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Is π the Same in Every Universe?
- deleted 5y ago[deleted]
- jiggawatts 5y agoIf you define pi based on physical circles, then it isn't even the "same pi" as the mathematical one in this universe. In fact, it changes locally from point to point. That's literally what General Relativity says: spacetime is curved!
- TheCraiggers 5y agoBut wouldn't the measuring device also be curved?
- skj 5y agoI measure distances in the time it takes light to traverse them, don't you?
- enkid 5y agoSomething being a circle or an oval would depend on your relative velocity, meaning the idea of a physical circle doesn't really make sense.
- Retric 5y agoNot quite. Observing an object with high relative velocity distorts the observed shape, but all observers can agree what the shape would be at zero relative velocity. For similar reasons astronomers say distant objects are redshifted rather than assuming they are the colors directly observed.
- ben_w 5y agoAlthough in the special case of a sphere, despite the length contraction, the observed behaviour is a rotation because light travel time perfectly balances length contraction: https://www.spacetimetravel.org/fussball/fussball.html https://www.spacetimetravel.org/fussball/fussball.html
- ithinkso 5y agoI don't think this works in GR the way you think it works (regardless of what 'physical circle' means), aren't you always able to change coordinates for the space to be locally flat? (This always confuses me, can someone that knows GR shed some light on this? I would greatly appreciate it. I've been trying to teach myself GR a while back from the lectures of Frederic Schuller and Alex Flournoy that are available on youtube but it's hard without being able to ask questions)
- krastanov 5y ago"Locally flat" means flat only very close to the point of consideration. That is the mathematical definition, the "practical" definition is "flat only as long as you do not stray further away than some characteristic measure of distance defining your problem and accuracy requirements". So as long as your circle is very small then the " experimentally measured" pi and the math pi are the same. The moment your circle becomes big enough to pass through interesting (measurable) gravitational effects, the experimental value will be different. If anything, experimentally measuring that difference is one way to experimentally measure spacetime curvature.
- ithinkso 5y agoThat's why I asked about GR specifically, to have a common framework to work with. Without it you can say words like experimentally, practical and talking about differences between different pis whatever that means but this will only lead to misunderstandings at best and nonsense at worst
- shadowgovt 5y agoNot in accelerating frames of reference, which includes both rotating frames (such as spinning objects) and frames with high gravitational distortion. Spinning black holes have a measured circumference that is not 2 x pi x r. (Consequences of the accelerating frame of reference include the spaghettification in a black hole... We can't use a mathematical conversion to go to a frame of reference where the subject isn't being spaghettified).
- wyager 5y agoPi is defined over a flat space with an L2 norm. It has nothing to do with physical circles except that physical space is locally flat and has an L2 norm.
- krastanov 5y agoBut you can define an experimentally measured pi. Define an experimental circle (the set of points experimentally measured to be at a given distance from a center) and then experimentally measure its circumference. The fact that this experimental measurement is different from the pi defined through an l2 norm in Euclidean space is certainly cool and interesting. It is basically a way to measure gravitational curvature.
- aj3 5y agoThat would be very unusual (unorthodox) definition of Pi though.
- TheOtherHobbes 5y agoJust because it's the same number doesn't mean it's the same relationship. Any definition that relies on trig (sometimes disguised as trig expansion) assumes the existence of a flat Euclidean space - which is a convenient mathematical abstraction in real-world terms, but not required in the general case. Most everyday physics assumes a flat space, so it's not surprising certain constants fall out of this. There are more complex and interesting definitions which rely on relationships defined by group theory. I don't know enough about those to say if they're just more complex ways to operate in a flat space, or more complex ways to disguise simple trig expansions, or if they're completely independent, or if something else is going on.
- rocqua 5y agoThat's not fully true. You can get trig functions without any kind of geometry.
- layoutIfNeeded 5y agoNo. Some universes use tau :^)
- labster 5y agoIn some alternate universes they use τ instead and I would rather live in those universes. https://tauday.com/ https://tauday.com/
- wyager 5y agoNever once while studying physics have I been bothered by constant factors, especially nice integer ones like 2. I don't really care if I have to multiply by 2. Feet aren't really different from meters from a physical perspective. The only choice of unit that really has any solid claim to superiority is the natural units.
- teruakohatu 5y ago> Feet aren't really different from meters from a physical perspective As a non American I can't help but think if you are doing anything that involves short and long distances feet must surley be painful. Either you use 5280 feet instead of a mile, or 0.000189 miles instead of a foot. And then you have yards... Madness!
- nomoreplease 5y agoAs an American, I have some distant hope that we will switch .. but no sooner than the year 2050
- quartesixte 5y agoYou just switch to miles for anything longer than a few hundred feet. Big units for big things, small units for small things. The flipside to this I’ve been encountering a lot is engineering drawings in metric for things that are several meters long and all the measurements are in millimeters! Why all the big numbers!
- teruakohatu 5y ago> measurements are in millimeters! Why all the big numbers! Precision, usually if you are getting something fabricated the numbers are quoted in millimeters if the millimeters is under 10,000. So a desk might be 1800x800x700mm. I don't think I have ever seen something listed as 120000x10000x10000mm. At that point they would switch to meters.
- blamestross 5y agoPi only exists in the context of euclidean space. We have never seen a real euclidean metric space in THIS universe. Pi literally isn't a universal constant. Unlike physical constants (All the force constants, speed of light ect) apparent values of Pi will vary wildly if you can even find situations where it makes sense to apply it. Concrete example: Assume you somehow magically can have a planet in a circular orbit around a star, surely you can use Pi to derive the circumference from the distance to the center of the star? Nope, the space in the system is distorted, Pi varies with radius and the mass of the star.
- wyager 5y agoSpace is locally euclidian. You can easily define pi as a universal constant in our universe, in terms of local flat regions.
- blamestross 5y agolocal euclidian-ness is an approximation. Nowhere is flat and it wont be until well after the heat death of the universe when every particle is outside the light cone of every other (making a lot of dubious assumptions about the eventual fate of the universe)
- nimish 5y agoNo, local euclidean-ness is (more or less) the equivalence principle. The locality could be very small though! But it would break GR if local flatness weren't possible.
- blamestross 5y agoEquivalence principal is about the fact we can't differentiate between a flat spacetime+force and a curved spacetime. It doesn't require actual flat spacetime to exist anywhere outside of a model. It only says you can't tell how curved it is based on the forces acting on you. Different argument to same end: as long as the universe is expanding, nowhere can be flat. Any 2 things have an inherent repulsive force due to spatial expansion.
- rssoconnor 5y agoIf you consider solutions to one of the simplest non-trival second-order differential equation, f''(x) = -f(x), then you will find that the solutions all have period 2π, no geometry needed.
- hatsunearu 5y agoWell, the thing about a circle is that the very core of the definition has the same sort of "information" as whatever abstract "information" f'' = -f contains.
- goldenkey 5y agoYeah, a circle is basically a particular exchange between the velocity in the x and velocity in the y direction, offset by a phase equal to half of the period. With different kind of exchanges, you can get squircles and all types of other pseudo-circular shapes.
- joppy 5y agoI guess this must be true, but from the sidelines it seems that “the set of points 1 metre from this particular point” and “the solution to this particular second-order differential equation” are quite different things. Any competent student of mathematics can connect them for sure, but it’s still quite remarkable they’re related isn’t it?
- gspr 5y agoOh I don't know about that. The solution involves trigonometric functions. Are they not geometric in nature? Said differently: periodic things in some sense express motions on a circle.
- mnw21cam 5y agoNo. Trig functions can be defined independently of geometry. One way is by considering the differential equation above. Another is through the Taylor expansion of the sin(x), which is periodic as a completely emergent property. Likewise, it's hard to imagine what would happen to that very famous equation e^iπ + 1 = 0 if π were something different.
- question000 5y agoRichard Feynman had a satirical theory that any complex mathematical problem/theory when stated in layman's terms becomes self-evident. This is the perfect example of this. There's only one universe, QED.
- historyloop 5y agoThere's a limit to how much we can hypothesize about the nature of other universes, given we don't know if they actually exist. It'd be foolish to claim "QED" either way.
- cowl 5y agoSince we can't possibly know anything about these universes for all intents and purposes we can say that they don't exist. This QED is more than acceptable.
- question000 5y agoI actually think what your saying is foolish. There's no "other universes." As I stated there's only one universe. Seems obvious to me.
- 0xdeadb00f 5y agoYou've made my day haha
- HPsquared 5y agoAsk any student of etymology and they'll give a similar answer.
- Enginerrrd 5y agoThere's only one directly observable universe* There may be other universes created by whatever mechanism created ours. It is possible also that in some such mechanisms, indirect effects of those universes may be observable.
- CapmCrackaWaka 5y agoReminds me of a short story by Liu Cixin, the author of the Three Body Problem. In it, a character discovers that, as a result of 11 dimensional string space, there is a finite number of possible initial configurations to the Big Bang, each one resulting in a deterministic universe with its own distinct combination of fundamental constants (speed of light, Coulomb’s constant, pi, etc.). He explores this quite a bit. It’s pure sci-fi, but fun to read.
- legobmw99 5y agoWhat’s the name of this short story?
- chrisjharris 5y agoIt's... it's not at all short.
- Phelinofist 5y agoI think it is The Mirror
- Koiwai 5y agoI read a lot (if not all) of his work, I'd like to say just ignore his physics, he's really not good at that. I'm not saying enjoying his story is a bad thing.
- gilbetron 5y agoYeah, the invocation of quantum entanglement in the 3 body problem was ... a problem. The best scifi of any "hardness" is by authors that know the limits of their own understanding of science and physics and beyond that point they don't try to explain things. I can accept a premise that violates physics, I have a tough time accepting an explanation that violates physics, if that makes sense. "Assume that the society actually found tachyons and figured out how to use them to communicate FTL" is fine, but "quantum entanglement really does let you communicate FTL" is hard to get past. Like you said, though, still great books and a great writer :)
- dhosek 5y agoI actually have used alternate metrics in production code: The problem was to be able to cluster points on a map. The naïve approach is to use Euclidean distance, d = √Δx² + Δy², but the problem here is that the neighborhoods you get are circles and we're looking at the map through a rectangular viewport. Clicking on a cluster to zoom in on it will not give an optimal zoom (and may include stray pins from other clusters as well). It turned out using a box metric of d = max(Δx, Δy) gives better results for our needs since the neighborhoods are now squares instead of rectangles (and has the added bonus of being easier to calculate).
- qqtt 5y agoThings in mathematics are definitionally true. That's the whole system of axioms and proofs. Pi is pi because of our axioms. If you define things differently, pi doesn't have to be discovered at all.
- ww520 5y agoSomeone should do a calculation of Pi in the hexagonal coordinate, good for gaming in general.
- potiuper 5y ago3 https://m.youtube.com/watch?v=_rJdkhlWZVQ https://m.youtube.com/watch?v=_rJdkhlWZVQ
- ww520 5y agoI meant in a hexagonal coordinate system, like the hexagon tiles in games.
- ben_w 5y agoIt’s still exactly 3, assuming you’re defining it by π´ = c/2r and c = [count of number of hexes that require exactly r hex-boundary crossings to be reached from the starting cell] 6 cells at a distance of 1 hex, 12 at a distance of 2, 18 at a distance of 3, and the underlying growth pattern requires adding 6 on every discrete ring.
- ww520 5y agoInteresting. Good direction in thought with counting the whole cell as a unit. In that case shouldn't the diameter be 3 instead of 2? See the grid in [1]. The center tile counts as distance 1, the left adjacent tile counts as 1, and the right adjacent tile counts as 1, so the total diameter d is 3. Circumference c = 6. c/d = 6/3 = 2. Pi = 2. Then the next ring has 12 tiles. c = 12. d = 5. c/d = 12/5 = 2.4. So Pi = 2.4 for the next ring! [1] https://www.gamedev.net/articles/programming/general-and-gameplay-programming/coordinates-in-hexagon-based-tile-maps-r1800/ https://www.gamedev.net/articles/programming/general-and-gam...
- ben_w 5y agoI’d count the central cell as distance = 0 for the purposes of board game mechanics, and use c=2πr rather than c=πD, but I guess that would make the formula fail my definition of circumference.
- historyloop 5y agoWell this blog post ended precisely how I expected it to end. "Maybe."
- lordnacho 5y agoThe beginning bit of the physics answer is the answer. Pi comes from logic, not nature. This is a bit like when you're trying to ask your kid what the area of the triangle is, and the kid tells you you've drawn the lines bent, or the corners aren't sharp. It's actually not terribly easy to explain to them that they're supposed to understand the idealized entity, and what exactly those ideals are, because you also can't actually draw a triangle with no width and then expect them to appreciate that the qualities you want to expose are somehow exposed when you're breaking your own rules.
- fooker 5y agoNow draw the same triangle on a curved surface. No matter how the idea of area comes from logic, the fundamental assumption for the commonly known formula to calculate area is that we are doing euclidean geometry. We do not yet know if there is some similar assumption hiding behind our ignorance of physics.
- HPsquared 5y agoEven in a non-Euclidean geometry, wouldn't π still emerge (or be approached in the limit) in "flat" areas?
- FreeFull 5y agoWith regular spherical and hyperbolic geometry, yes, it is flat-ish if you look at a small enough area. On the other hand, if you have a different distance metric, the ratio of a circle's diameter to circumference can be a different constant. For example, in taxicab geometry, it is 4
- DonHopkins 5y agoIn Uber Geometry, it can vary from 5 to 14 during surge pricing.
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- Animats 5y agoIt's interesting to think about what mathematics might have been like if geometry hadn't been developed so early. If algebra or number theory had come first, the history of mathematics might have been quite different. Alternatively, you could start from the Peano axioms and grind your way up to number theory. High school mathematics spends much time on plane geometry as a formal system for historical reasons, not because it's a particularly interesting or useful formal system.
- linspace 5y agoI think the reasons for using plane geometry are pedagogical. Also formulating mathematically real life objects is a nice skill to have. I would never teach mathematics axiomatically, it's extremely dull and I usually find the axioms as an approximation to the real thing and not the source of truth.
- slongfield 5y agoSome algebra was around very early--there are Cuneiform tablets inscribed with multiplication tables from ~2000 BC: https://www.metmuseum.org/art/collection/search/324587 https://www.metmuseum.org/art/collection/search/324587
- aj3 5y agoYou're definitely commenting on your own experience in high school and it's not universal. We were mostly doing analysis in high school with some analytical geometry thrown in.
- raducu 5y agoI sucked at algebra, and the math of physics until 7 or 8th grade, when geometry was introduced; I excelled at geometry and that completely changed my life, I later loved math.
- jmilloy 5y ago"algebra" in the gp comment is different from the algebra in early highschool.
- DonHopkins 5y agoI don't know, but I do wish π was the same in every font, so I could distinguish it from n, or a square missing its bottom edge. https://imgur.com/a/VBXLVA6 https://imgur.com/a/VBXLVA6
- xyzal 5y ago> The vast, vast, vast majority of the possible combinations of values of these physical constants produce boring universes, like a single huge black hole or just diffuse clouds of hydrogen without stars. I wonder if there any universes possible which do not end in a big collapse or heat death. That is, which would be able to support life for eternity.
- dave_sullivan 5y agoSometimes I think the whole universe is a cost function iteratively optimizing those constants.
- ginko 5y agoFor a while I wanted to write some basic 3d simulation app where the pi value used for all kinds of calculations (including inside standard libs) was configurable with a slider in real-time. Would be interesting to see how things would break.
- menzoic 5y agoWeekend project
- only_as_i_fall 5y agoSeems a bit silly to ask a mathematical question from the perspective of a physicist, but I guess as the comments point out speculation is the point here.
- johndoe42377 5y agoThis is a double barrel idiotic question. First of all, there is no other universes. More precisely, there is no way to know or even assume the existence of other. Second, pi is a notion (generalisation) of human mind, and does not exist anywhere outside of shared culture. It is just a relation of two abstract notions. Yes, other observers could come to the same notions and conclusions, but it is only a theoretical possibly.
- tpoacher 5y agoUnsurprisingly, like most such questions, the effective answer is: "Define π"
- ludston 5y agoAn interesting idea once presented to me, was that pi is not a number, but a function for producing a number to an arbitrary precision. The same could be said for any fraction that produces an irrational.
- deleted 5y ago[deleted]
- isaac21259 5y agoFor those wanting to read more about this this is exactly how the real numbers are defined in the lambda calculus. https://cs.stackexchange.com/questions/2272/representing-negative-and-complex-numbers-using-lambda-calculus https://cs.stackexchange.com/questions/2272/representing-neg...
- matthewfelgate 5y agoHow many constants that exist in nature are also in logic? Or is it just Pi and E?
- osigurdson 5y agoConsider the following alternate universe experiment: Wrap a piece of string around a circular object.Straighten it out and measure its length.Measure the distance between the center and the edge of the circular object.Then observe that the ratio between these two measurements is 12 How would observers in this universe recognize that this ratio is not correct?
- eckmLJE 5y agoA universe in which string cannot exist and objects freely pass through each other and through the fabric of spacetime, stretching through regions millions of light years apart
- mike_hock 5y ago> Then observe that the ratio between these two measurements is 12 Then their space isn't locally approximated by R^3, so what's a circular object? What does it mean to be straight? What's length?
- rocqua 5y agoObligatory rant that tau is better than pi. Tau (2 * pi) is a better constant. Especially for teaching people trigonometry. It makes radians so much more intuitive (90 degree angle becomes 1/4th tau radians). In general, mathematics characterizes circles by their radius, not their diameter. Tau codifies this practice by giving us the relation between a circles radius, and its circumference. This isn't a change that will make academic mathematics better. Its a change that will make high-school mathematics better. If anything, that makes it more important. /rant
- hutrdvnj 5y agoYes, because pi is not measured but mathematically defined. Same is true for all other constants, that are not measured.
- buescher 5y agoPi-the-number is usually lowercase (and frequently italic) pi, "𝜋" , almost never uppercase "Π", which is used for other things.
- RichardCA 5y agoThe difficulty with questions like this is that we don't know what we don't know. To put a fine point on it, we don't know if the concept of Pi is unique to us as humans. If we accept the premise that we are not the only species to evolve science and technology, we are still left with the question of how other sentient species might get their maths differently from how we did it. This idea gets explored a lot in science fiction. But we don't have enough of a handle on it to say anything interesting. It's just different levels of speculation and conjecture.
- kazinator 5y agoPi comes from its own universe: a universe which contains a point in a plane, and some equidistant points. When we work with Pi, we embed that universe into other universes.