what is a flat triangle on a round paper? — Winner568

Do you mean a three-dimensional spherical surface?

By furthering an idea of how a 2-dimensional flat-lander could realize how to calculate the hypotonus of a triangle. — Winner568

I don't have much experience at 3D geometry, but I don't think the flat-lander would be able to know that. I definitely might be wrong. She would measure the distance between the vertices of the triangle as straight lines. The Pythagorean Theorem would be the same as for a 2D surface. If she wanted to measure the length of the sides of the triangle on the sphere, she would need to calculate great circle lengths, which go through the center of the sphere. In order to do that, she would have to know the radius of the sphere. If I remember correctly, the sum of angles on a triangle drawn on a sphere is not 180 degrees.

I am really not sure this is true.

Just looked this up in Wikipedia:

The sum of the angles of a triangle on a sphere is 180°(1 + 4f), where f is the fraction of the sphere's surface that is enclosed by the triangle.

I think that means that the geometry of a triangle would depend on it's size. Not sure (at all).

I ask the question, what is a flat triangle on a round paper? — Winner568

No longer flat.

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I don't think a flat-lander will face any difficulty at all. Length, numbers, multiplication, and angle, all are 1 dimensional or 2 dimensional concepts.

The sphere is not flat but 2d. You can know your triangle is in 3d space by counting the sum of the angles.

Yes, you can detect intrinsic curvature on a sphere, even if it is not embedded in 3D space. Angles in a triangle won't sum up to 180 degrees.

http://www.thephysicsmill.com/2015/12/27/measuring-the-curvature-of-spacetime-with-the-geodetic-effect/

Yes, you can detect intrinsic curvature on a sphere, even if it is not embedded in 3D space. — SophistiCat

How do you envision THAT? A sphere with intrinsic curvature HAS to lay in an infinite flat space.

Yes, you can detect intrinsic curvature on a sphere, even if it is not embedded in 3D space. Angles in a triangle won't sum up to 180 degrees. — SophistiCat

After the discussion yesterday, I thought about this subject a lot, asking myself questions. The article you linked answered them all. Good article. Thanks.

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I think that it would be difficult to find the distance of the sides and the angles at the same time — Winner568

Look for the Pentagramma Merificum. You might be interested.

It's easier when the deficit angle is large - e.g. 90 degrees, as in the example given in the link above. If after making three right turns at a right angle you end up on the same spot where you started, there's no escaping the conclusion that you are living on a curved world!