π in Other Universes • Hacker News Board

Re: π in Other Universes

#31

Post by bmm6o » Mon, Oct 30, 2023, 3:48 AM UTC

jws wrote: post #26

Zanni wrote: post #25

Earlier quoted context omitted.

I 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.

Also, "broad reach" would like a word with lfnoise. (It's complicated.) https://physics.stackexchange.com/questions/186515/why-is-a-...

The 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.

Re: π in Other Universes

#32

Post by lachlan_gray » Mon, Oct 30, 2023, 3:57 AM UTC

IanCal wrote: post #2

> 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 n…

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

Re: π in Other Universes

#33

Post by jimmySixDOF » Mon, Oct 30, 2023, 4:03 AM UTC

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

Re: π in Other Universes

#34

Post by Filligree » Mon, Oct 30, 2023, 4:14 AM UTC

charlieyu1 wrote: post #21

azeemba wrote: post #11

Earlier quoted context omitted.

There 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, t…

How 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

Hmm. And if you keep increasing the radius, pi will shrink all the way to 0.

Re: π in Other Universes

#35

Post by svat » Mon, Oct 30, 2023, 4:26 AM UTC

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 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_...

Re: π in Other Universes

#36

Post by codeflo » Mon, Oct 30, 2023, 4:32 AM UTC

* 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.

Re: π in Other Universes

#37

Post by nlitened » Mon, Oct 30, 2023, 4:37 AM UTC

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.

> 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 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.

Re: π in Other Universes

#38

Post by dylukes » Mon, Oct 30, 2023, 4:39 AM UTC

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 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.

Re: π in Other Universes

#39

Post by nylonstrung » Mon, Oct 30, 2023, 4:45 AM UTC

lachlan_gray wrote: post #32

IanCal wrote: post #2

> 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 n…

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

I love this idea.

You might enjoy Stephen Wolfram's writing- it's exactly what you're talking about

Re: π in Other Universes

#40

Post by gumby » Mon, Oct 30, 2023, 4:48 AM UTC

When 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.