12 ms·
π in Other Universes
- TheOtherHobbes 3y agoAll of these assume your background metric is Euclidean. If your background 2D metric is a projection of a warped 3D space, you can make π as big as you want by tugging on the centre of the circle.
- azeemba 3y agoThere 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 π 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_Values_of_Pi https://www.researchgate.net/publication/353330827_Extremal_...
- charlieyu1 3y agoHow 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 3y agoHmm. And if you keep increasing the radius, pi will shrink all the way to 0.
- azeemba 3y agoThanks for your example! I have been thinking about it. Your observation is correct and the surface of the sphere is a metric. The ratio of radius to circumference is not constant with that metric though so I feel like something should disqualify it. But I am not sure how. So I think your observation shows that we need a stronger constraint than just being a metric. Other commenters have hinted that you need a normed vector space but I am not sure if that's sufficient.
- jimmySixDOF 3y agorelatively 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 3y agoThose 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-mathematically-abused-as-a-child/ https://raypatrick.xyz/blog/2023/10/27/were-you-mathematical...
- Miiko 3y agoIt'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 3y agoWell.... One still obtains pi for the ratio of circumference vs diameter in the limit that the diameter goes to zero.
- IanCal 3y ago> 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 3y agoThe same principle applies to programming. Functions that know less about their arguments are more generally applicable.
- lachlan_gray 3y agoIt 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
- gumby 3y agoThe 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 3y agoThis 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.
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- defrost 3y agoIF 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 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) https://en.wikipedia.org/wiki/Measure_(mathematics)
- thomasahle 3y agoYou 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 3y ago> 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 3y agoTake 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 3y agoNot 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 3y agoSorry, but you're wrong. Pi is 3. (more accurately, 3.2). https://cs.uwaterloo.ca/~alopez-o/math-faq/mathtext/node18.html https://cs.uwaterloo.ca/~alopez-o/math-faq/mathtext/node18.h...
- cvoss 3y agoWhy 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 3y agoYou can have any Pi_Earth you like where 0 < Pi_Earth < Pi_Euclidian
- quickthrower2 3y agoA flat earth would have pi at about 3.14159 though.
- tragomaskhalos 3y agoBased on the attitude of its advocates, a flat earth would lack science and mathematics entirely, so pi would be undefined?
- xigoi 3y agoYou're making the assumption that most flat earth advocates aren't trolls who actually know science very well.
- dclowd9901 3y agoI 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 3y agoI think you're right. Just apply a transform equal to the speed and direction of the wind.
- mannykannot 3y agoThere 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-beam-reach-the-fastest-point-of-sail-on-modern-sailboats https://physics.stackexchange.com/questions/186515/why-is-a-...
- quickthrower2 3y agoA 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 3y agoNote that even if another universe has a different π when it comes to geometry they are still going to also have an important constant that has the same value as our π. E.g., the zeros of the function defined by the series x - x^3/3! + x^5/5! - x^7/7! + ... are nπ where n is an integer and π is our π. Another place our pi will come up is in the exponential function. It's periodic with period 2πi.
- passion__desire 3y agowould their π 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 3y agoRight. Also (just a few more concrete examples): • the sum of the series 4(1 - 1/3 + 1/5 - 1/7 + …) will still be our π: https://en.wikipedia.org/wiki/Leibniz_formula_for_%CF%80 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 π²/6: https://en.wikipedia.org/wiki/Basel_problem 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/π² as N grows large • the product 2(4/3)(16/15)(36/35)(64/63)(100/99)… will still be our π: https://en.wikipedia.org/wiki/Wallis_product 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 π: https://en.wikipedia.org/wiki/Stirling%27s_approximation https://en.wikipedia.org/wiki/Stirling%27s_approximation (e.g. https://www.wolframalpha.com/input?i2d=true&i=N%5C%2891%29Divide%5B%5C%2840%29Divide%5BN%5C%2891%29n%21%5C%2893%29%2C%5C%2840%29%E2%88%9An+Power%5B%5C%2840%29Divide%5Bn%2Ce%5D%5C%2841%29%2Cn%5D%5C%2841%29%5D%5C%2841%29%C2%B2%2C2%5D%5C%2893%29+for+n+%3D+10000 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_involving_%CF%80&oldid=1179793348#Formulae_yielding_%CF%80 https://en.wikipedia.org/w/index.php?title=List_of_formulae_...
- NeoTar 3y agoIn 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 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.
- pr337h4m 3y agoThis is the sort of thing that makes me want to learn VR development.
- lfnoise 3y agoThis 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 3y agoWhat'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 3y agoI 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 3y agoAlso, "broad reach" would like a word with lfnoise. (It's complicated.) https://physics.stackexchange.com/questions/186515/why-is-a-beam-reach-the-fastest-point-of-sail-on-modern-sailboats https://physics.stackexchange.com/questions/186515/why-is-a-...
- bmm6o 3y agoThe 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.
- lfnoise 3y agoI also knew that someone was going to comment the the beam reach was not necessarily the fastest.
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- sepen77 3y agoI 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!
- daxfohl 3y agoThe 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 3y agoOh, I was measuring the sides in Euclidean length. In Manhattan length they're 2000 each, so area is 4 million.
- cyclotron3k 3y agoThe 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 3y agoThe 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 3y agoMy belief, which could be wrong, if we change π 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 3y ago* 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 3y agoI 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 3y agohttps://tauday.com/tau-manifesto#table-quadratic_forms https://tauday.com/tau-manifesto#table-quadratic_forms (Not to sound all Buzzfeed-y, but Table 3 makes a lot of sense)
- BlueTemplar 3y agoYes, and they actually keep using 2pi over and over in their examples.
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- gumby 3y agoWhen 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 3y agoOne 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 3y agoI 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 3y agoWell, 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 3y agoThis 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 3y agoMaybe 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 3y agoI 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 3y agoIs that true for the hexagon, or just very close ?
- i_am_a_peasant 3y agoDidn't 3blue1brown have a video on exactly this?
- lloeki 3y agoThere'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_properties https://en.wikipedia.org/wiki/P-adic_number#Topological_prop...
- kibwen 3y agoExcellent article, both informative and accessible, and the interactive visualizations are lovely.
- 3seashells 3y agoA 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?
- rstuart4133 3y agoOMG. After all this time, you're telling me the drafters of the Indiana Pi Bill [0] could have been right all along? It would mean that Indiana happened to be in a different Universe at the time, were: d=1/(2 √3) ∑( n=1…6 )∣∣x sin(3πn )+y cos(3πn )∣∣ [1] Well, whose to say otherwise? [0] https://en.wikipedia.org/wiki/Indiana_Pi_Bill https://en.wikipedia.org/wiki/Indiana_Pi_Bill [1] Poor man representation of the same equation in the article.