11011110

Linkage

2026-08-15T06:44:00+00:00

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 rotated permutations (starting with $1, 2, 3,\dots, n,$) and for each one in order, recursively generate the permutations of its length-$(n-1)$ suffix.

The link claims that this is the first method for systematically listing permutations (long before the discovery of the Steinhaus–Johnson–Trotter algorithm for generating a different ordering, by 17th-century bell ringers). However, I’m not entirely convinced: it’s not the same as an enumeration algorithm, but factorials and lists of permutations were known long before to the Jain mathematicians, to scholars of the Talmudic period, and to medieval Arab grammarians and mathematicians (see Wikipedia on the history of the factorial). They may not have described their rules for generating lists of permutations, but they did generate those lists, presumably by following some rule. Priority arguments aside, though, I still think it’s interesting.
Complexity Class of the Week: $\mathsf{L}_2^P$ ($\mathbb{M}$). This is a complexity class characterized by polynomial-time reductions to finding the minimum element of a comparator circuit that computes a total order, or finding a violation of total ordering for the circuit. It has interesting relations both to other complexity classes (e.g. $\mathsf{NP}\subset\mathsf{P/poly}$ collapses $\mathsf{PH}$ to $\mathsf{L}_2^P$) and to circuit complexity (it requires circuits of size $\ge n^k$ for all fixed $k$).
Terry Tao on recent problematic behavior in mathematics research ($\mathbb{M}$): AI companies generate a solution to a well-known problem, take the publicity for their solution, but don’t take the effort to digest and explain the solution in a way that makes sense to human mathematicians.
Transliteration of unicode text from one script to another through its Locale Data Markup Language is Turing-complete ($\mathbb{M}$, via). The proof is via Post tag systems. The widely-used ICU library guards against this by imposing a small limit on the number of times a unit can be rewritten, so there is no denial-of-service vulnerability in practice, but this limit is not in the spec.
Leslie Lamport’s dictum “State the problem before describing the solution” ($\mathbb{M}$).
Big spike in monthly submissions to arXiv Combinatorics ($\mathbb{M}$).
Q&A from the slop trenches ($\mathbb{M}$, via). Two human machine-learning researchers on their recent experiences refereeing LLM-generated slop with bad references, and then getting their reject recommendations overridden as accept for oral presentation with an unenforceable request to fix the slop.
Asymptotically attaining the Moore bound ($\mathbb{M}$), Wouter Cames van Batenburg and Samuel Korsky, on the largest graphs with maximum degree $d$ and diameter $k$, for fixed $k$ in the limit as $d$ grows.
The mathematical beauty of hyperbezier curves ($\mathbb{M}$), on a curve family with the same control points as cubic Bézier curves but with smoother curvature variation and a larger range of parameters for which the curvature is monotonic.
The only reason you’ll ever need not to write with AI ($\mathbb{M}$): because it could expose you to charges of plagiarism.
Illinois Tech lays off roughly 160 tenured faculty members including Lance Fortnow ($\mathbb{M}$).
A stroll through the pentagram zoo ($\mathbb{M}$). Richard Evan Schwartz explains the mathematics behind some of his mathematical artworks, based on the pentagram map, a dynamical system that repeatedly replaces the vertices of a polygon by the crossing points of its diagonals.
Slides for my two talks at CCCG’26 are now online ($\mathbb{M}$):
- “Minimum-weight Steiner triangulation of convex polygons requires interior Steiner points”
- “Tangent spheres and integer distances”
Liquids can shatter ($\mathbb{M}$).

Non-coplanar unit distances

2026-08-02T18:18:00+00:00

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.

The problem asks, for $n$ points in the Euclidean plane, how many pairs can be at unit distance from each other? When Paul Erdős posed the problem in 1946, he observed that the graph of unit distances cannot contain a subgraph of the form $K_{2,3}$, a complete bipartite subgraph with two vertices on one side and three on the other. One way to see this is to draw unit circles through the two vertices on one side of a supposed $K_{2,3}$ subgraph. These cross each other at most twice, and their two crossing points are the only points that can be vertices on the other side of the subgraph. Through reasoning later generalized as the Kővári–Sós–Turán theorem, Erdős observed that this forbidden subgraph implies an $O(n^{3/2})$ upper bound on the number of unit distances. More generally, the Kővári–Sós–Turán theorem implies that when $K_{a,b}$ is forbidden from an $n$-vertex graph, the number of edges in the graph is $O(n^{2-1/\min(a,b)})$. Subsequent researchers used different reasoning to reduce Erdős’s upper bound to $O(n^{4/3})$.

Erdős also found a set of points with a slightly superlinear number of unit distances, $n^{1+\omega(1/\log\log n)}$, by scaling a square grid with $n$ points in it by a factor of $1/r$, where $r$ is chosen somewhat smaller than $\sqrt n$ (so that a unit distance fits comfortably into the scaled grid) but so that $r$ has many prime factors congruent to $1$ mod $4$. This leads to the existence of many integer right triangles with side length $r$ and many copies of the hypotenuses of these triangles in the grid. For instance the image below shows many unit-length diagonal segments in an $8\times 8$ grid scaled by $r=5$, based on a $3$–$4$–$5$ right triangle; the axis-parallel unit-length segments are not shown. Recent combinations of LLM and human effort have improved this lower bound to $\Omega(n^{1.0358})$ (unverified, as of posting), disproving Erdős’s conjecture that the upper bound should be $n^{1+o(1)}$. But there is still a lot of room between the lower bound exponent of $1.0358$ and the upper bound exponent of $4/3$.

What about higher dimensions? In three dimensions there is another gap between a lower bound exponent of $4/3$ and an upper bound exponent of $295/197\approx 1.4975$. But there it stops. Already in four dimensions one can obtain arbitrarily large sets of points with dense complete bipartite graphs of unit distances, for instance by placing $n/2$ points each on two perpendicular circles of radii $3/5$ and $4/5$ (Pythagoras again!) in the $xy$- and $zw$-planes.

Or does it stop? One obvious feature of Erdős’s scaled grid example is that it contains many collinear points. It is not obvious why this should be helpful for creating many unit distances, because each point can only have two unit-distance neighbors on each line. The new improved examples for the unit distance problem take the form of projections of higher-dimensional lattices into the plane (with carefully chosen algebraic properties generalizing the existence of many integer right triangles) and these too have many collinear points. This naturally raises the question: when points in the plane have no lines through three or more points, how does this forbidden pattern affect the number of unit distances that these restricted point sets can have? It’s still nonlinear: one can project any hypercube into the plane so that all its edges have unit distances, producing for a suitable projection non-collinear point sets with $\Omega(n\log n)$ unit distances. I don’t know of any examples with asymptotically larger numbers than the hypercube, but perhaps I’m missing some known constructions.

In $d$-dimensional space, the analogous restriction is that no $d+1$ points lie on a hyperplane. The common neighbors of any two points $p$ and $q$ do all lie on a hyperplane (the hyperplane that perpendicularly bisects segment $pq$) so this restriction causes us to forbid the subgraph $K_{2,d+1}$. (We can rephrase this idea as the statement that, for any system of unit spheres whose tangencies contain a biclique $K_{a,b}$ with $\min(a,b)\ge 2$, the centers of the spheres on each side of the biclique must be coplanar; this is analogous to a statement on bicliques of non-unit spheres with $\min(a,b)\ge 3$ from my recent paper on sphere tangencies and unit distances, but much easier.) By the Kővári–Sós–Turán theorem, forbidding the subgraph $K_{2,d+1}$ implies that, for a $d$-dimensional point set with no $d+1$ points on a hyperplane, there can be at most $O(n^{3/2})$ unit distances.

Can a matching $\Omega(n^{3/2})$ lower bound be achieved, in high enough dimensions, or alternatively, is there some limiting exponent $c<3/2$ (perhaps even $c=1$) so that point sets with no $d+1$ points on a hyperplane have $n^{c+o(1)}$ unit distances? I don’t know.

(Discuss on Mastodon)

Linkage

2026-07-31T15:15:00+00:00

ACM considers allowing LLMs to read its digital library ($\mathbb{M}$) and requests feedback on this choice.
Daedalus’ labyrinth ($\mathbb{M}$), a puzzle game dual to loopy/slitherlink.
Voronoi go ($\mathbb{M}$), a variant of go where you can play anywhere on the board (not just intersections).
Human mathematicians are being outcounterexampled ($\mathbb{M}$). Kevin Buzzard of the Xena Lean formalization project on recent counterexamples to the unit distance problem and Jacobian conjecture.
3-boundary surfaces for 3-component links ($\mathbb{M}$) with crochet, 3d-printed, and vector graphics illustrations.
Innovations in Graph Theory added to DOAJ ($\mathbb{M}$).
zbMATH digitizes 50 years of its old reviews ($\mathbb{M}$).
Three new Wikipedia articles about mathematics books ($\mathbb{M}$), posted by Samuel Adrian Antz: Characteristic Classes (1974) by John Milnor and Jim Stasheff, Instantons and Four-Manifolds (1984) by Dan Freed and Karen Uhlenbeck, and The Geometry of Four-Manifolds (1990) by Simon Donaldson and Peter Kronheimer.
AI in scientific publishing: Slower, worse, and more expensive ($\mathbb{M}$, via). H. Holden Thorp notes that the rapid increase in AI-generated research is increasing rather than decreasing the need for human effort, in checking the research, and likens the resulting effects on human workers to those in the industrial revolution’s shift from craftspeople to factory workers.
The existence of designs ($\mathbb{M}$), posted to arXiv in 2014 by Peeter Keevash, has finally been accepted for publication in the Annals of Mathematics, 12.5 years later.
Three recent algorithms preprints on arXiv make some strong and interesting claims ($\mathbb{M}$):
- “Bellman–Ford in almost-linear time”, by Hair, Li, Li, and Zhang, arXiv:2607.19346, computes single-source shortest paths in a directed graph with real weights, allowed to be negative but with no negative cycles, in time $O(m^{1+o(1)})$, building on recent subquadratic breakthroughs by overlapping authors.
- “Splay trees are almost dynamically optimal”, by Chmel, Haeupler, Hladík, Koucký, Roeyskoe, Rozhoň, Sladký, and Tarjan, arXiv:2607.18498, proves a competitive ratio of $O(\log\log n(\log\log\log n)^3)$, the first nontrivial competitiveness for splay trees. Tango trees have a slightly better proven bound but unlike splay trees cannot be $O(1)$-competitive.
- “Shortest paths with linear edge weights”, by Chillara, Gajjar, and Raja, arXiv:2607.21055, considers shortest -paths in a directed graph whose weights vary linearly as a function of one or more parameters. Different parameter settings give different paths, but how many? The answer was known to be $n^{\Theta(\log n)}$ for a single parameter, with weaker bounds for more. The new preprint proves a bound of the same form for any fixed number of parameters.
You might have thought that such old and well-established topics in algorithms research as shortest paths and balanced binary search trees would be very stable, but instead things are still changing rapidly.
Wikimedia refuses to recognize a union for its employees ($\mathbb{M}$) after a supermajority of workers signed union cards requesting it.
A long thread about variants of Kőnig’s infinity lemma in relation to computability theory.
On using computational hardness to resolve a mathematical conjecture ($\mathbb{M}$). Noam Zeilberger points to “A system of interaction and structure III: The complexity of BV and pomset logic” (Lê Thành Dũng Nguyên and Lutz Straßburger, doi:10.46298/lmcs-19(4:25)2023) where an incompatibility of complexity classes for provability in two conjecturally-equal logics led to a counterexample to their equality.
I don’t know why it took me so long to realize it ($\mathbb{M}$), but my phone’s HP 16C programmer’s calculator app turns out to be surprisingly useful for mixing hex-coded html/svg colors.

Integer complexity and cographs

2026-07-19T12:38:00+00:00

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:

\[10 = (1+1+1)(1+1+1)+1.\]

The largest number with complexity $k$ can be obtained by breaking up the sequence of $k$ ones into subsequences of two and three ones (with as many threes as possible) and multiplying. For instance, for ten ones, you can’t do this with three groups of three (because you get an ungrouped one left over) but you can with two, giving

\[(1+1+1)(1+1+1)(1+1)(1+1)=36.\]

While looking at the integer complexity article on Wikipedia today, it occurred to me that I had seen the same formula for the maximum complexity before. It is the upper bound on the number of maximal cliques in an $n$-vertex graph. This upper bound was proven in 1965 by Moon and Moser, and in fact the OEIS sequence for the largest number with complexity $k$ cites Moon & Moser but without an explanation.

It turns out there’s a stronger connection, obtained through a class of graphs called cographs. These are the graphs that can be obtained from a single-vertex graph by operations that take the disjoint union of two smaller cographs, or that complement another cograph (replacing edges by non-edges and vice versa). The resulting structure can be represented by a “cotree”, a rooted tree with its leaves labeled by vertices and its interior nodes labeled by 0 or 1, with 0 meaning to take the disjoint union of the subtree graphs and 1 meaning to complement the disjoint union. Adjacent interior nodes with the same label can be merged, giving a unique cotree representation for which the labels alternate on root-to-leaf paths.

Every maximal clique in a cograph can be obtained recursively through its cotree. At a 1-node, choose a maximal clique in each child, recursively. And at a 0-node, choose a maximal clique in exactly one child, recursively. It follows that the number of maximal cliques is obtained by reinterpreting the cotree as an expression tree, multiplying the numbers of maximal cliques at the children of a 1-node, or by summing the numbers of maximal cliques at a 0-node. Each leaf node has only one maximal clique, itself. So for instance if I take the cotree above and interpret it as an expression tree with a sum for each 0-node and a product for each 1-node I get the expression

\[(a+bc+de)(f+g)\]

which (substituting one for each variable) evaluates to six. Through this correspondence, the integer complexity of $n$ is exactly the minimum number of vertices of a cograph that has $n$ maximal cliques.

This naturally raises the question: what is the minimum number of vertices in a graph that has $n$ maximal cliques, without requiring it to be a cograph? Is it ever smaller than the integer complexity? Yes! According to OEIS, the integer complexity of 23 is 11, as obtained for instance through the expression

\[\bigl((1+1+1+1+1)(1+1)+1\bigr)(1+1)+1.\]

But there is a 10-vertex graph with 23 maximal cliques: Just remove any two edges from the complete bipartite graph $K_{5,5}$. So the largest numbers with given integer complexity can be obtained by clique-counting in arbitrary graphs, and the integer complexity can always be obtained by clique-counting in cographs, but some integer complexities are not the same as what you get by clique-counting in arbitrary graphs.

(Discuss on Mastodon)

Linkage

2026-07-15T16:46:00+00:00

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 increased publisher interference with editorial decisions (the usual), but also editor dissatisfaction with the publisher’s editorial management software.
An American privacy emergency ($\mathbb{M}$): Cynthia Dwork on how new US government regulations forbidding the Census Bureau from masking its released data under differential privacy will give us less usable data, reduced protection against privacy-violating disclosures, or both. Cynthia also provides information about what you can do to help work against this.
A reduced planar body with area greater than $\pi\Delta^2/4$ ($\mathbb{M}$), new preprint by Scott Duke Kominers. Here, “reduced” is a concept for two-dimensional convex bodies that is closely related to having constant width. The directional width is the distance between parallel support lines, constant width means that all directional widths are the same, thickness means the minimum directional width, and reduced means that any convex body that is a proper subset has smaller thickness. So bodies of constant width are reduced but not necessarily vice versa. For instance both Reuleaux triangles and equilateral triangles are reduced; the first has constant width, the second does not. A structure theorem described in the paper states that reduced bodies have parts of their boundary with constant width and parts that are flat.

Anyway, it had been conjectured that the formula in the title was the maximum area for a reduced body of thickness , with bodies attaining that area including the circular disk and quarter-disk. As evidence for the conjecture, it is true both for shapes of constant width and for polygons. But the paper describes a shape resembling a sharper wedge of a disk than a quarter, with a rounded apex, that slightly betters this area.
Ed Pegg sent me the image below ($\mathbb{M}$) illustrating some of the minimal geometric independent dominating sets (small sets of grid points with no three in line to which no additional grid point can be added) from my EuroCG’21/CGTA’23 paper with Aichholzer and Hainzl, “Geometric dominating sets – A minimum version of the no-three-in-line problem”. See also Ed’s Wolfram community post about these sets.
Geometric models by A. Harry Wheeler in the Smithsonian Institution ($\mathbb{M}$). Another set of Wheeler models that for some reason doesn’t appear in the main list: Dissected polyhedra transformable into other polyhedra. For more on Wheeler, see Wheeler’s Wikipedia biography
Shachaf observes that, in the computable reals, you can find the values of a sorted list but you cannot determine its permutation, in response to another post claiming that sorting code that doesn’t explicitly discuss permutations is “awful”. But this raises questions about constructive type theory: how can you specify that this is a sorting algorithm without having a decidable notion of equality?
Does anyone know a reference for the following easy theorem about estimating area by counting lattice points ($\mathbb{M}$), extending Nosarzewska’s inequality from convex to simply-connected regions?

Let $J$ be a region of area $a$ bounded by a Jordan curve of length $p$. Then:
\[|a - \#(\mathbb{Z}^2\cap J)| = O(p+1).\]
Proof: sweep a unit square around the boundary of $J$; by Cavalieri’s principle the area of the swept region $B$ is $\le 1+p\sqrt 2$. Consider the Voronoi cells of the integer lattice; outside of $B$ they are completely inside or completely outside $J$. Therefore,
\[\begin{align}a-1-p\sqrt 2&\le \operatorname{area}(J\setminus B)\\&\le \#(\mathbb{Z}^2\cap J)\\&\le \operatorname{area}(J\cup B)\\&\le a+1+p\sqrt 2.\end{align}\]
Emacs org-mode adds support for using ltx-talk in $\rm{\LaTeX}$ to produce accessible slides in tagged pdf format ($\mathbb{M}$).
Two-sided ruler constructions ($\mathbb{M}$). A series of blog posts by David K. Butler on how to use an unmarked ruler with two parallel edges to do almost everything that you could do with a ruler and compass.
Reformatted schedule for the International Congress of Mathematicians ($\mathbb{M}$) with MathJax abstracts that don’t require clicky popups to read, by M.-J. Dominus.
OpenAI claims a very short proof of the cycle double cover conjecture ($\mathbb{M}$).
Chris Staecker live-tweets the figures for an in-progress digital topology book, deliberately omitting any explanations of the figures.
Terry Tao on Gilbreath’s conjecture ($\mathbb{M}$). The first difference sequence of the prime numbers starts 1, 0, 2, 2, 2, 2, 2, 2, 4, … . The second, third, and fourth difference sequences all start with 1, 2, 0, 0. The first numbers in each sequence must be odd and the rest even, but which odd number? Gilbreath and before him Proth conjectured that every difference sequence begins with 1. Merely having the same small gaps and parity properties as the primes does not suffice: see my old blog post “Anti-Gilbreath sequences” for sequences with these properties whose difference sequences have infinitely many non-1 starting values.

The primes are thought to behave similarly to random sequences, so researchers have attacked the problem by studying prime-like random sequences. Past work by Chase shows that sequences whose gaps between consecutive elements are random with very slow growth (slower than the primes) almost surely have all but finitely many difference sequences beginning with 1. Now a new preprint by Chase, Tao, and Zach Hunter, using a structural characterization of anti-Gilbreath sequences related to my post, extends Chase’s work to a model of random sequences with geometrically distributed random gaps of sizes matching the prime numbers. It still doesn’t address the actual prime numbers but I think by matching their distribution better it provides strong evidence for the conjecture.
How LLM nonsense is affecting Wikipedia: increased drama and increased volunteer workload triggered by “crap articles generated by LLMs, or people using LLMs to write extremely wordy, unhelpful replies to concerns about their behaviour”.

Packing Latin squares into sudoku puzzles

2026-07-07T18:17:00+00:00

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 implies an average case time for solving randomly chosen puzzles that is exponential in $n^4/\log n$, significantly better than the exponential in $n^4$ that one gets for formulating the problem as an exact cover problem without using this bound on blank cells or the exponential in $n^4\log n$ that one gets for a brute force search.

The formatting requirements for JCDCG³ are in one way quite free-form and in another way very strict: each submission can have only two a4 pages, and the font must be at least 10pt in size, but otherwise you can do what you want. I took advantage of this freedom to experiment with the LaTeX “Cochineal” font (\usepackage[cochineal]{newtx} in LuaLaTeX or XeLaTeX), not so much because it is more compact than Computer Modern (although it is), but because I was tired of the bulbous artificial-looking letterforms of Computer Modern and wanted an old-style font that reminded me more of the appearance of handwritten manuscripts. But even with a more compact font, and a two-column format, the paper is necessarily quite telegraphic and didn’t have room for illustrations. So I thought this posting would be a good place to illustrate a construction from the paper for packing $n^2$ disjoint Latin squares into an $n^2\times n^2$ sudoku puzzle. This is not needed for the main result, but helpful for small $n$. With this construction we find a large family of $9\times 9$ sudoku puzzle solutions that, regardless of how you specify clues with those solutions, require 18 clues (more than the minimum 17 clues for some sudoku puzzles), and $16\times 16$ sudoku puzzle solutions that require 80 clues (well more than the conjectured minimum 56 clues for some puzzles).

As a reminder, an $n\times n$ Latin square fills its $n^2$ cells with the numbers from $1$ to $n$ (or any $n$ distinct things) in such a way that each row and each column has one copy of each of these numbers, almost the same as sudoku but without the number of distinct values being square and without the additional requirement that square blocks of cells contain distinct values. Our construction finds sudoku puzzles of size $n^2\times n^2$ (for an arbitrary choice of $n$) within which one can pick out $n^2$ smaller non-overlapping $n\times n$ Latin squares of cells. One can then argue that each of these smaller Latin squares needs enough clues, just in its own subset of cells, to specify it unambiguously, forcing the larger sudoku puzzle to require $n^2$ times as many clues.

The construction begins by coloring the $n^2$ digits of the sudoku puzzle so that there are $n$ digits of each color. Each of the $n\times n$ Latin squares packed into the puzzle will use digits of a single color. Within each $n\times n$ block of the sudoku puzzle, each row of the block will use digits of a single color.

Next, group the $n\times n$ blocks of the sudoku puzzle into rows of blocks ($n$ contiguous rows of cells of the puzzle). For each row of blocks, choose an $n\times n$ Latin square whose $n$ values are our colors, and use this Latin square to assign colors to the rows of cells within each $n\times n$ block.

Finally, group the $n\times n$ blocks of the sudoku puzzle into columns of blocks ($n$ contiguous rows of cells of the puzzle). Within each column of blocks, each color is assigned to an $n\times n$ (non-contiguous) subarray of puzzle cells. Choose a Latin square whose values are the digits of that color. The illustration shows what this looks like for a single subarray, the yellow one in the rightmost column of blocks.

Repeat for each color in each column of blocks, and you have a solved sudoku puzzle packed with $n^2$ Latin squares, just waiting for you to select which of its cells will be revealed as clues and which will be left blank for the puzzle solver to deduce.

(Discuss on Mastodon)

Beachhenge linkage

2026-06-30T17:13:00+00:00

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 away.
Bipartite matching is in $\mathsf{NC}$ ($\mathbb{M}$). New preprint by Abhranil Chatterjee, Sumanta Ghosh, Rohit Gurjar, Roshan Raj, and Thomas Thierauf, solving a longstanding open problem. This is apparently enough to solve the related open problem of constructing depth-first search trees in parallel, but the non-bipartite version of matching is still open.
Illustration of ranked choice voting and how it worked in Maine’s gubernatorial Democratic primary ($\mathbb{M}$).
My father had a cat named Hillary, after Sir Edmund ($\mathbb{M}$). One of mine also likes high places. The two perches in these photos can be reached from partway up, but he can jump from the floor to almost the same height, well above my head.
Interactive demo of the Finsler–Hadwiger theorem ($\mathbb{M}$), that when two squares share a vertex, a third square is formed by their centers and the midpoints of segments connecting the vertices adjacent to the shared vertex.
Peter Rowlett reviews David Singmaster’s Mathematical Gazeteer of the British Isles ($\mathbb{M}$), “a treasure trove of information about local connections to the history of mathematics in the British Isles”.
Bridges 2026 Exhibition of Mathematical Art, Craft, and Design ($\mathbb{M}$).
“Möbius” sculpture, Masami Kodama, 2010.
Springer Nature retracts two old papers by Max Planck, one because it was also published elsewhere, another because someone else published something else with the same title ($\mathbb{M}$), and then tries to sell you the blank pdf of the retracted paper for $40. They claim that the retraction can only be discussed with Planck himself (long dead, of course). Who knows how many other publications by more-obscure authors they are also removing for equally spurious reasons?
Found at Little River Beach in Mendocino County, California ($\mathbb{M}$):
Why does paper fold so well ($\mathbb{M}$)? BBC CrowdScience, 26 minute audio link. I didn’t have much luck playing it in FireFox with an adblocker, and instead had to run Safari and sit through the same two ads twice (because it still didn’t work the first time through), but maybe the direct media file download (via) will work better.
In two-dimensional general relativity, particles form cone points with angular deficit proportional to their mass, so by Descartes’ theorem on total angular defect the total mass of a spherical space is fixed, and by Alexandrov’s theorem on polyhedra any such space has the metric structure of the surface of a convex polyhedron.

A counterexample for Steiner triangulation

2026-06-27T17:52:00+00:00

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 illustration using logarithmic coordinates but I didn’t find that entirely satisfactory so I’m trying again in a different way here.

The new paper is about a problem from a much older paper of mine, “Approximating the minimum weight Steiner triangulation” (SODA 1992 and Discrete Comput. Geom. 1994), on “Steiner triangulation”, adding points to a geometric input to reduce the total edge length of a triangulation of the input. The added points are called Steiner points. The old paper proved that one could get within a constant factor of the minimum possible total edge length, for various kinds of inputs including point sets and convex polygons, using a method based on quadtrees. For convex polygons, you do sometimes need to add Steiner points; for instance the trapezoid below is optimally triangulated with one Steiner point (blue) on its long parallel side.

However, in my earlier work I could only find examples of convex polygons whose optimal Steiner points were on the polygon boundary. When that is the case, the “weak dual” of the triangulation (the graph of triangles and their adjacencies with other triangles) forms a tree. This tree structure should make finding the best triangulation with this structure easier, amenable to dynamic programming algorithms, although there are some difficulties with numerics and nonlinear functions that would need to be overcome. In the older paper, I conjectured that interior Steiner points would never be needed and therefore that we could find optimal Steiner triangulations of convex polygons in polynomial time. The result of the new paper is that the conjecture is false: we construct a polygon whose optimal triangulation requires interior Steiner points.

To understand our counterexample, I think it might be helpful to start with a counter-counterexample, showing why it’s not easy to construct counterexamples. More precisely, it’s not possible to have an optimal interior Steiner point with at most six neighbors and a convex neighborhood. If you had a triangulation like that, you could find a better triangulation with fewer Steiner points by contracting the low-degree Steiner point into its closest neighbor.

In the example above, the Steiner point is $S$ and the closest of its six neighbors is $A$. Contracting $AS$ turns the blue shaded triangulation around $S$ into the red shaded triangulation, replacing the six edges $SA$, $SB$, $SC$, $SD$, $SE$, and $SF$ by the three edges $AC$, $AD$, and $AE$. Each added edge can be shown to be shorter than a pair of removed edges by using the triangle inequality and the assumption that $A$ is the nearest neighbor to $S$:

\[AC\le AS+SC\le BS+SC,\] \[AD\le AS+SD,\] \[AE\le AS+SE\le FS+SE.\]

Combining these inequalities gives

\[AC+AD+AE\le (BS+SC)+(AS+SD)+(FS+SE)\]

so the contracted non-Steiner triangulation is shorter than the Steiner triangulation.

This argument breaks down when a Steiner point has seven or more neighbors, because then there aren’t enough removed edges to pair them up and use the triangle inequality. We found our counterexample to the conjecture by looking for neighborhoods for which, instead, contracting the Steiner point to its nearest neighbor produces a worse triangulation. The resulting convex polygon has an approximate kite shape, with exponentially-spaced sequences of vertices along the short sides of the kite (which are rounded slightly to make the result convex). It has 28 vertices, most of which cluster together near the point where the two short sides of the kite meet:

The next image is a close-up of this cluster, scaled by a factor of approximately 40, with the Steiner point (blue) and its incident edges also shown.

Most of the edges incident to the Steiner point extend rightward, so contracting the Steiner point into its nearest neighbor to the left would lengthen these edges. The same contraction would eliminate the three edges connecting the Steiner point to its three nearest neighbors, but those eliminated edges are very short relative to the number of lengthened edges. A numerical calculation shows that, in fact, the Steiner triangulation is better than the triangulation obtained by connecting all vertices to the nearest neighbor of the Steiner point. More strongly, because of the exponential spacing of the points, that turns out to be the optimal non-Steiner triangulation. So for this polygon, there is a Steiner triangulation with an interior point that is better than any non-Steiner triangulation.

That’s not quite enough to disprove the conjecture. The harder part of the paper, extending to multiple pages of appendices, is proving that there is no good Steiner triangulation using only boundary Steiner points. Therefore, the Steiner triangulation with this one interior Steiner point is better than any Steiner triangulation using only boundary Steiner points. It’s (numerically) the optimal triangulation with a single symmetrically placed Steiner point. We didn’t prove that it’s the optimal Steiner triangulation, because (however implausible it might be) there might be some other way of placing interior Steiner points that is even better. But it’s enough to disprove the conjecture, and prove that some convex polygons require interior Steiner points in their minimum-weight Steiner triangulations.

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Ceramic orthogonal polyhedra

2026-06-23T16:59:00+00:00

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:

The blue one is an orthogonal realization of Boy’s surface, an immersion of the projective plane into three-dimensional space with three-way symmetry and a single triple crossing point. The idea to make an orthogonal version comes from Jean Pierre Petit’s Le Topologicon, where its relation to the usual curved version can maybe be seen more clearly than in my photographs.

The model hides a hidden graph embedding: the uncolored edges form the boundaries of a hemi-dodecahedron, an embedding of the Petersen graph with six pentagonal faces, each adjacent to all five others.

The other ceramic model, colored yellow, green, and red, is an orthogonal realization of the permutohedron or, almost the same thing, the Cayley graph of the four-element symmetric group generated by the three swaps of consecutive elements. Abstractly, it’s the same embedded graph as the paper kirigami model behind it in the family portrait, which I constructed and wrote about in 2009, but what this one loses in orthogonal nonconvexity it makes up for in bilateral symmetry.

Here are a few more views of it:

When viewed top-down their shapes almost look like writing to me. You can see a signature and date in the hollow of the Boy’s surface.

Richter also has a couple of papers on orthogonal polyhedra: “Generic Orthotopes” (arXiv:2210.12012) and “Ehrhart Polynomials of Generic Orthotopes” (arXiv:2309.09026).

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Impossible patterns of sphere tangencies

2026-06-17T17:27:00+00:00

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 theorem, any planar graph can be represented by interior-disjoint circles in this way, but here I’m not requiring disjointness. So, for instance, you can represent any unit distance graph in the plane (such as the Petersen graph below) by expanding each vertex of the graph into a unit-diameter circle.

In three dimensions, you can produce spheres whose tangencies form arbitrarily large complete bipartite graphs $K_{n,n}$ by placing $n$ spheres interior to a torus, like a ball-bearing race or those cat toys with balls rolling around a toroidal track, and another $n$ spheres tangent to them, centered on the axis of the torus. And by a form of Descartes’ theorem, any four mutually tangent spheres (a $K_4$ graph of tangencies) can be completed to a $K_5$ in two different ways, but the two added spheres are not tangent to each other, so one cannot form a $K_6$ graph.

Given these examples, you might think that it is the high local connectivity of the complete graph $K_6$ that makes it unrealizable, and the local independence of the complete bipartite graph $K_{n,n}$ (each vertex has an independent set of neighbors) that makes it realizable. But things are more subtle than that. For every odd $k$ I can prove the existence of a graph in which each vertex has an independent set of six neighbors, the shortest odd cycle has length $k$, and there is no realization as a subgraph of external tangencies in $\mathbb{R}^3$.

This follows from the main lemma in the paper: if a system of spheres or hyperspheres in $\mathbb{R}^d$ has external tangencies that include a subgraph of the form $K_{a,b}$, with both $a$ and $b$ having at least three spheres, then the sets of spheres on each of the two sides of this bipartite graph have centers that lie on a hyperplane. This is trivial in two dimensions (no such system can exist because it would lead to a planar drawing of $K_{3,3}$) but as we have seen, even three-dimensional spheres can have arbitrarily large complete bipartite graphs. Another famous system of spheres, Soddy’s hexlet, has a $K_{3,6}$ subgraph, and spheres with the tangency graph $K_{2,2,2,2}$ have many $K_{4,4}$ subgraphs. (The image below shows Soddy’s hexlet in a form where two of the spheres have degenerated to flat planes.) The lemma applies to these configurations and proves that the subsets of spheres on each side of each bipartition have coplanar centers. The paper then uses this lemma to prove that in localization from differences of distances to landmarks (as used by GPS), any four non-coplanar landmarks suffice, and to generalize the Erdős–Anning theorem, that Euclidean point sets with integer distances are finite or collinear, to arbitrary-dimensional hyperbolic spaces.

So let’s use the same lemma to construct a graph with no short odd cycles and small independent sets of neighbors for each vertex, that cannot be realized by spheres in $\mathbb{R}^3$. To do so, blow up each vertex of a $k$-cycle by replacing it with an independent set of three vertices. Here’s a confluent drawing for $k=5$ and $d=3$:

Suppose that this graph had a realization as the external tangencies of spheres in $\mathbb{R}^3$. No two spheres in a blown-up triple of spheres can be concentric, because concentric spheres could not be tangent to the same neighbors. Each triple $T$ of spheres (no two concentric), has a unique circle (or degenerate line) $O_T$ orthogonal to $T$, the circle or line through the centers of the spheres in $T$. Any Möbius transformation of space takes circles to circles and spheres to spheres and preserves tangency and orthogonality, so the transformation of the circle $O_T$ is the circle $O_{T'}$ for the transformed triple $T'$. We can perturb the realization by a Möbius transformation so that external tangencies remain external and so that, after this perturbation, the point at infinity does not lie on any $O_T$. After this perturbation, none of the $O_T$ degenerate to lines, and therefore the centers of each triple of spheres determine a unique plane. By the lemma from the preprint, each pair of triples two steps along the cycle from each other have centers that lie on the same plane, and because the cycle has odd length, all spheres have coplanar centers. But then this realization would continue to be valid in the plane of all the sphere centers, and we already know that the many $K_{3,3}$ subgraphs of this graph cannot be realized in the plane. This contradiction means that the supposed three-dimensional realization of our blown-up cycle graph cannot exist.

In four or more dimensions, the second part of the argument still shows that the $d$-blowup of a $k$-cycle has no realization with sphere centers in each blown-up $d$-tuple of spheres in general position. But the first part, where we use a Möbius transformation to perturb everything into general position, stops working. If one blown-up $d$-tuple of spheres has cocircular centers, like the ball bearing cat toy example in 3d, its two neighboring $d$-tuples can each be placed on lines orthogonal to the circle and through its center, but in four dimensions there is a whole plane orthogonal to a circle and through its center. We can choose two different lines on this plane, and then different circles orthogonal to each of these lines, etc., creating enough flexibility as one progresses around the cycle that a realization might be possible. But my powers of four-dimensional visualization aren’t strong enough for me to be certain that this works, and it might still be possible to prove that other locally independent graphs are unrealizable.

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