system
Given the following post, respond as if you're commenting on Reddit:
human
Would a being living in a hyperbolic space be able to tell that its space was a hyperbolic shape?

Also, if our hyperbolic being was a 18th century timid mathematician, would he be able to conceptualize euclidean geometry and see it as complex as hyperbolic is to us?
gpt
In hyperbolic space, curvature is negative. If the curvature is negative enough, the being would notice it, in several ways:

* If two people start out walking next to each other in parallel, and both keep walking straight ahead, then they will move away from each other, their paths will diverge. The amount of divergence is a measure for the curvature. (Note that on Earth's surface, which has positive curvature, the opposite happens: the paths will converge.)

* If you form a triangle from three straight lines, then the sum of the angles will be smaller than 180 degrees. The difference between the angle sum and 180 degrees is a measure for the curvature. (Again, triangles on Earth's surface have a degree sum bigger than 180 degrees.)

* The area of a circle of radius *r* will be larger than *r*^(2) π. (On Earth's surface, it's smaller.)

* The volume of a sphere of radius *r* will be larger than 4/3 π *r*^(3).

The mathematician would still have discovered Euclidean geometry (just like the ancient geometers concerned with Earth's surface discovered it) since in every space, no matter the curvature, Euclidean geometry governs the smallest scales. Tiny triangles always have an angle sum of about 180 degrees, tiny circles always have an area of about *r*^(2) π, etc.