In Euclidean geometry, the ratio of the circumference of a circle to its diameter is an irrational number, 3.14159 and so on. But if you change to non-Euclidean geometries, you get other values for that ratio. I would like to know if there are any geometries where the ratio is a rational number.
I found this similar question, but that question asked if pi had other values in non-Euclidean geometry… the answers given, although correct technically, all got caught up on the fact that pi is pi regardless of geometry in modern mathematics, so they ended up missing the real question being asked. Thus I am re-asking the question specifically about the circumference-diameter ratio.
Best Answer
Turns out the answer is "yes". If you have a spherical geometry, it is easy to find an example: Draw a circle around the equator of the sphere. Now draw a line across the diameter of that circle -- it will arc up through one of the poles and back down on the other side. The length of that arc is obviously exactly half the distance around the sphere, so the ratio of circumference to diameter will be 2, which is a rational number.
Outside of Euclidean geometry, different circles can have different ratios in the same geometry. There is no singular "circumference to diameter ratio".