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> Pi doesn't vary. The ratio of circumference to diameter of a circle may vary depending on the geometry.

Can you share some place where Pi isn't defined exclusively as being "the ratio of circumference to diameter of a circle"? I have never heard any other definition in my life, and couldn't find any other through the first few Google results

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"This definition of π implicitly makes use of flat (Euclidean) geometry; although the notion of a circle can be extended to any curve (non-Euclidean) geometry, these new circles will no longer satisfy the formula π = C / d"

https://en.wikipedia.org/wiki/Pi

Pi is the "the ratio of circumference to diameter of a circle in Euclidian space". Everyone agrees that. What people are arguing is that when you have a circle in non-Euclidian space, so that the ratio of it's circumference to diameter is different, do we still call this new ratio Pi.

Most people would argue that we don't. They say "the ratio is not Pi" rather than "Pi is a different value".

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