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π in Other Universes

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Re: π in Other Universes

#101

Post by zoomablemind »

xigoi wrote: post #99
zoomablemind wrote: post #89

Earlier quoted context omitted.

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.

> ...We know the value of π exactly.

Perhaps you could share that exact value with the rest of humanity. And I mean the number value, not the nominal value.

Re: π in Other Universes

#102

Post by jojobas »

xigoi wrote: post #98
jojobas wrote: post #80

Earlier quoted context omitted.

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.

Yep, silly me.

Re: π in Other Universes

#103

Post by one-another-dev »

gumby wrote: post #41
lachlan_gray wrote: post #32

Earlier quoted context omitted.

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

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,…

This reminds me of "The Road Not Taken" by Harry Turtledove. Awesome story!

Re: π in Other Universes

#104

Post by xigoi »

zoomablemind wrote: post #101
xigoi wrote: post #99

Earlier quoted context omitted.

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.

> ...We know the value of π exactly. Perhaps you could share that exact value with the rest of humanity. And I mean the number value, not the nominal value.

What do you mean by “number value”? There are many different ways to calculate π, the most famous (but converging very slowly) being

π = 4 atan(1) = 4 (1 - 1/3 + 1/5 - 1/7 + …)

Re: π in Other Universes

#105

Post by eternityforest »

tsimionescu wrote: post #58
chongli wrote: post #44

Earlier quoted context omitted.

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 Gödel [1]. [1] https://en.wikipedia.org/wiki/Gödel%27s_incompleteness_theor...

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 stri…

Is there no use for some kind of branch of mathematics that can prove false things?

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.

Re: π in Other Universes

#106

Post by rstuart4133 »

OMG. 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 [1] Poor man representation of the same equation in the article.

Re: π in Other Universes

#107

Post by zoomablemind »

xigoi wrote: post #104
zoomablemind wrote: post #101

Earlier quoted context omitted.

> ...We know the value of π exactly. Perhaps you could share that exact value with the rest of humanity. And I mean the number value, not the nominal value.

What do you mean by “number value”? There are many different ways to calculate π, the most famous (but converging very slowly) being π = 4 atan(1) = 4 (1 - 1/3 + 1/5 - 1/7 + …)

There are many ways to obtain inexact value of pi.

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.

Re: π in Other Universes

#108

Post by xigoi »

zoomablemind wrote: post #107
xigoi wrote: post #104

Earlier quoted context omitted.

What do you mean by “number value”? There are many different ways to calculate π, the most famous (but converging very slowly) being π = 4 atan(1) = 4 (1 - 1/3 + 1/5 - 1/7 + …)

There are many ways to obtain inexact value of pi. 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.

π has a finite amount of digits in base π. Why arbitrarily limit yourself to integer bases?

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.

Re: π in Other Universes

#109

Post by aesch »

codeflo wrote: post #36

* 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…

I wonder about the first point. As you explain in another comment the value of pi, 3.14159, can be derived from number theory alone but magically it plays a huge role in shaping the physical world we know.

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?

Re: π in Other Universes

#110

Post by thomasahle »

svat wrote: post #35
tzs wrote: post #16

Note 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.

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…

• The probability that a random walk with n steps return to the origin is 1 - π/logn + O(1/logn)^2: https://twitter.com/thomasahle/status/1719140649952571672
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