The concept of "circle" is a human invention, in that a perfect circle has never been observed in the universe. The value of pi follows immediately as a derivative human invention.
It doesn't have to be observed as some material manifestation to be a discovery. Notions can also exist outside human intervention and shape the universe.
Which is also why humans can discover such things (like pi) in parallel, without one discoverer ever having been in contact with the other.
Or why even aliens can discover the same concepts.
But if you require a more direct understanding I would say there is a perfect circle in every rotation
It would take an infinite amount of power to calculate every digit in pi, given pi itself is irrational, even while describing how to calculate it is rational.
I answered where circles are in our universe and you then try to change the original intent of omalleyt's question to mean a symbolic representation of pi
> It would take an infinite amount of power to calculate every digit in pi
Calculating pi is so simple life without nervous systems does it
Now, expressing that value in our limited yet overly verbose base ten symbolic representation would indeed require infinite recursion
It seemed to me you were changing the question that was being asked, and a question asked implies an answer, or 'goal post'
But if you were just developing new inquiry stead building off omalleyt then I apologise and would hope to encourage like critical thinking in the future
.. Thinking on it 'goal posts'.. and 'soap box' elsewhere in this thread.. lacks a timeless quality, I really should have left it out ;P
Representation by digits is just one representation. We have plenty of finite representations: http://mathworld.wolfram.com/PiFormulas.html
Base 10 notation also requires a calculation to get the concrete number: You have to sum and multiply and power a lot
(Imagine whole numbers described in base 10, now how you get that number of something?)
Yeah, that's not true. Many things in the universe have perfectly spherical or spheroid shapes, cross sections, or propagation patterns, like black holes or isotropic EM emissions or (more abstractly) light cones.
Pi is baked into the laws of physics in many locations as a consequence of the geometric behavior of our spacetime. It would also be baked into a lot of other plausible spacetimes. It's hard to quantify exactly what causes it to come up so much, but it's probably the fact that our universe is locally Euclidean and therefore uses the L2-norm for distance, which gives rise to circles/spheres as the unit shape and the extrema of a number of optimization problems.
Edit: "Scaled to the size of a beach ball, say scientists, the sun's equatorial bulge would be less than the width of a human hair"[1]
And, I assume the defects in the event horizon of a black hole are even more negligible.
[1]https://www.theguardian.com/science/2012/aug/16/sun-perfect-...
I'm not sure quantum mechanics and infinite precision are reconcilable
Basically trying to say that Pi, to me, doesn't appear to be a man-made construct. Any naturally occurring circle that's near perfect seems to illustrate that. Any deviation from a Pi-derived shape has plausible reasons for the defect, like the one you point out.