The value of pi is determined by the circumference of a circle.
Why is it any particular constant number? Would a circle as defined as a perfect circle in any universe lead to a different value of pi?
Would all universes where a circle could be constructed by "people" there also lead to the value of pi?
If it is true then it leads to the conclusion that pi is some sort of constant value constant to all universe. What is the meaning of that?

Science fiction references.
In science fiction pi sometimes has a different value in different universes, for example Greg Bear's "The Way", it says "Gates are capped with cupolas formed from Space-time itself. As distortions in space-time geometry, their nature can be calculated by 21st century instruments laid on their 'surfaces'. The constant pi, in particular, is most strongly affected.".
A message is found encoded within pi, in the novel by Carl Sagan, "Contact" "Ellie, acting upon a suggestion by the senders of the message, works on a program which computes the digits of pi to record lengths in different bases. Very far from the decimal point (1020) and in base 11, it finds that a special pattern does exist when the numbers stop varying randomly and start producing 1s and 0s in a very long string.".

Physically, the ratio of a circle's circumference to its diameter $C/d$ is not really $\pi$. General relativity describes gravity in terms of the curvature of spacetime, and roughly speaking, if you take $(C/d-\pi)/A$, where $A$ is the circle's area, what you get is a measure of curvature called the Ricci scalar.

But even if you're doing general relativity, you don't just go around redefining $\pi$. The thing is, $\pi$ occurs in all kinds of contexts, not just as $C/d$. For instance, you could define $\pi$ as $4-4/3+4/5-4/7+\ldots$, which has nothing to do with the curvature of space.

So if you define $\pi$ as $C/d$, you don't even get a consistent value within our own universe, whereas if you define it as $4-4/3+4/5-4/7+\ldots$, you get an answer that is guaranteed to be the same in any other universe.

Another way of looking at it is that $\pi$ is not the $C/d$ ratio of a physical circle, it's the $C/d$ ratio of a mathematically idealized circle that exists in certain axiomatic systems, such as Euclidean geometry. Viewed this way, it doesn't matter that our universe isn't actually Euclidean.
