Pi is the most infamous irrational number, being infamous for having an infinite number of non repeating digits and now calculated out to roughly 31.4 trillion places :D. Here is my hypothesis which is up debate, and since I am not a physicist or mathematician will likely have some errors in my underlying assumptions. Since pi is infinite and is defined as the ration of a circles circumference to it’s diameter, what I think this means for the real world is that making a perfect circle is not possible. Furthermore I think the reason it’s impossible is because there are no infinities in the real world. My guess is that if making a perfect circle were to be attempted, the fundamental problem would be that there is no such thing as an infinitely small point, but that rather there is such a thing as a smallest possible distance even if we don’t know what that might be. These thoughts come from my likely flawed understanding of things like the Planck length. As I said, I’m sure my understanding of these concepts is flawed, so I’m not claiming this to be correct, merely my theory on how things are. What assumptions do I have wrong, and is quantum theory and / or Planck constants a way to explain why infinities seem to not exist in the real world?
Taking pi out to 35 places past the decimal point is an almost trivial mathematical exercise. But doing so would define a circle around the entire universe accurate to less than the width of a proton.
The notion of a perfect circle in reality is therefore more or less meaningless. Circles can theoretically exist to such precision that no conceivable way of distinguishing them from perfect can be devised. True perfect circles don’t exist, though. They would be unstable. What could hold all the atoms in such perfect position over time? Magic?
Infinities have nothing to do with quantum mechanics per see. They pop up everywhere. The real world doesn’t have them because it would take infinite time or energy to achieve them. They can’t be achieved, therefore they don’t exist. Reality has enough problems of its own.
The Planck length is a red herring. That’s the smallest length our physics can say meaningful things about, but we have no way of knowing whether smaller things are real or possible or discoverable with different physics.
The OP is correct. We can’t make a perfect circle or any other perfect geometric shape in the real world. Not even a perfect straight line really.
No power on Earth, however great
Can pull a string, however fine
Into a horizontal line
That shall be absolutely straight.
That depends on what you mean by “make”. I can perfectly well say “Define C to be the set of all points in plane xy which have a distance of exactly 1.207 m from the origin”. There; C is now a perfect circle with a radius of exactly 1.207 m. I can’t draw it perfectly, because my pencil lead has a finite width, but is drawing it actually necessary?
Is it a coincidence today is also Pi Day? 
That’s what renormalization is for.
Yes, but can you draw seven red lines, all of them strictly perpendicular, some with green ink, and some transparent?
Stranger
The notion of the planck length puts to lie the idea that anything can be rendered in the physical world at infinite resolution - the only things that can be physically rendered accurately are ones whose definitions happen to resolve accurately at above planck resolution. This is not really a problem for a theoretical things, though, and the notion of something being infinitely indivisible is absolutely not a problem if we don’t have to do it in the real world. In the real world you’ve eventually sliced that block cheese so thin that the slices are one molecule thick and slicing again would cause molecular fission, but in the world of pure math the cheese can be sliced thinner forever.
I make a shaft of a hard metal, screw a bar to this shaft sticking out at ninety degrees and at the end of the bar I affix a small, very close to spherical, mass.
Then I slowly turn the shaft by some steady mechanical means. After a while this system is quite stable. What path is the center of mass for the small mass tracing out?
And if it’s not tracing out a perfect circle, what is the nature of the perfect circle you are measuring it against to decide it fails?
I am not sure what you mean by “pi is infinite”. How is it any more infinite than any other number? It is a precise value that is very difficult to express exactly with the decimal number system. But, you could construct a pi-based number system where every common number we use is irrational. All numbers are infinite in their uniqueness.
Barf.
It is impossible to express exactly in decimal, or (if I understand it correctly) any other natural base.
Here’s what I had in mind. Suppose you were able to get a pencil with a very fine point, just one carbon atom. If you had a really steady hand it would be theoretically possible to use such a pencil to draw a perfect square that was x number of carbon atoms on each side. With a circle it would be impossible to actually draw, since no whole number of carbon atoms, or neutrons, or quarks, or anything else, would fit into a perfect circle.
Sure you could. You just couldn’t draw such a circle, and for the same circle draw the radius. Either the radius, or the circumstance of a circle can be made of an integer number of atoms - just not both.
Carbon atoms are large enough to still be divisible into smaller components and perhaps weren’t the best example. Let’s say we’re dealing with quarks, or whatever particle is small enough that it can’t be divided any further. If you have 100 quarks lined up end to end, it would be impossible to make a circle around them. If you put 314 quarks around them there would still be a gap, which you couldn’t fill in with part of a quark since they can’t be divided up any smaller.
You can have a circle with any circumference. It could be exactly a thousand atoms, or exactly a million, or exactly 35. The only restriction that pi imposes is that (because pi isn’t a rational number) the radius of a circle with a circumference of (say) 642 atoms wouldn’t be an integer.
The same problem happens with a perfect square - if you have a perfect square with sides of a 1000 atoms, you won’t be able to draw a diagonal that has an integer number of atoms.
Let’s pretend that atoms are actually little balls. You could very easily group a countable number of them into a circle with seemingly no space in between. Any imperfections in that circle would be unmeasurable by any instrument. How could you ever prove that your circle is not perfect?
Ah, but my square is perfect, you might say. How would you know that? You might have the same number of atoms on all four sides, true, but how would you know that your little balls are all exactly the same size? Maybe they’re different in the 50 millionth decimal point. Or the 50th.
My point is that infinities belong to math. You can say nothing about them by resorting to pointing at real-world objects. It’s a pointless exercise.
I believe I put it inelegantly, but this is the point I was trying to make. Infinities don’t make sense in the real world, and the fact that pi is an irrational number is a piece of evidence that shows this to be the case.
But … but … Didn’t someone point out upthread that it’s just as hard to make a perfect square as to make a perfect circle?
I’m not sure how this follows.
Pi has an infinite decimal expansion. But so does 1/3 - it may be a rational number but it still has an infinite decimal expansion (though it doesn’t have an infinite expansion in base-3). But if you don’t like 1/3 for being rational, the square root of 2 is irrational and has an infinite decimal expansion. Both 1/3 and sqrt(2) ‘make sense’ in the ‘real world’.
As for utility, the irrationality of pi means you cannot take a real world circle and measure both diameter and circumference to infinite precision.
But so what? As mentioned above, you can’t do that with squares, either. You can’t measure both the length and perimeter of a real world square to infinite precision. That doesn’t mean numbers, whether rational or irrational, fundamentally don’t make ‘real world’ sense.
Even with your ‘1 carbon atom’ example, there’s fuzziness in where the ‘edge’ of the square is. Your precision is limited to “number of atoms”, which is itself an approximation of the actual precision of a perfect square, especially as there’s no way to verify these atoms all have the same size! That’s kind of important for building a square this way.
Math deals with idealized elements. No matter how fine your pencil is, you cannot draw a one-dimensional line. No piece of paper is ever thin enough to be two-dimensional. No two angles are ever perfectly alike; no two lines ever exactly the same distance apart for their entire length. Idealized quantities are not limited, therefore, to infinities. Nor are they limited to any real-world analogs. Math deals with million-dimensional spaces just as deftly as it does three-dimensional ones.
Physics is the science of dealing with the real world. People talk about quantum fuzziness and uncertainty all the time, but in fact quantum mechanics is the most accurate physical theory we’ve ever come up with. Measurements agree with theory down to ten decimal places. Relativity similarly makes extremely accurate predictions that are used in real-world situations. The Large Harron Collider accelerates particles to 0.999999991 c. That’s the speed of light minus 3 meters/second. Equations give us the energy equivalent we need to calculate possible outcomes.
Math can be adapted to our real-world, giving solutions which are so close to ideal that the differences simply make no difference. (We just had a threadabout being able to drop terms too small to matter.) Math also has 150 years of history defining and collaring infinities in ways that make them tractable. Pi is exactly c/d. Exactly. That we can’t write out in full the decimal equivalent of that ratio is not physically meaningful. That’s intellectually interesting, true, but if pi terminated or repeated after a million decimal places nothing would change in measurements or calculation because we can’t measure or in any way detect the millionth decimal point. We can nevertheless plug pi in millions of formulas and have them work.
You’re correct when you say there are no real-world infinities. But the real world doesn’t care, and the math world can handle them just as well as 2 and 4. I’m still not getting your point.