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π in Other Universes

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Re: π in Other Universes

#91

Post by nlitened »

jojobas wrote: post #80
nlitened wrote: post #62

Earlier quoted context omitted.

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 "2π" vanishes from most equations, and the complex math still works and all equalities hold.*

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.

Yep, you’re right, my bad.

Re: π in Other Universes

#92

Post by seanw444 »

jimmySixDOF wrote: post #33
TheOtherHobbes wrote: post #6

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 π as big as you want by tugging on the centre of the circle.

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

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...

Re: π in Other Universes

#93

Post by msds »

waveBidder wrote: post #56
msds wrote: post #42

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 se…

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!

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!

Re: π in Other Universes

#94

Post by thechao »

NeoTar wrote: post #64
svat wrote: post #35

Earlier quoted context omitted.

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 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 app…

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 ma…

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?

Re: π in Other Universes

#95

Post by NeoTar »

thechao wrote: post #94
NeoTar wrote: post #64

Earlier quoted context omitted.

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 ma…

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?

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.

Re: π in Other Universes

#96

Post by WorldMaker »

tsimionescu wrote: post #83
gumby wrote: post #81

Earlier quoted context omitted.

> 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.

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 math…

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.

Re: π in Other Universes

#98

Post by xigoi »

jojobas wrote: post #80
nlitened wrote: post #62

Earlier quoted context omitted.

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 "2π" vanishes from most equations, and the complex math still works and all equalities hold.*

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.

You forgot a minus before one of the equations, otherwise you get sinh and cosh.

Re: π in Other Universes

#99

Post by xigoi »

zoomablemind wrote: post #89
nlitened wrote: post #37

Earlier quoted context omitted.

> Another place our pi will come up is in the exponential function. It's periodic with period 2πi. 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 equa…

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.

The fact that π is irrational has absolutely nothing to do with physically measuring circles or with infinity. We know the value of π exactly.

Re: π in Other Universes

#100

Post by xigoi »

tragomaskhalos wrote: post #53
quickthrower2 wrote: post #14

Earlier quoted context omitted.

A flat earth would have pi at about 3.14159 though.

Based on the attitude of its advocates, a flat earth would lack science and mathematics entirely, so pi would be undefined?

You're making the assumption that most flat earth advocates aren't trolls who actually know science very well.
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