this post was submitted on 23 Jan 2025
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Pi as in math like Euler’s identity, cannot be changed. It arises from the definition of e and imaginary numbers, both of which arise from the natural numbers which arise directly from axioms.
Pi as in the ratio of a circle’s circumference to its diameter, however, could be changed, in which case you would change the fundamental geometry of space. This would be neither hyperbolic nor spherical space because those spaces still use the mathematical pi for determining angles (along with hyperbolic trig functions of course).
The geometry would likely be much closer to Chebyshev or Taxicab space since the ratio of circumference to diameter in those spaces is 4 (I think…). Because of this, I suspect that using a distance function like in Chebyshev or Manhattan but with a triangular grid instead of a square one would yield this exact situation where geometric pi=3. This would be confusing as hell but now I’m curious and have coincidentally already started exploring the concept of metric spaces so I’ll look into it. Though I’ll probably get distracted and forget…
Edit: Found it, Chebyshev distance on hexagonal grid would give a circumference/diameter ratio of 3. So a metric space with a distance function like that is the geometry you want.