Spheres are examples of surfaces with positive curvature. Negative curvature (where the angles of a triangle add up to less than 180 degrees) is represented by this saddle shape:
Yup! That saddle is a shape of constant negative curvature. On a toroid, the inside of the hole would have negative curvature and the outside would have positive curvature.
Same is true for a sphere. For example, if you draw a triangle with 1 vertex at the north pole and the other two on the equator, all the angles are 90 degrees. But if you move, say, the north pole vertex to be closer to the two in the equator, then you’d get a smaller sum.
Spheres are examples of surfaces with positive curvature. Negative curvature (where the angles of a triangle add up to less than 180 degrees) is represented by this saddle shape:
If you draw a triangle on different parts of a toroid, would you get different angles?
Yup! That saddle is a shape of constant negative curvature. On a toroid, the inside of the hole would have negative curvature and the outside would have positive curvature.
Wow. That would be truly bizarre kind of space to live in.
Same is true for a sphere. For example, if you draw a triangle with 1 vertex at the north pole and the other two on the equator, all the angles are 90 degrees. But if you move, say, the north pole vertex to be closer to the two in the equator, then you’d get a smaller sum.
Hmm… that’s a good point. Basically anything other than a flat surface will have these bizarre properties.