Constant Negative Curvature

One of my favorite events is the local annual Celebration of Mind gathering inspired by Martin Gardner’s immense contribution to recreational mathematics. I went to one last night in Brookline, MA sponsored by Eureka Puzzles and gave a short presentation on my interest in what I’m calling (probably incorrectly) a “unit hyperbola”. My talk was called “Horsing Around With Z=X*Y” and if this kind of thing interests you, you can download the PDF of it here.

Basically, I used OpenSCAD to model Z=X*Y for X and Y values ranging from -1 to 1. That generates a saddle-shaped half-cube (with Z value from -1 to 1). I found that you can take those half-cubes (or unit hyperbolae) and compose them into some very weird-looking shapes. Here are some examples:













I’ve convinced myself that every point on the curved surface sees constant negative curvature in all directions – hence the title of this post – but I’m no mathematician and it wouldn’t completely surprise me to learn I’m wrong. The basis of my conviction is empirical: I can slide the curved surface of one of the half-cubes over the curved surface of the more convoluted shapes while keeping the surfaces in full contact with each other at all times. Since I know the curvature is fixed for the half-cube, I conclude that the curvature must be fixed for the convoluted shapes too.

In any event, it’s clear that the “holey” shape repeats its pattern in all three dimensions so it should be relatively straightforward to define the minimal subset cube that “tiles” space accordingly – but I ran out of ideas when I was playing around in OpenSCAD. Now that I’ve had to organize my thoughts for the presentation, I think I’ll take another shot at it.