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We can consider the value of π as being roughly $3.1415927$, but π is defined as the ratio of the circumference of a circle to its diameter.

However, if the diameter is one Planck length, does it make sense to argue that the the circumference is not an integer? If it's not possible for "fractional" values of the Planck length to be measurable or have any physical impact on the universe, is the value of the circumference $3$ if the diameter is one Planck length? Or it is $4$?

If the circumference was $3$ or $4$, then as we have diameters get slowly larger, the ratio of the circumference to the diameter might start to converge on π, implying that π is a limit rather than a constant. However, at our scales, it would be impossible to distinguish between the two.

The implication to my mind is that many formulas in physics depend on π (Heisenberg uncertainty principle, Coulomb's law, and so on). Does π, in these contexts, somehow "know" the infinite precision of π, or is the representation of π in those equations really tied to the ratio of a circumference to the diameter? If so, at small enough scales, would it ever be possible to distinguish between the two?

Background: I started wondering about this due to papers such as Nicolas Gisin's paper Classical and intuitionistic mathematical languages shape our understanding of time in physics.

Qmechanic
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Curtis Poe
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    If you discretize spacetime as you posit, how can circles that small exist anyway? – J.G. Sep 29 '21 at 11:23

3 Answers3

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$\pi$ is a mathematical constant. Its value does not depend on the physical system we choose to use. If you have some weird system where for your circle the ratio of the circumference to the diameter is not $\pi$, then that just means you're "circle" isn't what we usually think of as a circle. But that doesn't change what $\pi$ is defined as. If you are actually thinking about a typical "circle", then the mathematics doesn't care about how small the circle actually is. If you are talking about trying to actually find and measure relevant lengths of a circle at this size, I think I can safely say at this point that you wouldn't be able to do this measurement to try to "validate $\pi$" at this scale.

As for your worry about getting "fractions of a Planck length", see this post discussing how the Planck length is really just a useful scale parameter, not a for sure limit on the size where reality (and mathematics) breaks down (that we know of :) ).

BioPhysicist
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    @CurtisPoe Physics is about modeling reality. "All models are wrong, but some are useful". Sure, the cross section of the rod I am using to build this structure is not perfectly circular, but I can approximate it as a perfect circle and then use that to get a pretty good idea of how large of a load my rod can support. – BioPhysicist Apr 14 '20 at 14:59
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A circle is a mathematical construct which does not exist in the real world. There is no circle of any size that physically exists and can be measured. The value of pi is an irrational number with an infinite number of decimal places, so you never have enough precision on any measuring device, whether you're measuring a dinner plate or a Planck circle. A circle is mathematically defined at an infinitesimal level of precision which simply does not exist in the real world. As you point out, at a certain scale, you can't even say a physical "circle" is here and not there.

If you try to derive pi from physical measurements, you will always be wrong. Your Planck ruler might have enough precision to say only that pi equals 3, while a meter stick might get you pi equals 3.14159, and atomic calipers could get you 100 decimal places. But they are all, to some extent, wrong - you will never arrive at an infinitely precise irrational number by taking the ratio of two physical measurements which must have finite precision. Pi is a fixed, theoretical value that does not change based on physical measurements. That would be like saying you disproved the Pythagorean theorem because you measured 3 sides of a triangle and found that there was some imprecision. Pi does not change because of your practical inability to measure radius and circumference.

Nuclear Hoagie
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    "If you try to derive pi from physical measurements, you will always be wrong."

    So what does π mean to equations in physics? If the π in those equations has no real correspondence to something in the world, how do the equations "get" this mystical value and somehow apply that to real world behavior?

     – Curtis Poe Apr 14 '20 at 14:46
  • @CurtisPoe We are modeling reality in Physics, so it's totally fine to use $\pi$ in our equations :) – BioPhysicist Apr 14 '20 at 14:54
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    what does the number 3.000000000000000000 mean in physics since you could not probably measure a distance in inches or m with this precision, still you talk of 3m length and do calculations with it. – trula Apr 14 '20 at 15:14
  • @trula 3.000000m and 3m are, in fact, different numbers. If I say an object moved 3m, I'm only really sure it moved somewhere between 2.5m and 3.5m. If it moved 3.000000m, I'm sure it's somewhere between 2.9999995m and 3.0000005m. This comes down to the issue of "significant figures" - in physics problems, you should only report as many decimal places as are reflected by the precision of your original measurements. When I say a length is 3m, I'm not implying infinitesimal precision, but rather that my precision is on the scale of a meter. – Nuclear Hoagie Apr 14 '20 at 15:21
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As far as I'm aware, there technically is no true circle in the entire universe. The Planck length is, on a technical basis, the "resolution" of reality. If you were to infer that looking at a circle on a screen could be considered a circle until closer inspection of said shape showed the pixelation of the shape to in fact make it a different polygon entirely then the same concept can then be applied to the scales at which the Planck constant becomes relevant. One Planck length in each of the 3 dimensions would make a cube, not a circle/sphere.

Vincent Thacker
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Donald Alonzo
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