system
Given the following post, respond as if you're commenting on Reddit:
human
At what point, specifically referencing Earth, does Euclidean geometry turn into non-Euclidean geometry?

I'm thinking about how, for example, pilots can make three 90degree turns and end up at the same spot they started. However, if I'm rowing a boat in the ocean and row 50ft, make three 90degree turns and go 50ft each way, I would not end up in the same point as where I started; I would need to make four 90degree turns. What are the parameters that need to be in place so that three 90degree turns end up in the same start and end points?
gpt
The answer to the title question is "always". The Earth is spherical. Period. Whether the spherical shape of Earth matters to you is dependent on the what you're measuring and your threshold for error.

As to your more specific question...

On a sphere, the area of a triangle formed by three geodesics (arcs of a great circle) is given by

> A = R^(2)(a + b + c -  π)

where *a*, *b*, and *c* are the interior angles of the triangle and *R*^(2) is the radius of the sphere.

If you want your triangle to have three right angles, then this formula reads:

> A =  πR^(2)/2

and, as a ratio of the total surface area of the sphere,

> r = A/(4πR^(2)) = 1/8

So if you want to make some sort of journey on the surface of Earth and get back to where you started by traveling along great circles and turning 90 degrees exactly ~~three~~ two times, then the surface area enclosed by your path must be 1/8 the total surface area of Earth. (That's about 3.7 times the land area of Russia.)

Of course, there's no reason you *have* travel along great circles. In that case, your triangle can have three right angles and enclose an arbitrary small area. But then the sides of your triangle would not be the proper analog of "straight line" for spherical geometry.