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.

You might find this interesting: Non-Euclidean Doom: what happens to a game when pi is not 3.14159…

The magic starts at 6:47

if pi was 3, the area of a circle would be equal to 3 squares with side-length of the radius. this means the definition of a circle would be different, since now a circle is 3 squares. This would break geometry. Since the definition of a circle is the foundation of geometry, and trigonometric functions sin and cos.

Hmm or maybe if the area of a circle A=r*r*PI and PI = 3 so A=r*r*3 you would just scale the radius by the factor of sqrt(pi)/sqrt(3) = 1.0233267079464885... to get to a correct circle again.

So you would just scale the universe by a factor slightly larger than 1, up-scaling everything.

But then the formula for the volume of a sphere V=4/3*pi*r*r*r would break ………

All circles would be slightly smaller. - source, am a Professor of Sizemology

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

I prefer π = 3.14 ± 0.14. Add a little chaos.

I suppose you could redefine pi to be exactly like 3, but you’d have to change the value of 1 to be equal to (π/3) which would make it very difficult to buy bananas.

Probably a lot like Indiana.

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’?”

Circles would be smaller

Probably a triangle.

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.

Probably the Distortion World.