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 negagive 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.
knightly@pawb.social 2 days ago
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:
Image
Hamartiogonic@sopuli.xyz 2 days ago
If you draw a triangle on different parts of a toroid, would you get different angles?
knightly@pawb.social 2 days ago
Yup! That saddle is a shape of constant negative curvature. On a toroid, the inside of the hole would have negagive curvature and the outside would have positive curvature.
Hamartiogonic@sopuli.xyz 2 days ago
Wow. That would be truly bizarre kind of space to live in.
WalrusDragonOnABike@reddthat.com 2 days ago
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.
Hamartiogonic@sopuli.xyz 1 day ago
Hmm… that’s a good point. Basically anything other than a flat surface will have these bizarre properties.