• The following lemma on distances between two out of three points around an obstacle is part of my new preprint on integer distances. I’m curious, has anyone seen it before?

    Lemma: Suppose that three given points in \(\mathbb{R}^d\) are disjoint from an open convex set \(U\). Then two of the points can be connected by a curve disjoint from \(U\) of length at most twice their Euclidean distance. (\(\mathbb{M}\)).

    For instance, in the plane, two nearby points can be separated by a thickened line or line segment, so they have no short connecting curve, but three points cannot all be made far from each other by a single convex obstacle. See the preprint for the (not very difficult) proof, whose illustration is below.

    Three points separated by a convex obstacle, and a short path between two of them

  • Particle swarms with one random friend and one random enemy per particle. The particles move toward their friend, away from their enemy, and toward the center of the screen. The resulting system shoots off arcs of particles that then slowly return to the center, “nicely chaotic behavior with no real physics”. Constructed and animated by Craig S. Kaplan.

  • Slides from my talk “Games on game graphs” (\(\mathbb{M}\)) at the AMS Special Session on Serious Recreational Mathematics at JMM.

  • Othello is solved to a draw by Hiroki Takizawa (\(\mathbb{M}\), see also). It’s still an unrefereed preprint, but Jonathan Schaeffer says he believes it.

  • My impression is that ACM is much less susceptible to sponsoring junk conferences than IEEE, but that does not mean they are immune (\(\mathbb{M}\)). Solal Pirelli reports on some “troubling ACM venues”.

  • Brewpub coaster showing a 13-tooth gear with nine evenly-spaced holes, noteworthy to a mathematician because neither of those things can be constructed with compass and straightedge.

  • On defunct journals with no-longer-accessible papers (\(\mathbb{M}\)), one of which, The Journal of Combinatorics and Number Theory, featured a paper by Bill Gasarch (fortunately also available on arXiv).

  • Two more new Wikipedia Good Articles (\(\mathbb{M}\)), one very basic, the other very technical:

    • Simple polygon: A shape bounded by line segments without any holes. A piecewise-linear Jordan curve. A connected 2-regular planar straight line graph.

    • Small set expansion hypothesis: A graph is a small set expander if every small-enough set of vertices has many edges out of it. In the simplified definition used by this article (following Barak), “small-enough” means at most and “many” means close to the degree of the vertices in the graph. The question is: suppose I give you a regular graph, and I promise you that either it is a small set expander, or it is very far from being a small set expander (there is a small set of vertices with very few edges out of it). Can you tell which? The small set expansion hypothesis is that this question should be \(\mathsf{NP}\)-hard. It’s closely related to the more famous unique games conjecture (it implies unique games and has been suggested to be equivalent), and like unique games provides a way of collecting together many conjectured inapproximability results.

  • Diagram of 3 non-degenerate and 9 degenerate quadrics, arranged to show which ones degenerate into which, by Rahul Narain.

  • Quanta on the Markov numbers (\(\mathbb{M}\), see also), solutions of the Diophantine equation \(x^2+y^2+z^2=3xyz\), the ternary tree of triples that links these solutions together, and recent progress on Frobenius’s uniqueness conjecture that each triple is uniquely determined by the largest of its three numbers.

  • Generating SVG for the Prime Knots (\(\mathbb{M}\)), Marcello Seri, using a circle-packing-based layout algorithm.

  • Mathematical generative art animated loops by Etienne Jacob (\(\mathbb{M}\)). See also Jacob’s mastodon account.

  • I found myself shut out of Google Scholar (again) because I did a big but entirely manual search on JSTOR (\(\mathbb{M}\)), collecting roughly 52 book review citations for a new Wikipedia article on Renaissance historian Margaret L. King. Although this did not cause JSTOR itself to show any annoyance, its back-and-forth with Google on each search step triggered Google’s “unusual robot activity” flags. Fortunately it only blocked one browser for a day, so I was able to make do with a backup browser, but it’s a very irritating aspect of JSTOR that normal use can have such consequences well beyond the site.

  • Uncountably many topological embeddings of the disk into Euclidean space (\(\mathbb{M}\)), obtained by adding connecting pieces to variant embeddings of Antoine’s necklace.