Linkage

Rigid heptagon linkage (\(\mathbb{M}\)). Improvements in the size of unit distance graphs whose unit distance representation is forced to contain a regular heptagon, from 59 to 35 edges. Based on a math stackexchange question. See also an old page on the same problem from Erich Friedman’s site, moved from its old location at Stetson University.

How Graphviz thinks the USA is laid out (\(\mathbb{M}\), via).

Adding comments to your static blog with Mastodon (\(\mathbb{M}\), via). It’s a cute method for using JavaScript to pull the comments onto the blog page itself, but the method I’ve been using (just add a link from the page to the Mastodon thread and let people follow it from there to Mastodon) seems simpler.

xkcd on depth-first and breadth-first search (\(\mathbb{M}\)), just in time for my lecture reviewing these algorithms.

The many ways of splitting a rectangle — or, how to use mathematics to make using Zoom much more complicated for no particular benefit (\(\mathbb{M}\), via). I am very fond of rectangle partition problems but I don’t want to have to think about them just to talk to multiple people online.

Doubly triangular number (\(\mathbb{M}\)), new Wikipedia article on triangular numbers with a triangular-number index. These numbers come up in counting pairs of pairs of things, for instance in the illustration, which describes colorings of the corners of a square (up to symmetry) as pairs of colorings of pairs of opposite corners.

How assigning impossible-to-solve problems can get mathematics students (or researchers) to take a step back and look at the bigger picture (\(\mathbb{M}\)). Patrick Honner in Quanta. I think it needs interactivity, though. It would be mean to do this on a problem set or exam.

Mathematical Art Exhibition from the January 2020 Joint Mathematical Meetings (\(\mathbb{M}\), via).

My latest puzzle (a belated Christmas present to myself) is the Hanayama Twist (\(\mathbb{M}\)). I’m very pleased with it: an elegant symmetric two-piece design, solid feel in the hand, and a solution that is surprisingly complicated but not tediously long. My only complaint is that the solution is very linear, with almost no ways to go wrong if you keep moving on from things you’ve already done. Anyway here’s a map I drew to help me, also used as an example of an implicit graph in my graph algorithms lectures.

In pre-covid lectures I hand-wrote notes as I talked, both to keep content fresh and to avoid racing through it too quickly. When we locked down I tried pre-recorded lectures with prepared slides, but it took too long to record and I missed the spontaneity. This term I’m back to live zoom lectures with prepared slides, and it seems to work: zoom chat keeps it spontaneous and I haven’t found myself going faster than when we were in person. Lesson learned, just in time to (I hope) unlock in fall. (\(\mathbb{M}\); see discussion for Pat Morin’s home whiteboard cat interruption.)

Two useful folds and a decorative one (\(\mathbb{M}\)). Paula Beardall Krieg. The useful ones seal a foil snack bag and wrap a sandwich securely in paper, without needing any fasteners.

Thoughts on the Pythagorean theorem (\(\mathbb{M}\)). Or, what did Euclid actually mean by saying that two squares are equal to a third square? And how does this view relate to type-theoretic foundations? From the xena automatic theorem-proving project. Both the name and the horrifying illustrations for the blog posts come from the author’s daughter. For some heavier going, see the post on perfectoid spaces and sphere eversion.

Regular star patterns for expanded US flags (\(\mathbb{M}\), via). I suspect that the likelihood of quickly adding Puerto Rico or the District of Columbia as states is not high, but just in case, there are some nice regular point arrangements available.

The figure below shows two pentagons inscribed in each other: each vertex of one pentagon lies on a line through a side of the other pentagon (\(\mathbb{M}\)). You can’t do this with quadrilaterals in the Euclidean plane, but you can in the complex projective plane. From a thread of fun math facts for Christian Lawson-Perfect’s birthday.

New preprint claims a proof of the Erdős-Faber-Lovász conjecture for all sufficiently large \(k\) (\(\mathbb{M}\), via), by Dong Yeap Kang, Tom Kelly, Daniela Kühn, Abhishek Methuku, and Deryk Osthus. The conjecture states that the union of \(k\) edge-disjoint \(k\)-cliques is \(k\)-colorable. See its Wikipedia article for a real-world-ish formulation with faculty committees as cliques and chairs in a common meeting room as colors.