
The formula for pi buried in a hydrogen atom - silasrude
https://www.rochester.edu/newscenter/discovery-of-classic-pi-formula-a-cunning-piece-of-magic-128002/
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segfaultbuserr
Original paper:

* Friedmann, T., & Hagen, C. R. (2015). _Quantum mechanical derivation of the Wallis formula for π_. Journal of Mathematical Physics

[https://sci-hub.tw/10.1063/1.4930800](https://sci-hub.tw/10.1063/1.4930800)

The result didn't come from experimental works, it's a pure theoretical
derivation for the sake of mathematical physics.

> _The existence of such a derivation indicates that there are striking
> connections between well-established physics and pure mathematics that are
> remarkably beautiful yet still to be discovered._

The paper only has three pages - it set up a particular calculation on the
energy levels of hydrogen and obtains a limit, thus recovering the Wallis
formula for π. If you know QM (I don't), the paper may be a fun read, it's a
short and understandable calculation.

~~~
rrss
The preprint is on arxiv too:
[https://arxiv.org/pdf/1510.07813.pdf](https://arxiv.org/pdf/1510.07813.pdf)

~~~
JadeNB
Routine reminder: please link to arXiv abstract pages, rather than directly to
PDFs. [https://arxiv.org/abs/1510.07813](https://arxiv.org/abs/1510.07813)

~~~
rrss
whoops, thanks for the reminder :)

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geoffmunn
I really love the idea that pure mathematics and nature are the same thing. On
a philosophical level, our daily lives are the expression of the differences
and inefficiencies of our systems compared to an optimal end-state. Also,
sometimes I think that religion and science have more in common than people
think.

~~~
user234683
> I really love the idea that pure mathematics and nature are the same thing

This might not be what you mean, but there is a philosophical position that
holds that all mathematical structures exist, and that our universe is simply
one of these structures:

[https://en.wikipedia.org/wiki/Mathematical_universe_hypothes...](https://en.wikipedia.org/wiki/Mathematical_universe_hypothesis)

~~~
gqcwwjtg
If you assume some notion of an object's description "really existing", it's
easy to then assume the Universe must have a zero information description,
since otherwise the description would be "outside" the Universe. And then
you've backed yourself into a corner: what could zero information describe
that also contains our observed universe? All mathematical structures it is.

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SonOfLilit
This brings to mind Richard Hamming's wonderful essay "The Unreasonable
Effectiveness of Mathematics", which really blew my mind ten years ago:

"But if you do not like these two examples, let me turn to the most highly
touted law of recent times, the uncertainty principle. It happens that
recently I became involved in writing a book on Digital Filters [8] when I
knew very little about the topic. As a result I early asked the question, "Why
should I do all the analysis in terms of Fourier integrals? Why are they the
natural tools for the problem?" I soon found out, as many of you already know,
that the eigenfunctions of translation are the complex exponentials. If you
want time invariance, and certainly physicists and engineers do (so that an
experiment done today or tomorrow will give the same results), then you are
led to these functions. Similarly, if you believe in linearity then they are
again the eigenfunctions. In quantum mechanics the quantum states are
absolutely additive; they are not just a convenient linear approximation. Thus
the trigonometric functions are the eigenfunctions one needs in both digital
filter theory and quantum mechanics, to name but two places.

"Now when you use these eigenfunctions you are naturally led to representing
various functions, first as a countable number and then as a non-countable
number of them—namely, the Fourier series and the Fourier integral. Well, it
is a theorem in the theory of Fourier integrals that the variability of the
function multiplied by the variability of its transform exceeds a fixed
constant, in one notation l/2∏. This says to me that in any linear, time
invariant system you must find an uncertainty principle. The size of Planck's
constant is a matter of the detailed identification of the variables with
integrals, but the inequality must occur.

"As another example of what has often been thought to be a physical discovery
but which turns out to have been put in there by ourselves, I turn to the
well-known fact that the distribution of physical constants is not uniform;
rather the probability of a random physical constant having a leading digit of
1, 2, or 3 is approximately 60%, and of course the leading digits of 5, 6, 7,
8, and 9 occur in total only about 40% of the time. This distribution applies
to many types of numbers, including the distribution of the coefficients of a
power series having only one singularity on the circle of convergence. A close
examination of this phenomenon shows that it is mainly an artifact of the way
we use numbers.

"Having given four widely different examples of nontrivial situations where it
turns out that the original phenomenon arises from the mathematical tools we
use and not from the real world, I am ready to strongly suggest that a lot of
what we see comes from the glasses we put on. Of course this goes against much
of what you have been taught, but consider the arguments carefully. You can
say that it was the experiment that forced the model on us, but I suggest that
the more you think about the four examples the more uncomfortable you are apt
to become. They are not arbitrary theories that I have selected, but ones
which are central to physics.

[http://inters.org/Hamming-Unreasonable-Effectiveness-
Mathema...](http://inters.org/Hamming-Unreasonable-Effectiveness-Mathematics)

I don't know any QM, but it does not surprise me in the least that, trying to
calculate something about rotation, you end up with a formula for Pi. Is it
there because of a connection to how the world works, or is it there because
we happened to ask a question where two effects of the real world canceled
each other out and all that remains is something about math? Sure, "no circles
were involved", but rotation was involved, and it is just as Pi-heavy a model
as anything involving circles. Also, QM is full of integrals of sines, for
reasons explained in the quote above, which introduce a bunch of Pi scaling
factors that have nothing to do with reality and everything to do with the
tools we chose to model it with, etc'.

(I'm not saying that the universe doesn't seem to have a lot of Pi factors in
its machinery, just that this specific case doesn't sound like one of them,
and that the distinction is interesting and mind blowing, to me at least).

~~~
WorldMaker
> rather the probability of a random physical constant having a leading digit
> of 1, 2, or 3 is approximately 60%, and of course the leading digits of 5,
> 6, 7, 8, and 9 occur in total only about 40% of the time

This is a part of the arguments used both pro/con of Tau over Pi. Constants
that start with 6 are "weird" because they are uncommon. They are uncommon
because "we" like to just halve them and use constants that start with 3 and
lots of factors of two when working with them. It's a weirdness entirely of
mathematics' own creation. Neither is more "fundamental", people just likes 3s
sometimes too much.

~~~
SonOfLilit
He's talking about physical constants, to which Benford's law applies (since
we expect the first digits to be distributed the same when stated in different
units, e.g. meters and feet, we expect there to be the same number of
constants that start with 1 in meters as those that start in 3, 4 or 5 in feet
(since 1m=3ft, 2m=6ft), etc').

In any unit you choose, about 30% of rivers on Earth and about 30% of public
company market caps will start with a 1. This has nothing to do with people's
preferences and everything to do with scale invariance.

~~~
WorldMaker
I certainly read it as applying to both. Certainly Benford's law is
demonstrable in many areas, but there's also an interesting sort of "Benford's
paradox" at play that when free choice is given between units options, people
seem to "prefer" the versions of constants with the lower starting digit. An
interesting question of whether one seems more "natural" than another simply
because Benford's law so often applies to similar situations.

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tobrien6
This is from 2015 FYI. Here's a more in-depth summary:
[https://www.forbes.com/sites/kevinknudson/2015/11/10/everyth...](https://www.forbes.com/sites/kevinknudson/2015/11/10/everything-
you-ever-wanted-to-know-about-pi-part-2-a-new-proof-of-the-wallis-formula-via-
physics/#3df5d3021150)

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doovd
The formula is also buried in a simple circle.

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FigmentEngine
no that's tau

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z3ncyberpunk
Tau is just pi * 2

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opless
PI is Tau/2

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perl4ever
Isn't there a page somewhere with a lot of physics formulas in terms of tau
instead of pi for comparison? I'm not very math or physics literate, but it
was interesting to contemplate.

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jacinabox
Clever of them to hide pi in there eh?

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irrational
Wasn't there something like this in the book Contact? Something hidden in pi?

