better my computer than me
My computer ran for eight solid days to extend this table from six rows — (2 3 7), (2 4 5), (3 3 4), (2 3 ∞), (2 ∞ ∞), (∞ ∞ ∞), each of which is (in some sense) minimal — to 106, by request. I don’t know why anyone would want all those others; I see no qualitative difference between any of them and one or more of the six.
Now that the run is done, I look again at my code and see where it could be made more efficient, by changing from complex to real arithmetic; I’ve already done that in my other hyperbolic programs, the ones that generate the ribbon patterns.
a bit of randomness
I made a minor improvement to my Doodles page: it now chooses the dragon fractal tile randomly from a series of 243 variants – one for each possible assignment of the five high bits to the three color channels.
soap films in curved space
A few of the many triply periodic minimal surfaces can be generated from a quadrilateral slice through a tetrahedron of mirrors, as refined by Surface Evolver. I had the idea that the same concept, applied to one of the analogous tetrahedra that tile spherical 3-space, could result in a pretty model.
To my disappointment, none of the non-prismatic kaleidoscopes — those that generate the six regular polychora — hosts a nondegenerate minimal surface (of this simple form); but each of the duoprism kaleidoscopes gives at least one.
but wait there’s more
This one is a projection of a design on (half of) the surface of a hypersphere: a ribbon spans two orthogonal great circles, wrapping three times around one and five times around the other.
Kitty teeth did break it at one point, but with redundancy that’s not a serious problem. Having got my pix I’ll keep the new toys away from them henceforth.
Baby’s First Klein Bottle™
Not only did it not fall apart in my hands …
first tangible result
The strand — for it is a Klein bottle woven of a single strand — is thinner than I expected, and has broken in at least two places. Ah well, live and learn; I’ve already redesigned it to be more robust.
I came up with this concept several years ago (I can’t tell exactly when, because I’ve mislaid the original code!) but never posted any of the images until now.
Some of you may like to guess what’s going on before I tell. For more hints, see these patterns’ 254 siblings.