If whales had invented geometry, the geometry they would have invented would be hyperbolic.
Suppose, for a moment, that you are a whale. Light is not very useful in the deep ocean, because the water is dark. So you mostly communicate and experience the world through sound. The shortest distance between two points in your world would be the path taken by sound waves. To you, this would be the analogue of a straight line.
Now here’s the catch. Sound does not travel at a constant speed in the ocean. Below a certain depth, roughly 2,000 feet (600 meters), it travels at a speed that is proportional to the depth below the surface. So the path that sound waves travel is not straight, but curved. A sound wave will get from whale A to whale B quicker if it goes downward, to exploit the greater sound speed at depth, and then comes back up. Thus, to a whale, what humans call a “circle” is actually a “line” (the shortest distance between two points).
|(courtesy of NYT's numberplay)|