For the new year, I’ve decided to try to get back into taking photos more frequently, and to make it lower-overhead I’m making individual Mastodon posts for some of them rather than writing a longer blog post for every batch of photos. So that’s why you see a couple of those images inline here.

• Linkage for the end of the year

• LaTeX, the game ( G+, via). It should be an even higher level to get the commutative diagram to format in Wikipedia’s lobotomized version of LaTeX.
• Motorcycle graphs and the eventual fate of sparse Life

The motorcycle graph is a geometric structure devised by Jeff Erickson as a simplified model for the behavior of straight skeletons, motivated by the light cycle game in the movie Tron. Its initial data consists of a set of points in the plane (the motorcycles), each with an initial velocity. The motorcycles leave a trail behind them as they move, and a motorcycle crashes (stopping the growth of its trail) when it hits the trail of another motorcycle.

• Circles crossing at equal angles

Let $A$, $B$, $C$, and $D$ be four circles, with pairs $AB$, $BC$, $CD$, and $DA$ crossing at equal angles (and no crossings among the other two pairs). Then it turns out that the two curvy quadrilaterals forming the inside and outside boundaries of the union of disks each have a circle through their four vertices:

This is my penultimate link roundup before I give up on Google+, rather than holding out for its rapidly-approaching demise.

• General-position hypercube projections

I recently posted about finding solutions to the no-three-in-line problem of finding large general-position subsets of grids, by using the probabilistic method or by throwing an integer linear program solver at the problem. Another potential method for finding solutions involves finding large general-position subsets in higher dimensions, where it’s easier (there’s more room to move the points out of the way of each other), and then projecting back down while being careful not to introduce any new collinearities.

• Triply-Hamiltonian edge colorings

Mark Jason Dominus recently made a blog post about the interesting observation that the regular dodecahedron can have its edges properly colored with three colors so that every two colors form a Hamiltonian cycle. Here’s another view of the same dodecahedral coloring:

In which I discover kramdown’s inability to pass raw vertical-bar characters to MathJax… (workaround: use \vert)
For the no-three-in-line problem, it has been known since the 1990s that $n\times n$ grids with $n\le 46$ have sets of $2n$ points with no three in line. Those results, by Achim Flammenkamp, were based on custom search software and a lot of compute time. I was curious to see how far one could get with more-modern but generic optimization codes, so this weekend I ran a little experiment.