A weak ordering (\(\mathbb{M}\)) is just a linear order with ties. On \(n\) items, there are obviously at least \(n!\) of them (don’t use ties); here’s a simple combinatorial proof that there are at most \((n+1)^{n-1}\), also provable using parking functions (below). In the complete graph \(K_{n+1}\), choose one vertex as root. By Cayley’s formula, it has exactly \((n+1)^{n-1}\) spanning trees. Each spanning tree can be used to weak order the non-root vertices by their distance from the root. Each weak order on non-root vertices arises from a rooted spanning tree in this way by setting the p ..read more

Michael Mitzenmacher is making an unusual request for publicity for his NON-participation in a conference (\(\mathbb{M}\)). It calls itself ICAIM, the International Conference on Artificial Intelligence and Mathematics, to be held in Nanjing in September, and it falsely lists Mitzenmacher as a conference chair, Mike Goodrich as program chair, and Jelani Nelson as a program committee member, among others. None of Michael, Mike, and Jelani have agreed to allow their names to be used in this way. The conference contact email goes to a known spam/phishing site. It appears to be a scam. Be aware ..read more

Last weekend I made a quick visit to my mother in Mendocino for a memorial service for my father, Tony Eppstein. He died last August, I think while I was on a plane to Tokyo, but I had seen him only days before and it was expected. My father (and my mother) both came from New Zealand, where they met at the University of Canterbury; they moved from there to England (where I was born) and then, when I was very young, to California. I think there’s much more to read about this time on my mother’s blog. He worked as an engineer on magnetic digital storage (first tape, and later disks). A guest at ..read more

As always, you can see these earlier and more spread out by following my Mastodon account. These linkage posts are just a redundant copy of what I post and boost there to make it easier for later me (and maybe others) to find the same content without having to scroll through miles of other posts to reach it. The “\(\mathbb{M}\)” links go to the Mastodon posts, where there may be more discussion. Wackiest use of SAT solvers I’ve seen so far (but I haven’t looked very hard): compressing CSS files by determining which selectors can safely be combined (\(\mathbb{M}\), via). It’s not clear to me ..read more

In the graduate version of my just-concluded graph algorithms course, one of the exam questions asked the students to find the faces of a topological embedding of the complete bipartite graph \(K_{2,3}\) (the embedding in which the two degree-three vertices have the same clockwise order of neighbors as each other), compute its Euler characteristic, and determine from the result whether the embedding is planar. Some of the students found it confusing (it was intended to be the hardest problem on the exam) so I thought I’d post an answer here. It turns out that there is a single 12-sided face. I ..read more

Recent edits to the Wikipedia article on toroidal polyhedra led me to a 1997 geometry.research discussion thread, “Polyhedra of positive genus”, in which John Conway describes a toroidal polyhedron with 36 equilateral-triangle sides, and suggests that this might be the fewest sides possible for a toroidal deltahedron. Conway’s description is clear enough: start with a central regular octahedron (eventually to become the hole in the torus). Find a cycle of six triangles separating two opposite faces of the central octahedron, and glue three octahedra and three tetrahedra in alternation around t ..read more

Robin Houston wonders about cuboid terminology. The specific question is whether it should mean a shape with six rectangular sides (as commonly taught in school) or a shape with six quadrilateral sides (as used in some research communities). Let’s not even speak about Branko Grünbaum’s use of it to mean a shape formed by gluing together a power-of-two number of six-quad-side shapes. Tiny period-15 glider gun and period-16 glider gun in Conway’s Game of Life (\(\mathbb{M}\)). Surprisingly good art exhibit on AI-generated and AI-manipulated photography (\(\mathbb{M}\)), at the California M ..read more

I have another preprint on the arXiv today with my student Hadi Khodabandeh: “Maintaining light spanners via minimal updates”, arXiv:2403.03290. Like my other work with Hadi, this is on geometric spanners: sparse graphs on a given set of points in a metric space, whose shortest path distances accurately approximate the metric distances between the points. There are a lot of detailed properties spanners could be expected to have, and a lot of existing constructions that achieve some (but not all) of those properties: The accuracy with which they approximate distances can be adjusted, to be wi ..read more

The mysterious math of billiards tables (\(\mathbb{M}\)), Dave Richeson in Quanta. Ph’nglui mglw’nafh Cthulhu R’lyeh wgah’nagl fhtagn! A figure caption from Erin Wolf Chambers’ doctoral dissertation gets cited in WikiQuote. The figure it describes could be of two Great Old Ones osculating. Non-adaptive Bellman-Ford: Yen’s improvement is optimal (\(\mathbb{M}\)), Jialu Hu and László Kozma. This new preprintimproves my paper from last year showing that a naive version of the Bellman–Ford shortest path algorithm that relaxes a predetermined sequence of edges without adapting to the results ..read more

Zig-Zag paths and neusis constructions of a heptagon and a nonagon (\(\mathbb{M}\)), new article by Dave Richeson and Dan Lawson. annie rauwerda tells the world why Wikipedia and Wikipedia editing is great in a letter to the editor in Harper’s. Reverse mathematics (\(\mathbb{M}\)). New entry in the Stanford Encyclopedia of Philosophy. CAT(0) Cube Complexes (\(\mathbb{M}\)). New book by Petra Schwer. I’m on the program committee for two recently announced calls for papers (\(\mathbb{M}\)): CCCG 2024, the 36th Canadian Conference on Computational Geometry, July 17–19 at Brock Univers ..read more