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One thing to be aware of is that if you actually made a circle and measured its radius and circumference you wouldn’t get pi. Not because your measurements would be off, but because the universe does not follow the assumptions mathematicians used to define pi—namely Euclidean geometry. Pi is mathematical, not physical. If real circles and real diameters don’t give you pi that is a problem for the universe, not a problem for mathematics.
Wat?
The universe is non-Euclidean, so no circle made in the actual geometry of the universe actually has the ratio of pi between its circumference and diameter.
Is that the part you are confused about, or did I write something else badly?
[finding people who don’t know that we live in non Euclidean space these days is like finding people who think the sun goes round the earth. But I guess if people can’t be bothered to learn 350 year old mathematics, they also can’t be bothered to learn 100 year old physics. Oh well.]
The universe being non euclidian needs some extraordinary evidence.
It's true! I just drew a circle and measured it. Turns out π≈5.
From your measurement of pi, we can deduce that you live in an anti-de Sitter space, so all the string theorists will now be sending you emails to test out their theories.
Overly snarky response: Uhhhm. Have you been asleep since, what, 1915 or something? We have extraordinary evidence, and everyone has accepted it, in so far as I know.
Less snarky response: the path on which light moves is the universes instantiation of a straight line. It is “the (locally) shortest path between two points”, the same definition you learned in geometry class. Yet in our universe, two straight lines can intersect each other twice. This is because our universe has at least some local curvature, meaning it is locally non Euclidean. In order to have a mathematically perfect circle you would need to live in a universe without any matter or energy, and with certain other properties.
Meh, spacetime curvature has so little effect that it actually does come down to measurement accuracy. And if you go to the scale of the whole universe it is flat as far as we currently know.
Now I wonder how LISA will handle curvature changes due to sun, earth and moon moving around, or if they won't make enough of a difference.
I mean, geodetic interferometers already exist and can measure very small deviations. Give them arms the length of the observable universe and they will increase in accuracy, not decrease in accuracy.
If you constructed a circle with the radius of the universe, then measured its circumference and radius measurement accuracy would easily be able to tell the difference between a real circle and a mathematical circle. That is because neither the perimeter of circle will nor the diameter of the circle will be through in empty space. They will be near enough to matter to measure detectable deflections.