Hyperbolic Geometry

by beeflavor

One of the oddest thing that happens in this world is coincidences. Can it be a mere coincidence that our solar system, the milky, the universe are held together in equilibrium (relative to our lifetime of course)? Physicists are interested in this question and developed theories to explain all this. More alarming to me is how the mathematical universe are held together. The existence of the hyperbolic geometry is remarkable. Even more remarkable is how it can be modeled inside our intuitive Euclidean geometry.

Sitting inside GL(2, \mathbb{R}) is the group of isometries on the hyperbolic plane, SL(2, \mathbb{R}).  When we first conceived the idea of matrices, it is highly doubtful that anyone had the slightest clue that a subset of matrices are actually distance preserving maps for a very bizarre geometry! After all, SL(2, \mathbb{R}) are linear maps on the plane that are possibly squeezes or stretches, it is astonishing that we can curve our space a slight bit and get isometries. If this isn’t coincidental, then I don’t know what is. It is almost a miracle. I imagine that when the mathematicians (Poincare?) discovered this, the feeling he felt was comparable to climbing to the top of Mount Everest, or being the first crew that flies to the edge of the galaxy. Mathematics is remarkable.