Although we can picture the surface of a sphere or doughnut, hyperbolic surfaces have such strange geometric properties that they’re impossible to visualize. But they’re also important to understand, ...

The product of their two curvatures at each point is -1 everywhere. So in a sense they are the opposite (or hyperbolic counterpart) of an ordinary sphere. Here is the simplest one: One nice thing ...

The product of their two curvatures at each point is -1 everywhere. So in a sense they are the opposite (or hyperbolic counterpart) of an ordinary sphere. Here is the simplest one: One nice thing ...