normal-distribution-manifold

The normal distribution, or bell curve, is ubiquitous in nature. Mathematically, though, normal distributions inhabit a strange world unlike our own three-dimensional universe. Instead of our three coordinates of width, length, and depth, normal distributions live in a world described by only two coordinates, mean and variance. In this unfamiliar world, two distributions that look close may in actuality be far apart, our eyes too accustomed to a lifetime of three-dimensional Euclidean geometry. Stranger still, whereas the shortest path between two points in our universe is a straight line connecting them, the shortest path between two normal distributions looks decidedly curved, at least until this world's hyperbolic geometry starts to make sense. Presented here is a visualization of this strange space: click to see an inhabitant of this world; drag to find the shortest path between two.