π in Other Universes
azeemba.com
azeemba.com
This is a really, really nice expression of something my mind's been hovering around for a while.
You might enjoy Stephen Wolfram's writing- it's exactly what you're talking about
Mathematics is also not provably internally consistent. This was famously shown by Gödel [1].
[1] https://en.wikipedia.org/wiki/Gödel%27s_incompleteness_theor...
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.
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.
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.
https://en.wikipedia.org/wiki/History_of_Lorentz_transformat...
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, Möbius geometry, and sphere geometry, which is connected to the fact that the group of motions in hyperbolic space, the Möbius group or projective special linear group, and the Laguerre group are isomorphic to the Lorentz group.
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.
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.
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?
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.
You would think that with how much math there is, there would be a whole field of working with uncertain proofs. I have no idea what for, but then again I'm not a math guy.
yet if we just tried, oh, making the unit circle a unit... ellipse... all of the epiphenomenal complexity that comes from remediating the pervasively accumulated 0.01% error in that fundamental assumption would instantly vanish.
They may not correspond to anything in our world, and then we usually discover something that does.
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.
> 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
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
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
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.
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.
• 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
• 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
• (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
• 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 (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_...
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.
Isn't it the opposite? As I understand, we (European civilization humans) historically _define_ our complex exponential function to have a period of 2πi 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 "360° 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.
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 π results.
f(0) = 1
f'(x) = i f(x)
So we can define a function with period of 1, and use it everywhere — then "2π" vanishes from most equations, and the complex math still works and all equalities hold.*
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.
No, it doesn't work in degrees.
The definition of e isn't that arbitrary.
2π 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)ᵇ = xªᵇ) and you find the function eˣ must be periodic with period 2πi.
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 2π.
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 2π seconds, if the acceleration is -1m/s² per 1m displacement.
This setup is called a harmonic oscillator.
In this way, 2π arises (and is measurable!) from physical properties of time, force and inertia, of things moving in straight lines.
No circles required.
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
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.
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.
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.
Perhaps you could share that exact value with the rest of humanity. And I mean the number value, not the nominal value.
π = 4 atan(1) = 4 (1 - 1/3 + 1/5 - 1/7 + …)
Being irrational number, there are no finite number of digits (e.g. in decimal form or other base) to represent the pi value exactly. Nor can such value be expressed as ratio of integers.
Likewise, in your representation the ellipsis hides away the infinity.
If you can compute a number to any desired precision, then you know its exact value.
And this still has nothing to do with measurement of the physical world. We cannot measure anything exactly.
* Manhattan (L_1): C = 8 R
...
* Euclidean (L_2): C = 2π R
...
* Maximal Distance (L_infinity): C = 8 R
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.
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!
* 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.
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.
(Not to sound all Buzzfeed-y, but Table 3 makes a lot of sense)
Would a different universe have a different number theory or is number theory something that is True regardless of the universe? What would an alternate number theory even look like?
https://physics.stackexchange.com/questions/186515/why-is-a-...
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.
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_...
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
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.
Math is far more elegant than public school allows it to appear.
https://raypatrick.xyz/blog/2023/10/27/were-you-mathematical...
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.
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?
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.
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.
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...
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"
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]
[0] https://en.wikipedia.org/wiki/Indiana_Pi_Bill [1] Poor man representation of the same equation in the article.
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.
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?
Pi is 3. (more accurately, 3.2). https://cs.uwaterloo.ca/~alopez-o/math-faq/mathtext/node18.h...
https://physics.stackexchange.com/questions/186515/why-is-a-...