Could there have been another value for Pi?

[musicgold]

Homework Statement

Not a homework problem; just wondering why the relation between the circumference and diameter of circle is so set in stone?

Relevant Equations

Does it have anything to with the properties of the fundamental particles? Given the shape of circle so abundent in this universe, could a universe have been created where the value of Pi was different? Why don't we see such fixed relationship with any other shape but the circle?

Thanks

 

[fresh_42]

Homework Statement:: Not a homework problem; just wondering why the relation between the circumference and diameter of circle is so set in stone?

Yes.

Relevant Equations:: Does it have anything to with the properties of the fundamental particles?

No.

Given the shape of circle so abundent in this universe, could a universe have been created where the value of Pi was different?

No. It is defined per the construction of a (flat) circle.

Why don't we see such fixed relationship with any other shape but the circle?

We do. The relation between the circumference and the diagonal of a square is ##2\sqrt{2}.##

Thanks

 

[FactChecker]

The ratio between the diameter and circumference of a circle is determined by the geometry of the space it is in. A circle drawn on a curved surface like a globe or a saddle would have a different ratio. A lot of geometric constants would be different. For instance, the sum of the angles of a triangle would not be 180.

 

[Delta2]

The ratio between the diameter and circumference of a circle is determined by the geometry of the space it is in.

So if we had a universe with a different geometry of its space, then that universe would have different value for ##\pi## right?

To put a slight objection with what @fresh_42 said, some of the properties of the fundamental particles depend on the pi, for example the hypothetical classical radius of the electron is $$r_e=\frac{1}{4\pi\epsilon_0}\frac{e^2}{m_ec^2}$$
https://en.wikipedia.org/wiki/Classical_electron_radius

 

[PeroK]

So if we had a universe with a different geometry of its space, then that universe would have different value for ##\pi## right?

If the universe had a wildly different geometry, I.e. not locally approximately Euclidean, then intelligent life may not have invented Euclidean geometry. But, if they did, then ##\pi## would still be ##\pi##

To put a slight objection with what @fresh_42 said, some of the properties of the fundamental particles depend on the pi, for example the hypothetical classical radius of the electron is $$r_e=\frac{1}{4\pi\epsilon_0}\frac{e^2}{m_ec^2}$$
https://en.wikipedia.org/wiki/Classical_electron_radius

The equation assumes spatially Euclidean geometry. In the hypothetical universe above, then these sort of formulas could be very different. In any case, if ##\pi## appeared in these formulas it would be the same mathematical constant. That's not to say a different mathematical constant would appear instead.

 

[Delta2]

If the universe had a wildly different geometry, I.e. not locally approximately Euclidean, then intelligent life may not have invented Euclidean geometry. But, if they did, then π would still be

Er, sorry if the universe was not Euclidean, the ratio of the circumference of a circle (I am not sure how we would define a circle in that universe) to the diameter of a circle would still be 3.14159...?

 

[PeroK]

Er, sorry if the universe was not Euclidean,

The universe is not Euclidean. That doesn't mean there isn't (still) Euclidean geometry.

 

[Delta2]

Yes it's not Euclidean (though locally it is approximately euclidean, at least here in planet earth).

 

[PeroK]

Yes it's not Euclidean (though locally it is approximately euclidean, at least here in planet earth).

##\pi## is fundamentally defined as the ratio of the circumference to the diameter of a circle in a Euclidean plane. It's not the ratio of the circumference to the diameter of a circle in any geometry.

 

[Delta2]

##\pi## is fundamentally defined as the ratio of the circumference to the diameter of a circle in a Euclidean plane. It's not the ratio of the circumference to the diameter of a circle in any geometry.

Now you talking... It depends how we would define a circle in a different geometry .

And how we would define pi...

 

[PeroK]

And how we would define pi...

Then it's not ##\pi## any more.

 

[Delta2]

Then it's not ##\pi## any more.

Yes ok, but in the spirit of the OP he defines ##\pi## as the ratio of a circle in a hypothetical universe with a hypothetical other than euclidean geometry.

 

[PeroK]

Yes ok, but in the spirit of the OP he defines ##\pi## as the ratio of a circle in a hypothetical universe with a hypothetical other than euclidean geometry.

See post #3. We define ##\pi_0## in terms of a Euclidean circle. And we define ##\pi_1## in terms of a great circle on a sphere. Then ##\pi_1 = 2##. That doesn't change ##\pi_0##. And, it doesn't mean that ##\pi_1## replaces ##\pi_0## in mathematical formulas such as:
$$e^{i\pi} + 1 = 0$$And, it doesn't make ##2## a fundamental constant with the numerous properties that ##\pi_0##, the real ##\pi##, has!

 

[Delta2]

Yes ok i get your points, euler's identity and other properties hold for the pi of the euclidean geometry and not for the pi of the hypothetical universe.