by Jonathan Kujawa
Human beings are tightly bound by the limits of our intuition and imagination. Even if we grasp an idea on an intellectual level, we often struggle to internalize it to the point where it becomes a native part of our thinking. Rather like the difference between being able to comfortably converse in a foreign language by translating on the fly and being fluent enough to think in the language like a native. Or, as the philosopher Stephen Colbert explained, it's the distinction between truth and truthiness.
We struggle to imagine things much different from what we see around us. This failure leads one in four Americans to believe the Sun goes around the Earth. It means we can't truly grasp the staggering, mind-boggling length of a billion years and this fuels skepticism about evolution. And for science fiction readers it leads to raging internet arguments about whether the authors have any imagination at all.
When it comes to geometry our everyday intuition tells us that we live in the planer geometry of good old Euclid. The angles of triangle add up to 180 degrees, parallel lines will never meet, and the shortest distance between two points is a straight line. But intellectually we know we live on the sphere called Earth, and that the geometry of the sphere leads to triangles whose angles sum to 230 degrees, parallel lines which meet, and flight paths between cities which follow “Great Circles”.
Media portrayals to the contrary, mathematicians are human, too. From Euclid until the first half of the 19th century, everyone was on board with Euclidean geometry. After all, that was what their gut told them geometry should be. But then Bolyai and Lobachevsky showed us that there are more things in heaven and earth than Euclid could dream of. In two dimensions there are also hyperbolic and spherical (elliptic) geometry. In higher dimensions the possible geometries multiply like rabbits and Einstein's theory of relativity tells us that the geometry of our universe isn't Euclidean [0].
How can we free our feeble minds from their Euclidean prison and develop an intuition for these new geometries?
