Curvature is probably one of the hardest concepts of differential geometry to grasp (I'm not sure if it really is, because I don't really understand it...). But here's a demonstration of how it works. You can also see it as a demonstration of holonomy or the Gauss-Bonet theorem, if you prefer those. The great thing about it is that it is equally impressive if you know nothing at all or a lot about differential geometry.

The demonstration shows how moving in straight lines on a sphere causes rotation! Note that this is emphatically not true on the plane. Here, "moving" means what is known as parallel transport, equivalent to moving your hand along a surface without turning it.

Let's start with the plane. Put your hand on the table, and trace it (without turning it!) along a triangle. Note how it comes back to rest at the same angle it started off with. Not very exciting...

If you've done it all correctly (and it's just 3 movements), your hand returned to its initial position, but your palm is now pointing forwards! Moving in a closed path along a sphere has rotated your hand!

For added effect, note that your hand described a triangle with angles summing to 270o degrees (this is possible on non-flat manifolds). The excess over the "correct" value of 180o, 90o, is exactly what your hand rotated by. If you make some of the angles in the triangle smaller, the equality of the excess and the rotation of your hand is preserved!

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