Rahul Narain @narain@mathstodon.xyz

Animation of the minimum-weight matchings of increasingly many points of two colors in a unit square: https://twitter.com/thienan496/status/1446021847292669953

Because the color densities fluctuate, the matching develops regions of many parallel long edges transporting excess density from one place to another. This is reflected mathematically in the fact that the expected length is \(\Theta(\sqrt{n\log n})\) compared to \(\Theta(\sqrt{n})\) for non-bipartite matching; see https://doi.org/10.1007/BF02579135

Hanlon's razor for identical incorrect exam answers: never attribute to collusion that which is adequately explained by synchronised stupidity.

Because of Euler we know that for polyhedra without holes we have
V+F=E+2. As a result we can't have a polyhedron made entirely of
hexagons, and need at least 12 pentagons or an equivalent set of
shapes. Etc.

But things change if we allow intersecting faces. Clearly we have
a lot of scope for variations, but we can always retain limits to
make things reasonable.

But ...

Can we make a "polyhedron" consisting only of hexagonal "faces"?

The four points, two distances problem: https://www.theguardian.com/science/2019/oct/21/can-you-solve-it-the-four-points-two-distances-problem

Can you find all of the ways of arranging four distinct points in the plane so that they form only two distances? The link is not a spoiler but it has a separate link to the solution. "Nearly everyone misses at least one" says Peter Winkler; can you guess the one I missed?

Robert MacKay's Chaos Machine. Its configuration space is a genus three surface. The dynamics of the machine are equivalent to geodesic flow on the surface, which is Anosov, hence chaotic.

Compact packings of the plane with three sizes of discs: https://arxiv.org/abs/1810.02231, Thomas Fernique, Amir Hashemi, and Olga Sizova

Here, "compact packing" means interior-disjoint disks forming only 3-sided gaps. The circle packing theorem constructs these for any finite maximal planar graph, with little control over disk size. Instead this paper seeks packings of the whole plane by infinitely many disks, with few sizes. 9 pairs of sizes and 164 triples work. Here's one from Fig.3 of the paper.

Here's a 1965 RAND report demonstrating a conversational interface with an algebraic processing system. They determine that a conversational user interface vastly improves usability in their tests.

Full PDF: https://www.rand.org/content/dam/rand/pubs/papers/2008/P3146.pdf

New blog post: Motorcycle graphs and the eventual fate of sparse Life, https://11011110.github.io/blog/2018/12/27/motorcycle-graphs-eventual.html