# Why does PI occur so much in physics?

• imiyakawa
In summary, PI appears frequently in physics due to its connection to circles and spheres, which are often used to describe the geometry of objects in the physical world. It also arises in the study of periodic motion and has important implications in various mathematical equations. Its value remains constant, regardless of the number of dimensions or the type of geometry being used.

#### 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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imiyakawa said:
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.

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.

PI is NOT a universal constant.

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

PI is NOT a universal constant.

Really?

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.

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

Really?

Yes, Really

jmatejka said:
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.

KrisOhn said:
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".

KrisOhn said:
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
jmatejka said:
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
vanesch said:
...

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...

vanesch said:
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?

jmatejka said:
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.

KrisOhn said:
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.

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.

to denote the operation of integration, it is used for short hand methods of notation. the term PI for the ratio of circumference of a circle to its diameter the letter I is the square root of minus one. And the bringer of this notation is Leonard Euler. And why do you see this in physics, because physics is more about mathematics than anything.

Because simple waves usually follow a sine pattern. These are calculated in radians, and radians are built around a circle...

Why does PI occur so much in physics?

Because we can't use units in which pi = 1.

## 1. Why is PI used so often in equations and calculations in physics?

PI, represented by the symbol π, is a mathematical constant that is used to represent the ratio of a circle's circumference to its diameter. It is used frequently in physics because many physical phenomena involve circles or circular motion, such as planetary orbits, wave propagation, and rotational motion. Additionally, PI is an irrational number, meaning it has an infinite number of decimal places, making it a useful tool for making precise calculations.

## 2. How is PI derived and calculated?

PI is a mathematical constant that has been known and used since ancient times. It is derived by dividing the circumference of a circle by its diameter and has been calculated to over one trillion decimal places. However, since PI is an irrational number, it cannot be expressed as a finite decimal or fraction, and its exact value cannot be determined.

## 3. Are there any physical phenomena that do not involve PI?

While PI is used frequently in physics, there are some physical phenomena that do not involve PI. For example, some equations in classical mechanics and thermodynamics do not involve PI, such as the ideal gas law and Newton's laws of motion. However, PI is still a fundamental constant in physics and plays a crucial role in many other areas of study.

## 4. Can PI be used in other fields of science or mathematics?

Yes, PI is a universal constant that is used not only in physics but also in other fields of science and mathematics. It is commonly used in geometry, trigonometry, statistics, and engineering to name a few. PI is also used in computer science and technology for various calculations and algorithms.

## 5. Is there a physical significance to PI?

While PI is a fundamental constant in physics and used in various equations and calculations, it does not have any physical significance. It is simply a mathematical concept that is essential for understanding and describing many physical phenomena. However, the value of PI has been a topic of interest and debate among mathematicians and scientists for centuries.