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.

More Prototypes

Though I haven’t blogged in about a year, at least I have been making progress on my puzzles. Pictured here are Model #808 (the rectangular one with two holes), Model #518 (the multi-colored cube), Model #921 (the circular one), and Model #360 (the thicker rectangular one with the red logo).

The internal design of Model #808 (formerly known as Model #873) continues to evolve and I’m hoping to get a prototype today that will confirm the reliability of the new design. I guess it’s no surprise but getting a puzzle mechanism to work in theory is easy, getting it to work once or twice in an actual mechanism is doable, but getting it to work dependably and reproducibly is really hard. I’m on Rev. 9 on this one.

Model #360 turned out to be easier to get working than I expected and it may be the first one to reach production in aluminum. Models #518 and #921 still have a ways to go.

In other news, I now have an FDM type 3D printer (the i3PRO from MakerFront) to shorten the cycle of revise design / make new prototype / test new prototype. I’m still learning how to get good quality prints from it but I look forward to being able to get a physical prototype of a puzzle design within hours instead of days. I’ll post some videos of my experiences with it in case anyone else is contemplating buying one of these things.