Linkage

Video by Steve Mould on flexible polyhedra (\(\mathbb{M}\)).

Paper fortune teller (\(\mathbb{M}\)), cootie catcher, chatterbox, or whatever else you want to call it, now another Wikipedia Good Article.

Tokarsky’s unilluminable room (\(\mathbb{M}\)), via Joe O’Rourke. This is a polygonal room with two designated points so that, if the walls of the room are lined with mirrors, one of the points cannot see any reflected image of the other. For rooms with curved boundaries you can make regions with nonzero area that cannot see reflections of each other, but for rooms whose angles are all rational fractions of a circle such as Tokarsky’s, only a finite set of pairs of points can be mutually invisible. That leaves an intermediate case, polygons with irrational angles, for which the question appears to remain open; see Wikipedia on the “illumination problem” for more.

Product structure extension of the Alon–Seymour–Thomas theorem (\(\mathbb{M}\)), update to a preprint I previously discussed, now showing that \(K_{3,t}\)-minor-free graphs can be expressed as subgraphs of a product of a treewidth-2 graph and a complete graph of order \(O(t\sqrt n)\).

Clément Canonne observes that academics do not “forget” to answer emails. They remember and feel bad about them.

In March or April Zoom changed its terms of use (\(\mathbb{M}\)) to allow Zoom content to be used to train AI models, with no opt-out, and to push legal responsibility for any resulting copyright violations onto its users. After a big online outcry, and some discussion of alternatives, Zoom updated their terms again to drop the idea of using content for AI.

A brief illustrated guide to scissors congruence (\(\mathbb{M}\)). The Conversation discusses geometric dissection, its use to axiomize 2d area, Dehn’s invention of the Dehn invariant to prove that similar uses fail in 3d, and higher-dimensional generalizations of the Dehn invariant.

Disproof of the local-global conjecture for integer Apollonian circle packings (\(\mathbb{M}\), via). An Apollonian packing is a fractal pattern generated from an initial 4-tuple of mutually tangent circles (one outside and three inside it) by repeatedly placing another tangent circle into each triangular gap between three circles. When the curvatures (1/radius) of the first four circles are all integers, so are the rest. It was conjectured that, for any such packing, the curvatures include all but finitely many of the numbers in six or eight of the residues modulo 24. A summer research project for two students, Summer Haag and Clyde Kertzer, has shot this conjecture down.

Why those mini-golf volcanos are so difficult: John Baez explains that for the usual shapes of domes, it’s impossible to roll a ball with the right momentum to reach a dead stop at the top in finite time. But a special shape, Norton’s dome, does allow this. The time-reversed path of the ball is non-deterministic: it is consistent for the ball to sit at the peak for any finite length of time before starting to roll down.

A nicely made video about the lengths of longest increasing subsequences in random permutations and the shapes of random Young tableaux (\(\mathbb{M}\)).

All back issues of the now-defunct journal Mathematical Spectrum (\(\mathbb{M}\)), aimed at high school and undergraduate mathematics students.

While in Montreal for WADS and CCCG a couple of weeks ago, I took time to photograph some of its abundant street art (\(\mathbb{M}\)). My photos are now online. They fall into three portions: the neighborhood of Concordia University (where the conferences were held; 12 photos), a bike ride from there along Blvd De Maisonneuve through the Latin Quarter to Le Plateau-Mont-Royal (18 photos), and Ave du Mont-Royal (16 photos).

On Google Maps failing to display certain street names no matter how far you zoom in. Apple Maps is seemingly better but still far from perfect in this respect.