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Pi only exists in the context of euclidean space. We have never seen a real euclidean metric space in THIS universe. Pi literally isn't a universal constant. Un
by blamestross 5y ago
Pi only exists in the context of euclidean space. We have never seen a real euclidean metric space in THIS universe. Pi literally isn't a universal constant. Unlike physical constants (All the force constants, speed of light ect) apparent values of Pi will vary wildly if you can even find situations where it makes sense to apply it.
Concrete example: Assume you somehow magically can have a planet in a circular orbit around a star, surely you can use Pi to derive the circumference from the distance to the center of the star? Nope, the space in the system is distorted, Pi varies with radius and the mass of the star.
- wyager 5y agoSpace is locally euclidian. You can easily define pi as a universal constant in our universe, in terms of local flat regions.
- blamestross 5y agolocal euclidian-ness is an approximation. Nowhere is flat and it wont be until well after the heat death of the universe when every particle is outside the light cone of every other (making a lot of dubious assumptions about the eventual fate of the universe)
- nimish 5y agoNo, local euclidean-ness is (more or less) the equivalence principle. The locality could be very small though! But it would break GR if local flatness weren't possible.
- blamestross 5y agoEquivalence principal is about the fact we can't differentiate between a flat spacetime+force and a curved spacetime. It doesn't require actual flat spacetime to exist anywhere outside of a model. It only says you can't tell how curved it is based on the forces acting on you. Different argument to same end: as long as the universe is expanding, nowhere can be flat. Any 2 things have an inherent repulsive force due to spatial expansion.
- keithnz 5y agoas per the article, in this universe, and potentially others as long as they have some notion of smooth space, we infer an abstract ideal of flatness and from there derive Pi.
- blamestross 5y agoThe notion of smooth space is an approximation, it doesn't exist. So yes, you could find a space with an "apparent pi" within a given delta of the ideal, but it won't be "ideal pi"
- AnimalMuppet 5y agoYou either missed or ignored keithnz's point. There are plenty of spaces that are close enough to smooth that we can deduce the properties of a perfectly smooth space, and from that, the value of a perfect pi. So we can't find a space that has that same perfect pi. So what?
- blamestross 5y agoMy point is that the argument is wrong. "ideal flat space" isn't a property of our universe. That "perfect pi" is a property of the mathematical abstraction of "smooth space". Our universe might asymptotically flirt with such a state but "flat space" isn't real not in the sense that the speed of light and force constants are descriptions of things that actually happen and actually exist.
- keithnz 5y agoThere is no real argument, essentially the article says well, "it depends" and basically we have no idea, but, given a few assumptions, is it reasonable that other universes would have denizens that conceive of circles and calculate the same value of Pi? and they postulate that if that also had a notion of flat space (not that the universe comports to that) then they too probably would come up with the same value of Pi
- 5y ago
- yakubin 5y agoYou don't need any geometry to define pi. At uni, I had it defined as the smallest positive zero of sin(x). And sin(x) was defined axiomatically (in terms of real analysis) in conjunction with cos(x) (and then shown to be equal to its Maclaurin expansion).
- HPsquared 5y agoPi comes out in the limit of flat spacetime.
- Akronymus 5y agoI'd think the construct of "The shape of something where all points that are x distance away from a certain point" is something that would be universal. (Thinking of spheres rather than circles.) AFAIK pi crops up in any dimensionality in any type of space for spheres.