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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.

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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".

picture of sphere showing equatorial circle and its diameter

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    $\begingroup$ In non-Euclidean geometries it's misleading to talk about the circumference-diameter ratio, because it changes from circle to circle. $\endgroup$ – Robert Israel Apr 21 '14 at 2:16
  • $\begingroup$ @SRM It would be nice if your description is accompanied by a picture. $\endgroup$ – Mick Apr 21 '14 at 8:46
  • $\begingroup$ Robert: Good point. Mick: I'll look into building one. $\endgroup$ – SRM Apr 22 '14 at 15:06
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In the so-called Taxicab geometry, where the distance function is given by the sum of the absolute differences of the coordinates, "circles" are squares with sides at 45 degrees to the axes. The ratio of the circumference to the diameter in this geometry is 4.

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Consider the radius of the circle not as a straight line distance but an arc equal in length and curvature to 1/6 of the circumference. Each of such sixths was formed by stepping round the circumference using the same compass aperture with which the circle was constructed in the first place.

Pi will then be the ratio of the circumference to the sum of two of these radii, the diameter. That is 6/2, exactly 3.

The ratio only increases to 3.14159 etc if we pull the radii straight, resulting in shorter lengths from centre to perimeter.

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