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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).

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  • $\begingroup$ Comments are not for extended discussion; this conversation has been moved to chat. $\endgroup$ – JDługosz Apr 13 '17 at 14:53
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    $\begingroup$ I created an Answer based on our discussion in the comments, since nobody’s tried answering yet! $\endgroup$ – JDługosz Apr 13 '17 at 14:59
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Your answer depends on what is meant by “notice”.

Consider a parking lot. Perhaps a pedestrian won’t see the curvature by eye as he walks through it. But the contractor who paves it will consume some amount of material, and his tolerance in that depends on the accuracy and uniformity of his paving technique. Maybe ±half a wheelbarrow of concrete will cause concern; maybe differences in a truckload or two are chalked up to the crew’s raking technique and go unremarked.

You should make a chart (program a spreadsheet) showing the ratio of area to radius, percentage difference from Euclid's, and absolute difference; for patches of different sizes in a logarithmic progression. Another pair of columns shows the relative and absolute difference for expected lengths of the sides of a triangle, for each patch size.

Then, use this chart to decide whether someone will notice based on specific context of the activity.

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    $\begingroup$ Areas aren't the only thing which is relevant. It's a lot easier to get lost in hyperbolic space as well, depending on curvature. $\endgroup$ – PyRulez Apr 13 '17 at 15:35
  • $\begingroup$ You would definitively notice that. Hmm, maybe that's what happens when I can’t find my way home? I left out angles from the chart; harder to say how they “line up” especially if light follows the geodesics. $\endgroup$ – JDługosz Apr 13 '17 at 15:46
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    $\begingroup$ I don't think you would get lost in the grocery store more easily than in flat geometry. If you drive in a random direction for an hour on the other hand ... $\endgroup$ – PyRulez Apr 13 '17 at 15:48
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Using this formula we can calculate the curvature $$K \approx -0.313339 \text{ per mi}^2$$

Using this, we can the radius of a Euclidean disk v.s. a disk in this world.

       Area        Euclidean Radius   Actual Radius  
 ---------------- ------------------ --------------- 
  1 mi^2           0.56 mi            0.56 mi        
  10 mi^2          1.78 mi            1.72 mi        
  100 mi^2         5.64 mi            4.42 mi        
  1000 mi^2        17.84 mi           8.26 mi        
  10000 mi^2       56.42 mi           12.34 mi       
  100000 mi^2      178.41 mi          16.44 mi       
  1000000 mi^2     564.19 mi          20.56 mi       
  196900000 mi^2   7916.77 mi         30.00 mi       

This means that over the size of a building, distortion will not be noticeable. In a small town, probably only builders will notice. In a small city, everyone will start to notice, but it will still be mostly Euclidean. In a bigger city, the curvature will be quite noticeable, and people will have to take into account if they do not want to get lost. At the size of countries, you will need to follow a navigation system very carefully if you want to get to the right place, and it will very non-euclidean.

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