicated.)
|
| https://physics.stackexchange.com/questions/186515/why-
| is-a-...
| bmm6o wrote:
| The polar diagram shown there is what should replace the
| ellipse in TFA. It's far more complicated than a simple
| geometric shape since it has to account for such
| practicalities as sail inventory.
| sepen77 wrote:
| I didn't know anything about sailing, but your one comment made
| me search up point of sail and now you've opened my eyes to
| something that was a mystery to me for all my life -- how
| sailboats can "course made good" against the wind. Thank you..
| this stuff is amazing, and sailing is an incredible science!
| joelthelion wrote:
| What does the "circle" look like with correct assumptions?
| mnw21cam wrote:
| A beam reach isn't necessarily the fastest point of sail. It
| depends on the boat, the efficiency (lift/drag ratio) of the
| sail, and the efficiency of the centreboard/keel (again,
| lift/drag ratio), but a reach of some kind is likely to be the
| fastest - it just won't be exactly perpendicular to the true
| wind direction. It'll also vary with the wind speed, wave
| height, weight distribution, etc.
| daxfohl wrote:
| The area of the circle in Manhattan distance comes out to 2
| million, but pi * r^2 is 4 million. What am I doing wrong?
| daxfohl wrote:
| Oh, I was measuring the sides in Euclidean length. In Manhattan
| length they're 2000 each, so area is 4 million.
| cyclotron3k wrote:
| The boat analogy seems particularly poor.
|
| a) Comparing a sailboat on a windy day to a sail boat on an
| [implied] non-windy day? Surely the boat with no wind wouldn't
| even have a circle.
|
| b) I'm no boatologist, but if the wind is X knots, then the boat
| can travel downwind at a rate of X knots, but contrary to what
| the article states, the boat would be able to travel cross-winds
| at some multiple of X. So you would get something resembling an
| oval, but in the opposite orientation as depicted.
|
| Also, it's worth pointing out that it's perfectly possible for a
| boat to travel "into" the wind via "tacking and jibing"
| skykooler wrote:
| The hexagonal metric at the end uses pi in its definition - is
| this our value of pi, or the value of 3 that that metric
| provides?
| passion__desire wrote:
| My belief, which could be wrong, if we change p and distance
| metric i.e. definition of unit distance between 1 and 2, 4 and
| 5, 10 and 11 to be their unit distance, all the equations
| involving numbers and pi would come out to be same. e.g. basel
| problem etc.
| codeflo wrote:
| * pi = 3.14159... appears in analysis and by extension
| statistics, independent of geometry. So aliens in these other
| universes would know this value, they'd just have a different
| constant for circles. Since they wouldn't use Greek letters
| anyway, we'd have to translate, and it would be a bit silly to
| equate their 3.757... with "pi" instead of their 3.14159...
|
| * Personal aside: Of course, whether 3.14... (pi), 6.28... (2pi)
| or even 0.785... (pi/4) should be the fundamental constant is
| debatable, and aliens might have different ideas about that.
|
| * The article introduces the concept of metrics to explain that
| there could be different circle constants in other universes. But
| arbitrary metrics don't necessarily have linear scaling or
| translation invariance. You need stronger assumptions than a
| metric to meaningfully define a circle _constant_ at all, like a
| normed vector space. AFAICT, all of the given examples are in
| fact normed vector spaces, not just metric spaces.
| dylukes wrote:
| I don't find the first point surprising. (Our) pi is the one
| tied to the only metric where the unit circle is perfectly
| continuous, differentiable, etc.
|
| The 2-norm is _very special_ for many reasons I won 't
| enumerate... and it seems apropos that its corresponding
| constant (pi)... for relating a distance from a point (wlog
| 0,0) to the result of integrating a constant around the path
| those points occupy/form/consist in... would itself tend to be
| found more than others.
|
| Perhaps this is simply because without that continuity and
| differentiability everywhere of the corresponding path
| generated by the metric's unit circle, many other pieces would
| fall like dominoes.
|
| There is something uniquely central about a concise relation
| between a point, a distance, and a path.
| PennRobotics wrote:
| https://tauday.com/tau-manifesto#table-quadratic_forms
|
| (Not to sound all Buzzfeed-y, but Table 3 makes a lot of sense)
| BlueTemplar wrote:
| Yes, and they actually keep using 2pi over and over in their
| examples.
| gumby wrote:
| When I was a kid I liked to muse about relationships like these.
| Since I was a kid I imagined that there might have been a god
| that created the universe, and imagined that they were a bored
| kid like me perhaps making it as a school assignment.
|
| So what if the god had turned the pi or e knobs to a rational
| number (presumably in a god's universe knobs can be turned to
| precise irrational values). Would it have made our lives easier
| or harder (probably easier...?). Or what about the apparent size
| of earth/moon/sun when viewed from earth? It's a great clue, but
| perhaps we would have known more about astronomy if that
| coincidence had not existed? (We would have missed out on that
| fabulous Connie Willis story though).
|
| Maybe all those weird cosmological QM oddities and (literally
| obscure) imbalances needing mysterious dark matter are just due
| to bugs in a kid's rushed assignment and actually don't make
| sense?
|
| But the irrationals...they led to the most musing.
| msds wrote:
| One thing this doesn't touch on is that there are multiple
| meaningful definitions of pi-like constants for the p-norm unit
| circle that don't necessarily agree with each other in p != 2.
| Defining pi as the area of the unit circle gives an entirely
| different set of values that satisfying some wonderful properties
| - in particular, that definition of pi turns out to be the
| periodicity constant for a (arguably) natural set of
| trigonometric functions for the p-circle. Furthermore, pi(p) = 2
| Beta(1/p,1/p)/p...
|
| However, this (circumference/arc-length based) definition of pi
| does have a fascinating property for conjugate p,q: pi(p) = pi(q)
|
| "Squigonometry: The Study of Imperfect Circles" is a very fun
| reference for this sort of stuff.
| waveBidder wrote:
| I wonder whether not being a Hilbert space has any awkward
| implications for geometry. I guess we have to chuck out the
| Polarization identity, which probably has implications for
| parallelograms, though I'm not sure quite what. anyway, thanks
| for the rec!
| msds wrote:
| Well, there isn't a meaningful inner product, so how can you
| speak of parallelograms? The geometries are definitely weird!
| Once you leave p=2 and break the rotational symmetry around
| the origin, the only isometries in your geometry are signed
| permutation matrices - so geometry "over here" looks
| different from "over there". Angles aren't really meaningful,
| I guess.
|
| The other interesting thing is that duality kicks in (or
| maybe becomes non-trivial, since it's always there) and
| derivatives naturally start to live in a different space. If
| you take the particularly natural definitions of general
| cos_p and sin_p I alluded to, you get a nice parameterization
| of the unit p-circle as (cos_p(t), sin_p(t)) - but if you
| differentiate this wrt t, the resulting tangent vectors don't
| lie on the p-circle. Instead, they form a parameterization
| for the q-circle!
| WiSaGaN wrote:
| This largely depends on how one defines pi. I believe that the
| concept of R^n (Euclidean space) exists even in entirely
| different physical spaces. This is because Euclidean space
| represents a universally recognized idea of simple space in terms
| of curvature. For instance, in any world, the concept of '0'
| represents simplicity. In this context, pi will always remain
| constant.
| nologic01 wrote:
| Maybe worth pointing out that there are countless other weird
| Universes where "Pi" retains its standard value.
|
| This is the domain of differential geometry where the relation of
| circumference and radius holds only in the limit of
| infinitesimally small.
|
| By all accounts our own Universe is of such a deformed-in-the-
| large but Euclidean-in-the-small variety. At least for as far we
| understand geometry in the quantum realm.
| wcoenen wrote:
| I noticed that all the "circles" for alternative metrics are
| aligned with the coordinate system. For example, the one for the
| Manhattan distance has its corners on the coordinate axes.
|
| What if we added an additional condition that a distance metric
| should not change when the orientation of the coordinate system
| is changed? Could we still have different values for the pi
| constant then?
| BlueTemplar wrote:
| Is that true for the hexagon, or just very close ?
| i_am_a_peasant wrote:
| Didn't 3blue1brown have a video on exactly this?
| lloeki wrote:
| There's this fun space made of p-adic numbers upon which you can
| define a simple distance, and then circles have mind bending
| properties like the diameter (max edge to edge distance) and
| radius (distance from edge to center) being equal to each other.
|
| Quirky stuff happens to disc area and perimeter as well, and open
| discs are also closed. The equivalent of Pi there is nuts.
|
| Sadly I can't recall the details (it was a 2000-ish exercise on
| my maths course).
|
| https://en.wikipedia.org/wiki/P-adic_number#Topological_prop...
| kibwen wrote:
| Excellent article, both informative and accessible, and the
| interactive visualizations are lovely.
| 3seashells wrote:
| A circle is a pillar for a 3 dimensional universe in a 2
| dimensional universe. So I guess every dimensional jump has one
| and the binary one is the origin constant?
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