You universe’s pi doesn’t equal pi… it equals 3…
But can it run Doom?
https://youtube.com/watch?v=_ZSFRWJCUY4&t=450s
Yes. Mostly.
Spatial cohesion worsens as pi diverges from… pi.
This question does not make sense. π is an abstract mathematical constant whose value has absolutely nothing to do with the physical world. It’s like asking “what would the universe look like if the word ‘fish’ started with ‘p’?”
This doesn’t directly answer your question, but things would probably get very weird compared to our universe. Here’s an interactive visualization of a different weird universe with two time dimensions, Dichronauts by Greg Egan:
https://www.gregegan.net/DICHRONAUTS/02/Interactive.html
He really goes through the math on that site, so you might get some insight into how other topologies would look
Circles would be smaller
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.
