Linkage

Bridges between descriptive set theory and the theory of distributed computing through coloring geometric graphs (\(\mathbb{M}\)), Quanta, based on Anton Bernshteyn’s “Distributed algorithms, the Lovász Local Lemma, and descriptive combinatorics”, Inventiones Math. 2023.

Icosahedral dice from Ptolemaic Egypt in the Met. These may be among the ones Knuth discussed in his 2016 Christmas lecture Hamiltonian paths in antiquity noting that in many such dice the alphabetical order of faces describes a Hamiltonian path.

Dante’s cosmology as a 3-sphere, the three-dimensional topological space analogous to the two-dimensional surfaces of spheres. For Dante, the universe was arranged as a series of concentric spheres, with the spheres shrinking inwards from the earth’s surface to Satan at the bottom level of Hell, but then shrinking outwards again to God at the center point of the heavens. That’s a 3-sphere!

Conway’s 99-graph problem and the Paley 9-graph (\(\mathbb{M}\)). The Paley 9-graph has the property that each two vertices belong to either a unique triangle or a unique 4-cycle (but not both). There are only six possibilities for the numbers of vertices in such a graph, for which only three are known to work: 3, 9, and 243. The 99-graph problem is the first unsolved case.

Basic accounting is just graph theory (\(\mathbb{M}\), via).

Texas A&M University bans philosophy professor from covering Plato (\(\mathbb{M}\), via), deeming his Symposium as containing forbidden “race ideology and gender ideology”.

Translating machine learning into smoothed particle hydrodynamics. Denoising autoencoders, that convolve point samples with a Gaussian in order to be able to define and calculate gradients of density, use the same trick as particle-based simulation of physical systems for which the particles used in the simulation are far coarser than the ones being simulated.

Venezuelan news reports that the Mathematical Center of the Venezuelan Institute for Scientific Research was destroyed by the recent US invasion (\(\mathbb{M}\)). In Spanish.

A relation symbol not in Unicode: the kite, from a part of Ziegler’s lecture notes on convex polytopes involving circle packing.

Illustration by Augustin Hirschvogel, from Geometria (1543) (\(\mathbb{M}\)). The illustration shows a tetrahedron and octahedron, their inscription within spheres, and their unfolded nets. This is the publication that gave us the word “net” for an unfolded polyhedron.

Paper Folding for the Mathematics Class, Donovan A. Johnson, NCTM 1957, and Mathematics through Paper Folding, Alton T. Olson, NCTM 1976 (\(\mathbb{M}\)). Taken from a discussion of a school homework problem asking students to visualize how a net for a cube folds up, about which my feeling is that as a way of encouraging students to play with paper and folding, it’s a fine problem, but if they’re expected to do it only through mental visualization it’s very unfair.

Packings of circles with relative radii 1, 2, and 3. As well as fitting inside a bigger circle in various ways, these also form infinite packings of the plane that can be periodic or that can have a limited amount of non-periodicity.

JMM 2026 Mathematical Art Exhibition (\(\mathbb{M}\)). from the Joint Mathematics Meetings last week in Washington DC. They selected three works for awards: “Square Root of Two” by David Reimann, an image of a branching structure playing with the ideas of a tree root, the diagonal of a square, and the decimal representation of \(\sqrt2\); “A Knitted, Braided Table of Knots” by Katherine Vance, representing braid diagrams of knots by actual braided textiles; and “Hexastix Knot 4,24”, a glass structure by Anduriel Widmark.

On convex 4-polytopes that cannot be realized with rational coordinates (see also; I’ve posted both Mastodon and Bluesky links because the conversations there continued disjointly). In his 1996 book Realization Spaces of Polytopes, Jürgen Richter-Gebert constructed an example with 33 vertices. Moritz Firsching has enumerated 274148 combinatorial types of 9-vertex 4-polytopes and found rational coordinates for all of them, so any irrational 4-polytope needs at least 10 vertices. There is still a big gap between 10 and 33, though.