Comment by kannanvijayan
6 days ago
I don't have an answer to your questions, but I think these thoughts are not uncommon for people who get into these topics. The relationship between the reals, including Pi, and the countables such as the naturals/integers/rationals is suggestive of some deeper truth.
The ratio between the areas of a unit circle (or hypersphere in whatever dimension you choose) and a unit square (or hypercube in that dimension) in any system will always require infinite precision to describe.
Make the areas between the circle and the square equal, and the infinite precision moves into the ratio between their lower order dimensional measures (circumfence, surface area, etc.).
You can't describe a system that expresses the one, in terms of a system that expresses the other, without requiring infinite precision (and thus infinite information).
Furthermore, it really seems like a bunch of the really fundamental reals (pi, e), have a pretty deep connection to algebras of rotations (both pi and e relate strongly to rotations)
What that seems to suggest to me is that if the universe is discrete, then the discreteness must be biased towards one of these modes or the other - i.e. it is natively one and approximates the other. You can have a discrete universe where you have natural rotational relationships, or natural linear relationships, but not both at the same time.
Easily fixed! I choose 1 dimension. :)
Hah, nice find :)
Good show, and I appreciate your sentiment about the "messiness" of pi.
There's a unit-converting calculator[0] that supports exact rational numbers and will carry undefined variables through algebraically. With a little hacking, you can redefine degrees in terms in an exact rational multiple of pi radians. Pi is effectively being defined as a new fundamental unit dimension, like distance.
Trig functions can be overloaded to output an exact representation when it detects one of the exact trigonometric values[1] eg cos(60°) = 1/2. It will now give output values as "X + Y PI", or you can optionally collapse that to an inexact decimal with an eval[] function.
That's the closest I got to containing the "messiness" of pi. Eventually I hit a wall because Frink doesn't support exact square roots, so most exact values would be decimals anyway.
Still, I can dream!
[0] https://frinklang.org/
[1] https://en.wikipedia.org/wiki/Exact_trigonometric_values
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The universe, being a physical entity not bound by the rules of human logic doesn't have to be either.
Our logical approximation of the universe might need to be, assuming that we don't add more axioms to our system of logical reasoning.