At what scale would negative curvature be noticeable (and a nuisance to the flat space society)?

After playing some HyperRogue, I got interested in the use of hyperbolic geometry in WorldBuilding, in particular how an area grows exponentially with respect to its radius.

Let's say that the universe is a hyperbolic space. The world itself is a disc with a radius of 30 miles and an area with 196.9 million square miles. This would mean that you can get anywhere within an hour if you travel in a straight line at 60 mi/hr, and the area is as large as the surface area of our whole entire earth!

This world is necessarily non-euclidean, of course, since it has negative curvature. The angles of triangles will add up to less than 180 degrees, for example. It will appear euclidean at sufficiently small scales though (the angles of the triangles will add up to only slightly less than 180 degrees). It will be noticeable at large scales (the sum of a triangles angles will be close to 0 degrees).

My question is, at what scales will the negative curvature of the space be noticeable to humans? Will the non-euclidean geometry only be relevant to a world traveler, or would an artist have to be familiar with it while painting, or somewhere in between? (If you wish to add flavor to your answer, you can present it as problems for the flat space society, which denies the reality of non-euclidean geometry).

posted about 7 years ago

PyRulez‭

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