# Why does PI occur so much in physics?

imiyakawa
What's so special about PI? Why does it appear in so much in physics?

Einstein's field equations, Coulomb's Law, Kepler's Third law constant, uncertainty principle....

This wouldn't be a co-incidence. Surely there's an underlying reason for this that we don't know yet?

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Staff Emeritus
2021 Award
Surely there's an underlying reason for this that we don't know yet?

What makes you think we don't know this?

You get pi when you have circles and spheres.

Gerenuk
Most (if not all?) occurences of pi in physics are due to geometry of spheres and circles.

My question would be: Why is pi sufficient for spherical objects in all dimensions?

Without knowing the advanced maths, I get the feeling that pi's origin is whenever an infinitesimal step on something periodic (for example revolution on the circle) is considered.

jmatejka
PI is NOT a universal constant.

Phrak
I think this is a very nice question to ponder. In the realm of physics pi seems to result universally from space or spacetime, or various manifolds and how we impose a measure upon a point-set topology to obtain a topological space. I'm not equipped to comment on Hilbert space...

Can anyone think of a counter example, where pi shows up in physics, where our imposed methods of measuring space and spacetime do not enter into it?

I suspect I haven't been imaginative enough to find any. In mathematics there are many ways to express pi without reference to geometry. But what of physics?

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Blenton
From the top of my head I can't currently remember any, but I could've sworn ive seen equations use pi without having anything to do with circles or whatnot.

PI is NOT a universal constant.

Really?

Rajini
hi phrak,
PI more or less get introduced into physics when ever we study periodic motion ..
a simple e.g., convertion of Hz to per second.

jmatejka
From the top of my head I can't currently remember any, but I could've sworn ive seen equations use pi without having anything to do with circles or whatnot.

Really?

Yes, Really

KrisOhn
Yes, Really

Nice explanation

I'm sure you mean it is not because of the differences in the value for pi depending on if you deal with Euclidean or non-Euclidean geometry.

jmatejka
Nice explanation

I'm sure you mean it is not because of the differences in the value for pi depending on if you deal with Euclidean or non-Euclidean geometry.

More or less, yes, the way it was explained to me was something along these lines, "The properties of "space" in the universe are not so homogenous that pi is a "universal" constant. It was years ago when the Physics Chair Richard Blade(University of Colorado, Colorado Springs) explained this to me.

With regard to Pi, some people marvel at remembering many decimal places, I wonder at which decimal place does the "number" become meaningless, or variable, even for our "seemingly" stable corner of "space".

Staff Emeritus
Gold Member
Nice explanation

I'm sure you mean it is not because of the differences in the value for pi depending on if you deal with Euclidean or non-Euclidean geometry.

Even in non-Euclidean geometry, pi still has the value of pi

Certain relationships between geometrical quantities might have a number different from pi, there where in Euclidean space, that number would be pi, but that doesn't mean that the value of the NUMBER pi changed.

Not any more than that the value of the number 2 changes, if you go from 2 to 3 dimensions. In 2 dimensions, the number 2 indicates the number of dimensions, while in 3 dimensions, 2 doesn't indicate the number of dimensions. But even in 3 dimensions, the number 2 still has the same value as before...

Merlin3189
Staff Emeritus
Gold Member
With regard to Pi, some people marvel at remembering many decimal places, I wonder at which decimal place does the "number" become meaningless, or variable, even for our "seemingly" stable corner of "space".

After how many decimal places, the decimals (they are all 0, or all 9, depending...) of the integer 2 become meaningless ?

Merlin3189
KrisOhn
...

Ahh yes, but that's basically a discrepancy over words. In my post I essentially used the word pi to signify the ratio $$\frac{circumference}{diameter}$$. I was referring to the fact where in elliptical geometry, the value of that ratio with diameter 1 is less than pi. While in hyperbolic geometry, the value of that ratio with diameter 1 is greater than pi. It's interesting...

jmatejka
After how many decimal places, the decimals (they are all 0, or all 9, depending...) of the integer 2 become meaningless ?

I'm sorry, you lost me. Could you clarify?

KrisOhn
I'm sorry, you lost me. Could you clarify?

I can't really speak for him, but I believe he is saying that no matter how many decimal digits there are, they never become meaningless. I could see them becoming less important than earlier decimal digits, but not ever meaningless.

jmatejka
I can't really speak for him, but I believe he is saying that no matter how many decimal digits there are, they never become meaningless. I could see them becoming less important than earlier decimal digits, but not ever meaningless.

OK, thanks, perhaps poor choice of words on my part.

Gerenuk
I think the origin of pi is always something periodic. For me the most (mathematically) natural definition of pi is the solution to
$$\lim_{N\to\infty}\left(1+\frac{a}{N}\right)^N=1$$
$$\therefore |a|=2\pi$$
where you can see that pi connects an infinitesimal addition with an infinite exponentiation to yield 1 again.

Maybe someone can tell the algebraic requirements for this solution to hold.