Geometry, graphs, algorithms, and more
A permutation generation algorithm in the work of 13th-century Kabbalist Abraham Abulafia (\(\mathbb{M}\), via). The resulting permutation sequence is the one you get by reversing suffixes whose lengths form the sequence \((((2, 3)^2, 2, 4)^3, 2, 5)^4, \dots\) but that’s not the generation rule. Instead the rule is: to generate the permutations of \(1, 2, 3,\dots, n,\) form its \(n\) cyclically…
Reports that LLMs have killed the Erdős unit distance problem turn out to be greatly exaggerated. There is still plenty not yet understood about the problem.
ACM considers allowing LLMs to read its digital library (\(\mathbb{M}\)) and requests feedback on this choice.
The integer complexity of a number \(n\) is the minimum number of ones needed to express \(n\) as a parenthesized combination of sums and products of ones. For instance, 10 has complexity 7 as it can be expressed using seven ones, but not fewer:
Another mathematics journal leaving its commercial publisher (\(\mathbb{M}\)), but with a twist: usually this is accomplished by a mass resignation of the editorial board. But in this case, Communications on Pure and Applied Mathematics is owned by the Courant Institute and was published by Wiley, so taking it in-house is just a matter of not renewing the contract. The causes of friction were…
I have another new preprint, the result of a research project with UC Irvine undergraduate Cindy Zhang: “Sudoku grids that require many clues” (arXiv:2607.05728, to appear at JCDCG3 2026). The main result is, I think, surprising: When generalized to \(n^2\times n^2\) grids, almost all sudoku puzzles must be almost entirely covered by clues, leaving only a logarithmic fraction of cells blank. This…
On the scale of stupid things the US government is doing this is pretty small, but they appear to be banning the census bureau from any effective methods of privacy-preserving information release, and in particular from adding noise to their data to help create differential privacy (\(\mathbb{M}\), via). Sadly, taking away valuable disclosure avoidance tools doesn’t make fundamental trade-offs go…
I’ve been hesitating in writing up a blog post about my latest preprint, “Minimum-weight Steiner triangulation of convex polygons requires interior Steiner points” (with my student Zahra Hadizadeh, arXiv:2606.25302, to appear at CCCG) because, despite being a completely concrete two-dimensional construction, its exponential scale makes it difficult to visualize. In the paper we included an…
David Richter, a mathematician at Western Michigan University, recently found himself with a surfeit of ceramic orthogonal polyhedra and, knowing of my own interest in orthogonal polyhedra, generously offloaded two of them to me. They fit nicely in my office together with the paper and crochet orthogonal polyhedra I already had:
My latest preprint, “Tangent spheres and integer distances” (arXiv:2606.18569, to appear at CCCG), involves the patterns of external tangencies of circles, spheres or higher-dimensional hyperspheres. You can make a graph whose vertices are a given set of spheres and whose edges are pairs of externally-tangent spheres, and I’d like to understand which graphs are possible. By the circle packing…