Curvature −1: A surface of constant negative curvature −1, also has the same local geometry everywhere, with the hyperbolic cosine playing the same role as the cosine in for the Pythagorean theorem. It seems natural to search for an analog of the sphere, a complete surface of constant negative curvature, closed and without singularities. But in 1901 David Hilbert proved there is no such surface. However, here are some interesting surfaces (these illustrations are from a book by Alfred Gray). In all cases you can see boundaries or creases that illustrate Hilbert's theorem.
The square: There is another such surface which struck me as obvious, but which I have not seen. Start with the simple surface z = xy. By manipulating the surface, it should be possible to make the curvature constant while retaining the symmetries. Here is a picture of what I call the hyperbolic square. I wanted to find a closed form representation of this surface, or come as close as possible.
One representation: You may notice that the surface twists in such a way that it cannot be represented overall as the graph of a function like the original z = xy. However, around the origin, this type of representation is possible. Using the analytic expression of curvature for the graph of z = f(x, y), (f11f22 − f122)/(1 + f12 + f22)2, it is possible to write a program to calculate terms of its Taylor series. It is not obvious that the series has positive radius of convergence, but it does. There are interesting patterns to the coefficients, but I have not found a closed formula for them. One pattern is that the coefficients of the terms involving the first power of x are the same as the series for the tangent function.
Intro Curvature −1 Coordinates θ Complex θ More Equationss Trick General Entire
