11011110

Linkage

2026-04-30T18:19:00+00:00

Retraction Watch reports on the mass resignation of the editorial board of Elsevier’s Journal of Approximation Theory ($\mathbb{M}$). This follows in close succession from the mass resignation from Taylor & Francis’s Communications in Algebra.
On Max Bill’s gelbes feld ($\mathbb{M}$), Barry Cipra, Notices of the AMS. Analysis of this artwork reveals that it depicts a magic square, with dice-like dot patterns encoding its digits, and with equal numbers of dots in each row or column of dots. You can do this with every magic square, but the diagonals are more problematic.
One week ahead of its announced deadline for major institutions to make all online content meet WCAG 2.1 A/AA accessibility standards, the US government kicked the can down the road instead, extending the deadline to April 26, 2027 ($\mathbb{M}$). Although I was more or less on top of getting my 1600 pages of old university-hosted html content accessible, I also have a couple hundred old pdf files (for instance of papers and talk slides) that are difficult to convert, and are fortunately grandfathered by the requirements. Nevertheless I would like to make them as accessible as possible, eventually. I have found that it is often possible, if tedious, to convert old pdf files to tagged and alt-textified pdf within Acrobat.

However, I hit a roadblock with some old pdf files, consisting purely of vector graphic artworks with no text. The accessibility checkers all suspect that these are secretly “image-only pdfs”, scans of text that need OCR to make them accessible to non-sighted readers. They are not scans. They are not written in any language. They are purely vector graphics. It does not work to add tags labeling them as figures, to add alt text to the figure tags, nor to set the document language to “None”: the accessibility checkers are still convinced that there must be secret hidden text somewhere in all that line art and complain that I haven’t told them what that supposed text says. Does anyone know how to tag or otherwise annote these files with the information that they contain no text in a way that will make the accessibility checkers shut up about them?
The Centrality Fallacy and ACM ($\mathbb{M}$). Moshe Vardi in CACM protests ACM’s claims that they are making their Digital Library open-access while at the same time paywalling all of its metadata features. His post links to a petition to reopen the metadata.
Peter Brass, Former NSF Theory Director, on the NSF budget ($\mathbb{M}$).
Does anyone know what was the significance of the stella octangula to André Breton ($\mathbb{M}$)? In case anyone near Paris wants a mathematically-themed excursion, one of these shapes ornaments Breton’s tomb in Batignolles.
There’s no “age verification”, there is only “identity verification that includes age”, and the system doing that verification is not just a privacy-invasive user tracking system but a remotely controlled off switch for anyone of any age.
A knot invariant that can be computed in polynomial time, separates more knots than many other invariants, likely gives a genus bound, and produces fun “QR code” images representing each knot ($\mathbb{M}$, via Quanta). A recent arXiv preprint by Dror Bar-Natan and Roland van der Veen.
On coloring Penrose rhomb tilings. A recursive substitution system can simultaneously generate a Penrose tiling and color its tiles with four colors. But a graph degeneracy argument and the De Bruijn–Erdős theorem on coloring infinite graphs together imply that every rhombus tiling, and in particular the Penrose tiling, has a 3-coloring. Can it be generated by a substitution system?
Trump fires entire 24-member National Science Board ($\mathbb{M}$, via). This board oversees the National Science Foundation’s funding of US science and advises the government on science policy, and the move is “widely seen as [Trump’s] latest move to erase NSF’s independence”. The National Science Foundation has also been lacking a director for the past year. Trump has proposed to cut the budget of the NSF by another 55% for the coming year (at the same time as saddling it with expensive white elephant projects).
Book of New Patterns by Hokusai, 1824 ($\mathbb{M}$, alt. link, via). The via-linked discussion led me to the story of the 1986 rediscovery of the wood blocks for printing this book in the Boston Museum of Fine Arts. The blocks were then brought back to Japan and new prints made from them.
The number of ISO A(n) sheets that fit orthogonally into an A(0) sheet ($\mathbb{M}$) might be the least satisfying sequence in the whole OEIS. It should just be powers of two, but no: rounding sets in.
Arizona State University professors disturbed to find their lectures chopped up and turned into AI slop ($\mathbb{M}$), by a new platform introduced by Arizona State using content scraped from its courses’ Canvas sites, without warning the faculty or offering any opt-out.
The product of two zero-dimensional schemes can be infinite-dimensional. The example involves $\mathbb{R}\otimes_{\mathbb{Q}}\mathbb{R}$, the same space in which Dehn invariants of polyhedra live.

Cubic salespeople revisited

2026-04-28T19:09:00+00:00

Good news: Email from Knuth. Bad news: It begins “is there a bug in your paper?” The paper in question is “The traveling salesman problem for cubic graphs” (WADS 2003 and JGAA 2007). Of course the bug report is accurate, but fortunately it can be easily patched.

My paper has algorithms for finding a minimum-weight Hamiltonian cycle (when it exists), in an $n$-vertex graph of maximum degree three, in time $O(2^{n/3})$, and for listing all Hamiltonian cycles in time $O(2^{3n/8})$. Both have since been improved by others. Maciej Liśkiewicz and Martin R. Schuster gave an algorithm for the minimum-weight cycle with time $O(1.2553^n)$ in their paper “A new upper bound for the traveling salesman problem in cubic graphs” [J. Discrete Algorithms 2014], and Heidi Gebauer gave an algorithm for listing all cycles in time $O(1.276^n)$ in her paper “Enumerating all Hamilton cycles and bounding the number of Hamilton cycles in 3-regular graphs” [Elect. J. Combinatorics 2011]. Despite these improvements, I think it’s interesting enough to examine what the issue was with my paper and how to work around it.

The bug is in the case analysis for the algorithm to list all Hamiltonian cycles. The minimum-weight algorithm includes a case where a triangle is contracted into a single vertex. However, for some reason the cycle-listing algorithm omitted that case, and then later tried to assume there were no triangles, but without a valid justification for this assumption. It’s not impossible that the rest of the analysis goes through without using the assumption of no triangles, but working out the details looks messy.

Both algorithms in the paper work with a more general problem in which, together with an input graph, one is given “forced edges” which are required to be included in any Hamiltonian cycle. This allows certain simplifications in which, for instance, paths through degree-two vertices or paths of multiple forced edges can be compressed into single forced edges. It’s important when listing cycles that the input is a simple graph, not a multigraph, because multigraphs can have as many as $O(2^{n/2})$ Hamiltonian cycles. So, when this sort of compression produces a forced edge that’s parallel to an unforced edge, the unforced edge can be deleted, and when it produces two parallel forced edges, there can be no Hamiltonian cycle and the algorithm backtracks. After these simplifications, one can assume that the graph is simple and 3-regular rather than merely having maximum degree three, and that no two forced edges are adjacent. But it might still have some triangles.

With that as background, it is indeed possible to eliminate the triangles. Here are the missing cases to do so.

If a triangle includes a forced edge, then the opposite edge entering the triangle should also become forced. (The paper included a case with the opposite implication, but this direction is needed to preserve the forcing information after the contraction of a triangle, below.) So in the image below, if either of the two thick edges is forced, the other one should be too.

If there exists a triangle that does not share any of its edges with another triangle, contract it. Hamiltonian cycles in the contracted graph correspond one-for-one with Hamiltonian cycles in the uncontracted graph, and (with no shared edges) this contraction will not create any parallel edges.

If there exist two triangles that share an edge, then there are two paths through their four vertices that might extend to Hamiltonian cycles. If only one of these two paths is consistent with the forced edges within this subgraph, add the rest of its edges to the forced set; their contraction will eliminate the triangles. Otherwise, make two recursive branches, each with one of the two paths added to the forced set. Each branch eliminates four previously-unforced edges, consistent with the analysis of the other cases in this part of the paper.

As far as I know, another problem from the same paper remains open. The paper constructs a family of cubic graphs that have $2^{n/3}$ Hamiltonian cycles, but the cycle-listing algorithm proves only that the number of cycles is always $O(2^{3n/8})$, a bigger number. Gebauer’s paper improves the upper bound, but it still does not match the lower bound. Is $2^{n/3}$ the maximum number of Hamiltonian cycles in an $n$-vertex 3-regular graph?

(Discuss on Mastodon)

Linkage

2026-04-15T17:17:00+00:00

Congratulations to Ken Clarkson and the rest of the newly-named SIAM Fellows ($\mathbb{M}$)!
Richard Elwes on Numberphile talking about big numbers ($\mathbb{M}$). The twist: these ones are from many centuries ago, from Jain mathematicians. They got at least as far as tetration.
How wireless origami cushioning could improve logistics ($\mathbb{M}$), by providing shipping damage alerts in real time.
The long-predicted cleveref apocalypse is upon me ($\mathbb{M}$). For the most part it works to just replace
```
\usepackage[capitalize,nameinlink]{cleveref}
```
by
```
\usepackage{zref-clever}
\zcsetup{cap}
\let\cref\zcref
```
I have been using an “observation” theorem type, and for this I need to tell zref-clever what to call them:
```
\zcRefTypeSetup{observation}{Name-sg=Observation,Name-pl=Observations}
```
If you want to have different theorem types share the same numbers, things get more complicated. I haven’t needed that yet, but there’s guidance in Section 10.2 of the zref-clever user manual.
After a long-ish quiet period, quite a few new solutions to the no-three-in-line problem have been discovered this year ($\mathbb{M}$).
A medieval woman teaches geometry to a group of students ($\mathbb{M}$). From an early 14th-century manuscript of Euclid’s Elements. Sadly, most online sources make the sexist inference that a woman couldn’t possibly have been a geometry teacher then, and therefore that she must instead be a “personification of geometry”.
Deterministic primality testing for limited bit width ($\mathbb{M}$). Precomputing a small set of good bases gives a fast derandomization of the Miller–Rabin primality test.
The Music of the Spheres ($\mathbb{M}$), sequence from the Saturday Morning Breakfast Cereal comic strip, coauthored by Terry Tao, on what mathematicians do and the unreasonable effectiveness of mathematics, as illustrated through the sphere packing problem.
Alison Martin folds a single sheet of paper into an umbilic torus (see also).
Cathedral building geometry basics: how to find the center of a circle with compass and straightedge ($\mathbb{M}$). It might not come as a surprise that people who repair stonework on old cathedrals with handheld chisels and hand-poured molten lead (the rest of the content on this YouTube channel) might also prefer to make their architectural drawings and templates by hand instead of with a CAD system.
Tariq asks: What do people recommend for publication quality plots of mathematical functions?
Many sequences in OEIS come from various real-world applications. Jeremy Kun looks for the opposite: open-source code that has applied OEIS sequences in unusual ways ($\mathbb{M}$). These include music synthesis, pen tool size settings, and geocaching.
Through a new Wikipedia article I learn that Uruguay has a museum devoted to origami ($\mathbb{M}$), claimed to be the only one in the Americas.
Cats on a lintel ($\mathbb{M}$):

H-trees are not trees

2026-04-01T21:09:00+00:00

A sheet of a4 paper has aspect ratio $1:\sqrt2$. This means you can crease it along the shorter of its two midlines to get two rectangles with the same aspect ratio (of a5 size) and continue in the same way recursively.

Now, whenever you subdivide one rectangle into two smaller rectangles in this recursion, connect the centerpoints of the two new rectangles by a line segment. The union of all of these line segments is a fractal, the H-tree.

It comes arbitrarily close to any point of the initial rectangle, so its closure is the whole rectangle and its fractal dimension is two. But the points that belong to the H-tree (normalized with one corner of the rectangle at the origin) all have at least one coordinate that is a dyadic rational multiple of one of the rectangle sides. So as a subset of a countable union of axis-parallel lines, the H-tree has measure zero.

At each finite level of recursive construction (using closed line segments) the union of the segments is a kind of topological tree called a dendroid. Roughly this means that its a connected compact set with a unique arc connecting each two points. The full H-tree is not compact (it is not closed: its closure is a different set, the whole rectangle). But this is not the only way that it is not tree-like: it no longer has the property of having a unique arc connecting each two points! This is because it has certain simple closed curves as subsets, within which you can connect pairs of points in two different ways around the curve.

In this it differs from some other fractal space-filling trees like the one below, which might be called an X-tree. An X-tree can be formed by recursively subdividing a square into four smaller squares, adding an open line segment for the diagonal of each square. Or, instead, you could form an X-tree the way my code drew it: start with a grid of a power of two size, draw the diagonals of the whole grid, and then remove (or redraw as white) the grid points whose coordinates have binary representations ending in differing numbers of $0$ bits. You can find arcs of the X-tree whose two ends get arbitrarily close to each other, but they do not form simple closed curves because their limit point is missing. Instead, in the H-tree, a branch of the tree can have an interior point of another segment as its limit point, so the limit point is not missing.

Looking again at the H-tree, one can see additional tree-like structures in the negative space where the H-tree isn’t.

It’s common to see planar surfaces divided up by pairs of interdigitating trees: the two trees of a river network and of the ridgelines separating the river’s branches are a familiar example.

I’ve used this idea of interdigitating trees in several of my research papers and in several old posts here. The complement of a tree would have a tree-like skeleton (for instance the X-tree has a $45^\circ$-rotated X-tree-like dual skeleton). But because the H-tree is not a tree, its interdigitated forest has many separate fractal trees. These trees are not exactly the same as the connected components of the complement of the H-tree: like the H-tree itself their points have one coordinate that is a dyadic rational multiple of the initial rectangle side, so they miss many points of the outer rectangle. But I think they have the same closures as the connected components.

(Discuss on Mastodon)

Linkage

2026-03-31T18:22:00+00:00

Firefox testing reveals as many as 5% of crashes are caused by hardware memory faults.
Useless advice about making technical online content accessible, from my university’s head bureaucrat for online content accessibility bureaucracy ($\mathbb{M}$): “I would recommend avoiding PDFs altogether. They are extremely difficult to make accessible. If PDF format is absolutely necessary, create a document in Microsoft Office and then save it as a PDF.”

Meanwhile, TeX Live 2026 + ltx-talk has been working for me in making pdf-format slide decks that pass all Acrobat accessibility checks, and I have moved on from experiments in using it to using it in production for all my course lecture slides. There is no real alternative to some form of TeX for content involving mathematics. This works. And if you’re already using LaTeX + beamer, it’s not difficult.
Knuth’s packing puzzle ($\mathbb{M}$), a variant of Hoffman’s packing puzzle with 28 rectangular cuboids of carefully chosen shapes that should fit into a cube. Like Hoffman’s, this is difficult.
White faceted aluminum flows through branching curved surfaces to form the pillars and dome of an architectural folly by Marc Fornes ($\mathbb{M}$).
Wikipedia prohibits the use of LLMs to generate or rewrite article content ($\mathbb{M}$).
Folded cyanotype art by Fritz Horstman ($\mathbb{M}$). More description.
Sumerian clock ($\mathbb{M}$).
Goran Konjevod’s curved and pleated origami vases ($\mathbb{M}$).
Two years ago in connection with SAT-solver optimization of cascading stylesheet files I briefly mentioned the possibility that CSS might be Turing-complete, with a link to some attempts at demonstrating this via simulation of the Rule 110 cellular automaton ($\mathbb{M}$). These attempts were unsatisfactory for a couple of reasons: Rule 110’s completeness requires an infinite array of cells and a mostly-repeating pattern of initial cell values, the demonstrations had only finite arrays of cells of fixed size implemented as html objects, and each step of the simulation required some user interaction. Since then Clement Cherlin has found a better solution: a CSS Turing machine simulator whose only interaction requirement is that you move the mouse to a starting position within 5 seconds of opening the page. It still appears to use html elements as tape cells, so the tape has a predetermined size, but having a fixed and finite tape is less of a problem for Turing machines than for Rule 110. You can still do arbitrary computations for which you already know how much tape you’re going to need (which I guess can be described as Turing completeness). But determining whether the computation terminates is not an undecidable problem, because with a fixed tape size the total number of machine–tape states is finite.
Women talking about their experience in mathematics ($\mathbb{M}$).
Unexpected cutbacks in international student visa approvals by the Canadian government (far beyond their projected cutbacks) lead to program cuts and faculty layoffs at Canadian universities ($\mathbb{M}$, via). According to an auditor report, the Canadian immigration department “did not know why approval rates were lower than projected”. The story focuses on BC but it appears that the effects are nationwide.
A report on the huge amount of bot traffic putting strain on Wikipedia’s infrastructure, and the steps taken to counter it ($\mathbb{M}$).
ICML’26 had two groups of reviewers one for which LLM use was forbidden and another for which it was not. Reviewers could state their preference and only those who claimed to be ok with forbidding LLM use were assigned to that group. But then, ~500 of the reviewers in the LLM-forbidden group were caught using LLMs through hidden watermarks in the papers they reviewed ($\mathbb{M}$, via). As a consequence, ~400 of those reviewers had their submissions desk-rejected (~500 rejections).
On the perspective of Manet’s Bar at the Folies-Bergère ($\mathbb{M}$). The face-on view of a waitress blankly staring across an empty bar while her reflection interacts with a customer seems like an impossible piece of artistic license, but Malcolm Park made a photographic reconstruction without even an in-shot camera reflection.
Maybe everyone else already knew this geometric food trick ($\mathbb{M}$), but it took me years to discover it: subdivide your artichoke hearts into quarters or sixths before removing the choke, not after. That way, the nearly convex shape of the pieces, compared to the bowl shape of a whole heart, makes it much easier to remove the choke with a straight cut of a sharp knife.
An improved algorithmic 4-color theorem ($\mathbb{M}$). Previous proofs could only produce a single reducible subconfiguration at a time; with linear time to find and handle each subconfiguration, they led to algorithms for 4-coloring planar graphs with $O(n^2)$ total time. This new preprint claims to find a linear number of disjoint reducible subconfigurations, leading to an $O(n\log n)$ algorithm.

Linkage with plum blossoms

2026-03-15T18:47:00+00:00

Remembering Joe Halpern ($\mathbb{M}$) focuses on Joe’s pivotal role in founding and guiding the CS section of the arXiv.
A regular pentagon has five symmetry axes through one corner and its center point ($\mathbb{M}$). Its five diagonals cross to form a smaller nested pentagon. Kevin Grace has called these ten lines (symmetry axes and diagonals) and eleven points (nested pentagon corners and center) the “Betsy Ross configuration” because of the five-point stars on the US flag. Its construction necessarily involves the square root of five, because the diagonals of a regular pentagon are longer than its sides by a factor of the golden ratio, $(1+\sqrt5)/2$. It is “projectively rigid”: every ten lines and eleven points with the same pattern of point-line incidences comes from a projective transformation of the regular pentagon. Therefore, in any other drawing of points and lines in this pattern, $\sqrt5$ still appears, in the cross ratio of distances among four collinear points. Points with rational numbers as coordinates would have rational cross-ratios, so the Betsy Ross configuration cannot be drawn with rational coordinates.

If you remove from this configuration one symmetry axis and the two pentagon corners that it passes through, the remaining nine points and nine lines form the Perles configuration. It is again projectively rigid and is the smallest system of points and lines that requires irrational coordinates. It was used by Micha Perles to construct 8-dimensional convex polytopes that also require irrational coordinates; other applications involve counting point-line incidences in points with forbidden configurations, the complexity of recognizing visibility graphs of point sets, and proving irrationality for certain graph drawing problems.

Now a Good Article on Wikipedia.
Chemistry in pictures: Droplet origami ($\mathbb{M}$). Lowering and then raising the temperature of a spherical droplet of hexadecane causes it to take the shape of an icosahedron, a flat hexagon, and a six-pointed star.
MacTeX TeX Live 2026 now available ($\mathbb{M}$). You probably need this if you use Macs and are working on generating tagged pdf from LaTeX for accessibility. You might want to avoid this if you rely heavily on cleveref, which is broken in recent TeX Live releases.
Chalkdust Book of the Year 2025 shortlist ($\mathbb{M}$).
Indonesian government blocks Wikipedia editors from logging in over lack of official registration of the site with the government ($\mathbb{M}$).
It’s that time of the year when the plum blossoms in the alley behind my office catch the late afternoon sunlight ($\mathbb{M}$).
Using Voronoi diagrams and minimum cuts to fill in the gaps in a partial outline of the British coastline
Counting tetromino tilings of $2\times n$ rectangles ($\mathbb{M}$). The answer turns out to be a squared Fibonacci number!
In search of falsehood ($\mathbb{M}$). Tristan Stérin leverages LLMs to search for soundness bugs in the kernels of the Rocq and Lean proof assistants, that would allow these kernels to verify a proof of falsehood.
Het Ding ($\mathbb{M}$, see also), a piece of 50-year-old guerilla artwork on the campus of Twente University in the form of an enormous 6-bar tensegrity structure.
3d-printable truncated octahedron 72-pencil pencilholder ($\mathbb{M}$), for all your hexastix pencil structure construction needs.
Semi-automated visualization of finite state transducers for aperiodic tilings constructed by clustering a larger graph, using dotty to draw the condensed graph of the clusters, expanding the clusters, and tweaking the positions manually. The resulting layout is noticeably nicer than the one found by dotty from the unclustered graph.
Lightning Calculator Quincunx WD-7 ($\mathbb{M}$). Video by Chris Staecker on Galton’s bean machine, in a version used to demo statistical concepts in industrial engineering, and its connection to the birth of eugenics.

Making accessible LaTeX talk slides with ltx-talk

2026-03-01T14:52:00+00:00

US-based academics are being required by the government and by their universities to make all online content accessible by April 24, and I think many of us have been running around trying to figure out what that means and how to do it. My university has been especially unhelpful, demanding compliance in vague terms but not telling us what standard they’re using and pointing only to Microsoft and Google’s web sites for guidance on how to improve accessibility for Microsoft and Google products. The relevant standard appears from the government links to be WCAG 2.1 AA. For those of us who have been using the LaTeX beamer package to create mathematical course lecture notes as pdf files, beamer cannot do this. The accessibility standards require tagged pdf and beamer does not have code to generate the tags correctly. But there is a solution: a replacement package, ltx-talk, that is mostly compatible with beamer. After some effort, I have succeeded in using ltx-talk to create slide decks that closely resemble my old beamer decks both in appearance and coding and that pass all automated accessibility checks in Acrobat. That makes now a good time to record what I have learned about going from beamer to accessible ltx-talk, before I forget it all again.

Online guides

I have found three helpful guides to accessible LaTeX (not specifically about ltx-talk): “Using LaTeX to produce accessible PDF” from the LaTeX tagging project, “A quick primer on modifying existing LaTeX for digital accessibility” by Richard Wong at Rice University, and “Creating accessible PDFs in LaTeX” from Overleaf. If you are using some unusual LaTeX packages or document classes, the LaTeX tagging project also maintains a useful list of the tagging status of LaTeX packages and classes. This includes the information that beamer cannot generate accessible tagged pdf but ltx-talk can. These links got me from a state of having no idea how to generate accessible pdf files from my slides to a state where I thought it might be possible, and helped me start setting up my files. Most of the rest was from carefully reading log files and accessibility reports and then experimenting to figure out what to change in order to make the errors and warnings go away.

This posting is not intended as a substitute for these guides, but rather as a collection of tips and tricks for converting beamer slide decks into accessible ltx-talk slide decks.

Compilation

To compile ltx-talk files and produce accessible tagged pdf files, you need to run lualatex instead of pdflatex. Sometimes you need as many as four runs: one run of lualatex to get an aux file with bibliography items, a run of bibtex, and then three more runs of lualatex before it stabilizes and stops telling you to run again because the labels have changed or because it miscalculated page numbers and added a dummy page. Check the log files for these things.

You also need to be running a very recent version of LaTeX, dated November 2025 or more recent. TeX Live 2025, released in March 2025, will not work. The new TeX Live 2026 pretest will. I use MacOS and have not figured out whether there is any way to access the TeX Live pretest on a Mac. Instead I have been using either Overleaf (through its Overleaf Labs feature) or a linux install on some departmental machines accessible to me via ssh. This already includes the ltx-talk package; you do not need to install it separately.

There are apparently incompatibilities between recent releases of LaTeX and the cleveref package. Fortunately, my slide decks do not use cleveref.

Preamble

The magic incantation at the start of your file is different, of course, because it’s a different package but also because tagged pdf needs a metadata command before the document class to set it up. The old incantation (for, say, a $16\times 9$ target aspect ratio) was:

\documentclass[aspectratio=169]{beamer}

Instead, now it is:

\DocumentMetadata{
  lang          = en,
  pdfstandard   = {ua-2,a-4f},
  tagging       = on,
  tagging-setup = {math/setup=mathml-SE} 
}
\documentclass[aspect-ratio=16:9]{ltx-talk}

Change the lang = en line if you are writing in a different language than English. Wong suggests instead tagging-setup={math/setup=mathml-SE,math/alt/use}; I have no idea whether that would be an improvement.

The other important change in the LaTeX preamble works around what I think is a bug in ltx-talk. By default, it tags frame titles with an h4-level header tag. But there are no h1, h2, or h3-level header tags that it produces. This causes Acrobat’s accessibility checker to complain about header nesting. Maybe the right thing to do is to add some h1, h2, and h3-level tags, for instance on the title page, but I haven’t figured out how to do that. Instead, I changed the frame titles to h1, with

\tagpdfsetup{
  role / new-tag = frametitle / H1
}

The default appearance produced by ltx-talk is close to beamer with the structure skin, but not quite the same. You can change it in your LaTeX preamble but the documentation for how to change it is somewhat lacking. What worked for me is to look for code like DeclareTemplateInterface in ltx-talk.cls and to use \EditInstance to change it. For instance I use the code

\definecolor{ksblue}{RGB}{0,129,205}
\EditInstance{header}{std}{
  color            = ksblue,
  left-hspace      = 0cm plus 1fil,
  right-hspace     = 0cm plus 1fil
}

to change the color of frame titles and center them. I also use

\renewcommand{\labelitemi}{ {\footnotesize\color{ksblue}$\blacktriangleright$}}
\renewcommand{\labelitemii}{ {\scriptsize\color{ksblue}$\blacktriangleright$}}

to make big triangular itemize bullets like beamer, instead of the default little circular ones. (So far I have only converted slides with two levels of itemize nesting.)

Figures

In the article content, the easiest change to explain but the most time-consuming one, for me, is adding alt-text to all images. The syntax is straightforward: \includegraphics[alt={Alt text goes here}]{filename}. I have seen big online debates on what alt-text should describe and how detailed it should be. The important thing to remember is that you’re not trying to describe the image in vivid-enough detail that an AI image generator could make a copy of it. The purpose of these things is to substitute for the image when people use a screenreader to convert your slides into spoken text. So it should be concise enough that it doesn’t interrupt the flow of the text when spoken but informative enough that people using a screenreader don’t miss out on the meaning. For complicated mathematical examples that can be a big ask.

I had one figure, created as a pdf file by Adobe Illustrator, that triggered a flate decode error in lualatex. The same image had worked correctly in pdflatex and beamer. The compilation continued with a warning but the figure did not render correctly in the compiled file. I could not figure out why. The only workaround I could find was to save it as a different pdf version in Illustrator.

Environments

Getting ltx-talk to run required a few changes elsewhere in my LaTeX files. I was using \begin{frame}{Title of frame}, and ltx-talk has an option to make that work, but by default it doesn’t. Instead you need to use \begin{frame}\frametitle{Title of frame}. Using unicode characters such as an en-dash in a frametitle leads to a “unreferenced destination” error; code them in ASCII (in this case a double hyphen) instead.

Columns need to be delimited by \begin{column}{size}…\end{column} instead of starting them by \column. (Also the spacing between columns is different between beamer and ltx-talk so you may need some reformatting to get them to look good.) And the same thing about using an environment rather than a command applies to some formatting things like \flushright: it won’t cause a LaTeX error but it will cause the tagging to get mismatched with a warning message at the end of the log. Then you have to work through your file trying to figure out which slide caused the mismatch.

If you use verbatim environments, beamer needed to mark this by \begin{frame}[fragile]. This still works in ltx-talk but with a different syntax, \begin{frame*}. If you use tabular environments, you need to tell LaTeX which rows of your table are header rows. See the accessibility guides linked above.

Formulas

There is lua code somewhere that generates mathml tags for the mathematical formulas (I think this is the main reason that you need to use lualatex). It is not as robust as LaTeX itself. Using code like \bigl and \bigr will cause a tagging mismatch; use \left and \right instead and let LaTeX choose for itself how big to make the parentheses. Using \dots inside math often does not work at all, and in one case using $\dots$ inside a tabular environment caused lualatex to crash hard. Use \ldots or \cdots instead. Also, I think using mathematics inside \text inside more mathematics caused a tagging mismatch; don’t nest them like that. This code will also generate files with names of the form *-luamml-mathml.html; add that pattern to your .gitignore file.

I had one deck (discussing the Ackermann function) containing the expression $\approx 2\times 10^{19728}$. This caused lualatex to freeze. The only hypothesis I can find is that it looks like a formula whose numerical value can be calculated and that the mathml conversion code was trying to calculate it, using a slow exponentiation algorithm.

Sections

If your deck has 21 or more slides, Acrobat will complain if it doesn’t also have bookmarks for navigation within sections of the deck. I did this using \pdfbookmark[1]{Bookmark name}{Bookmark name} at the start of each section. Maybe there is a better way. This would be a good place to put more higher-level header tags than the H4 frame titles, if I knew how.

Many of my slide decks have bibtex bibliography sections. I don’t generally show these when speaking but I want them to be part of the pdf files of my slide decks that I distribute. I like to use natbib for this (plus the doi package to make the external links and dois work properly). But natbib never worked correctly in beamer; I needed to add the code \newcommand{\newblock}{} to the preamble to make it work. In ltx-talk, I also needed to add \newcommand{\thebibliography}{\relax} before loading natbib. Then you can just put your bibliography within one of the frames. A problem I have not found a good workaround for is that beamer allows multi-frame bibliographies with \begin{frame}[allowframebreaks]. However, ltx-talk does not and reading its documentation reveals that its developer is very hostile to long bibliographies (such as would arise in a talk incorporating a literature review). The only workaround I have found is to use a small-enough font size (in some cases \tiny) to allow the whole bibliography to fit on one slide. This is mildly problematic with respect to accessibility but better than truncating the bibliography because it overflows the slide.

(Discuss on Mastodon)

Linkage

2026-02-28T18:07:00+00:00

Joe Halpern (1953–2026) ($\mathbb{M}$). A leader in the mathematical reasoning about knowledge, founder of the Computing Research Repository (later the CS branch of arXiv), and recipient of the Gödel Prize and Dijkstra Prize.
A single-line drawing that is simultaneously an open knight’s tour of a 99x99 chessboard and a 3x3 Latin square.
The circle packing theorem ($\mathbb{M}$): every planar graph can be represented by the tangencies of a system of non-overlapping circles. This theorem was proved by Koebe in 1936, and popularized in the 1980s by Fields medalist William Thurston as a discrete analogue to conformal mapping and uniformization. Its Wikipedia article was created by Oded Schramm in 2008, not long before his untimely mountaineering death. In his own research, Schramm found deep analogies between random walks on circle packings and Brownian motion. My interests in circle packing relate to its use in drawing graphs, constructing polyhedra for given graphs, modeling soap bubble foams, and finding planar separators. And others have found even more varied applications from the study of discrete symmetry groups of hyperbolic space to methods for visualizing the functional areas of the human brain, spread out into a flattened map. Now a Good Article on Wikipedia.
Discussion involving Terry Tao on security vulnerabilities in the Lean theorem prover ($\mathbb{M}$). Importing an untrusted module can result in running arbitrary code.
Two new papers on near-linear shortest path search in dense graphs with negative edge weights ($\mathbb{M}$). I don’t know the story here, but when parallel papers claiming the same strong result come out simultaneously on arXiv, it’s usually not a coincidence. The papers are:
- “An $n^{2+o(1)}$ time algorithm for single-source negative weight shortest paths”, Sanjeev Khanna & Junkai Song, arXiv:2602.16638
- “Bellman-Ford in almost-linear time for dense graphs”, George Z. Li, Jason Li, & Junkai Zhang, arXiv:2602.16153
I recently posted about archive.today (also archive.is, archive.ph, archive.fo, archive.li, archive.md, and archive.vn) using its archive links to launch a ddos attack against a blogger they accused of doxing them ($\mathbb{M}$). That attack triggered Wikipedia (at least, the English part) to discuss banning archive.today links, and the ensuing discussion turned up evidence that (as part of the same dispute with the same blogger) archive.today had also tampered with its archived content to falsify certain names in old archived links. This led to a quick close of the discussion and a consensus to remove all archive.today links from Wikipedia. For the same reasons I have removed all archive.today links from my blog, where I had been occasionally using them as a convenient way to access paywalled content. I suggest that others remove their links as well, lest you unwittingly become part of additional ddos attacks and falsification.
National Institute of Standards and Technology appears to be squeezing out “foreign-born researchers” ($\mathbb{M}$). The language used here is especially concerning: we should not be hobbling our research institutions by limiting their researchers to being US citizens, but requiring US birth goes far beyond even that.
Subdivisions of a triangle into smaller similar triangles lead to new substitution tilings of the plane based on the plastic and superplastic constants ($\mathbb{M}$). Ed Pegg, Wolfram Insights.
One-line 3d-printable Schönhardt polyhedron ($\mathbb{M}$).
“Google just keeps getting dumber”. The link goes to a screenshot of a Google Books page for the book Topics in Topology by Stevo Todorčević, displaying the plot summary “a thirteen-year-old with a talent for throwing loops and who lives on a ranch with his father and grandfather yearns for a roping horse”.
According to the Bonnet theorem ($\mathbb{M}$), describing the surface distances and principal curvatures of a smooth 2d surface is enough to determine a local embedding of the surface (an immersion) into 3d. A related result by H. Blaine Lawson and Renato de Azevedo Tribuzy shows that using mean curvature instead of the principle curvatures is almost enough: for a smooth compact surface and non-constant mean curvature, there can be at most two immersions. The recent paper “Compact Bonnet pairs: isometric tori with the same curvatures” (Bobenko, Hoffmann & Sageman-Furnas, Pub. Math. de l’HÉS 2025) shows that the case of two immersions can actually happen: there are pairs of immersed tori in 3d with different shapes in 3d but the same surface distances and mean curvatures. Recently described in Quanta: “two twisty shapes resolve a centuries-old topology puzzle”.
Flip distance of triangulations of convex polygons / rotation distance of binary trees is NP-complete ($\mathbb{M}$). A new arXiv preprint by Joseph Dorfer answers a well-known problem that was implicit in the STOC 1986 work of Sleator, Tarjan, and Thurston on the extreme values of flip distance / rotation distance and already explicit by 1988 in the (incorrect) claim of a polynomial time algorithm by Křivánek (see Theorem 7).
Josh Millard investigates the number of tilings of a square by 1x2 rectangles of varying sizes.

Linkage

2026-02-15T11:52:00+00:00

Morphing between a polyhedron and its dual using 3d-printed scissor-link edges ($\mathbb{M}$) raises the question of which polyhedra can be realized in this way. The linked paper by Liao, Kiper, and Krishnan suggests that they need to be midscribed and have equal-length edges to avoid binding during the morph. A much weaker but obvious necessary condition is that for each pair of dual edges the lengths are equal. But it is unclear which polyhedra have realizations with this property.
Frank Vigor Morley’s realizations of regular polygons using knotted paper strips ($\mathbb{M}$). Morley was the son of Frank Morley, a geometer known for Morley’s theorem on triangle trisectors.
Type 8 pentagonal cobblestone tiling in Kleve, Germany.
The Brouwer fixed-point theorem in action, as exhibited by Github labels ($\mathbb{M}$, via): If you try to define a continuous function from text colors to contrasting background colors, there will be some text colors whose background is the same as the text rather than contrasting with it. Discontinuity is necessary to avoid this. This is from 2023 so I suspect the specific buggy behavior is long fixed but the phenomenon will recur for any attempt like this one to define a formula using only continuous building blocks.
Complexity in Computer Science ($\mathbb{M}$), free online textbook, about to be published through Cambridge University Press.
Archive.today / archive.is / archive.ph etc initiates a DDOS attack against a blogger they accused of doxing them, sparking calls to blacklist them from reference link archiving on Wikipedia ($\mathbb{M}$).
The Lambek–Moser theorem ($\mathbb{M}$) is a bijective equivalence between two different-looking mathematical objects: partitions of the positive integers into two disjoint subsets, and pairs of almost-inverse monotone functions from positive integers to non-negative integers.

Real monotone functions are inverse when their graphs are mirror reflections across the diagonal line $y=x$. We can define something like a graph for an integer function, a staircase curve whose lowest point above any integer on the $x$-axis gives the function value; then two integer functions are almost-inverse when these curves are mirror reflections. The image below shows these two reflected staircase curves for the prime-counting function and its almost-inverse. This pair of functions corresponds to the partition of positive integers into prime and non-prime (composite or one).

By using this equivalence to go from functions to partitions and back, you can sometimes get amazing formulas for sequences of integers that you might not expect to have a formula at all. For instance, the $n$th number that is not a $k$th power (for integer $k>1$) has the formula:
\[n+\left\lfloor\sqrt[k]{n + \lfloor\sqrt[k]{n}\rfloor}\right\rfloor.\]
Now a Good Article on Wikipedia.
Lisa Dusseault crafts a cellular automaton lace scarf ($\mathbb{M}$).
When science discourages correction: How publishers profit from mistakes ($\mathbb{M}$). Douglas Sheil and Erik Meijaard argue that the profit motive disincentivizes publishers from maintaining the integrity of the scientific record.
As part of the Integrated Explicit Analytic Number Theory network, Terry Tao finds a need to revive tables of logarithms, with machine-verified proofs of their accuracy ($\mathbb{M}$).
The Music of the Primes, Corinthia Beatrix Aberlé ($\mathbb{M}$). On connections between musical tuning systems and the Riemann zeta function.
CCC vs GCC ($\mathbb{M}$). An LLM-generated compiler passes all unit tests but somehow manages to generate code that runs complex SQLite queries 158,000 times slower than GCC.
I have been experimenting with ltx-talk as a replacement for beamer as a way to use LaTeX to make lecture slides ($\mathbb{M}$), in preparation for the April 24 doomsday in which all online content of all US universities is required to meet WCAG 2.1 A/AA accessibility standards (including properly tagged pdf). My test slide deck is based on beamer slides on union-find and its analysis from my graduate data structures course. It sort of works but is still obviously a work in progress.

Minor issues:
- This requires lualatex 2025-11-01 or more recent. I can run this in Overleaf Labs but cannot find a recent-enough MacTeX.
- The LaTeX preamble is different and the formatting changes it makes were largely undocumented.
- I needed a hack to make natbib work. Actually two hacks but one was already there for beamer.
- I had to change \begin{frame}{Title} to \begin{frame}\frametitle{Title} everywhere.
- I had to change \column{size} to \begin{column}{size} ... \end{column} everywhere.
- The tabular environment needs to be tagged for whether it is really a table (and which of its rows are headers) or just for formatting; see the LaTeX tagging project’s accessible formatting tips.
Major issues:
- Compiling $\approx 2\times 10^{19728}$ ($\approx 2\times 10^{19728}$) caused lualatex to get into an apparent infinite loop and time out. Is it trying to calculate $10^{19728}$??
- Some expressions with $\dots$ produce empty output. This would be a minor issue but having this empty output in the cells of a tabular environment caused lualatex to crash with the code “error: (nodes): fuzzy token cleanup in whatsit node with type whatsit and subtype 29”. Using $\ldots$ or $\cdots$ as appropriate worked.
- beamer has an option \begin{frame}[allowframebreaks] that paginates overlong frames, necessary for bibtex bibliographies. It does not work in ltx-talk and I cannot find a replacement for this functionality. The test deck uses a tiny font for the bibliography but this is not very satisfactory and will not scale to larger bibliographies.
It’s good to see there are still people online crazy enough to think that emulating a GameBoy using arbitrary precision arithmetic implemented purely through compass and straightedge constructions is a good idea ($\mathbb{M}$).
The cleveref apocalypse is on us ($\mathbb{M}$). The cleveref LaTeX package is long-unmaintained and breaks on recent LaTeX versions, and an arXiv update to TeXlive means that we can no longer keep limping along using old-enough versions of TeX to avoid the problem. I haven’t yet tried it but my bookmarked solution is to switch to zref-clever.

Linkage with bulging mussels

2026-01-31T18:15:00+00:00

When I upgraded to MathJax 4.0 last October, I failed to notice that its mathematics expression explorer prevented the $\mathbb{M}$-links to my Mastodon posts from working. Mouseover would show the url for the link (which is why I didn’t notice) but if you clicked on it you couldn’t get anywhere unless you knew to disable the speech and Braille options in the explorer first. I’ve updated the site to disable those options by default so that the links work. You can still re-enable speech and Braille, in the context menu on any mathematical expression, under Accessibility → Speech → Generate and Accessibility → Braille → Generate (the same options you previously needed to disable to get links to work).

Anyway, on to the links:

175 years of 3d viewing failure, starting with the 1851 Brewer Stereoscope, in honor of the big mid-January layoffs and cuts at Meta’s Reality Labs and Metaverse divisions.
The joys of data archaeology ($\mathbb{M}$): I happened to be looking up one of my old papers, “On triangulating three-dimensional polygons” (with Gill Barequet and Matt Dickerson, SoCG 1996 and CGTA 1998) when I realized that, although both published versions are open-access, the journal version has figures that have somehow been made completely illegible (all the grayscale is blacked out) and the conference version’s figures weren’t well reproduced either. My ftp links for alternative copies were long dead (nobody uses nor provides ftp any more). I do still have LaTeX source for the final version, but I couldn’t find all the figure files (probably buried as attachments in an associated unix-mail archive). And when I tried to compile it, even without having found all of the figures, the idiosyncratic LaTeX formatting from however long ago caused the text to keep switching mid-paragraph between two very different font sizes. Fortunately I was able to recover a preprint of the full journal version from one of those academic-web paper scrapers. The text is a little spindly because of the old formatting, but the figures are better. I put it back online and linked it from my publications page.

The main result of the paper, by the way, is that it is NP-complete to test whether a 3d polygon has a non-self-intersecting triangulation. Knotted polygons obviously don’t, but the polygons from the NP-completeness reduction are unknotted. Unknotted curves always form the boundary of a disk, but triangulating this disk might require many internal vertices, and the triangulations studied in this paper are allowed to use only the given vertices.
Purdue rescinds departmental graduate acceptance letters to over 100 students ($\mathbb{M}$, via), mostly from China, under an unwritten university policy formulated in response to Trump administration pressure. A faculty quote: “when we’ve asked to get it in writing, [administrators] say they haven’t done it because then somebody could sue us.” As usual don’t read comments on the Y.
Nice profile of Robert Lang from 2024 by NASA ($\mathbb{M}$), explaining how his methods for designing the folding patterns for his artistic origami came out of methods he had been applying to pack components into integrated optical circuitry, and then later how he brought the same expertise back to NASA in work for them on packing components into space missions.
Warm winter weather and a holiday weekend made for a busy day at our local beach ($\mathbb{M}$). A few more photos.
A surprising fact about polyhedral self-duality ($\mathbb{M}$): There exist self-dual polyhedra for which the duality is not a self-inverse permutation of the set of vertices and faces. Robin Houston makes a 3d printable model.
An old Mastodon thread about triangular billiards and a new Wikipedia article about triangular billiards. The question is: if you made a billiards table in an arbitrary triangular shape, would it always be possible to find a starting position and direction for a billiards ball to travel so that (with ideal reflection and no friction) it would return to the same position and direction after finitely many bounces? It’s solved for triangles whose angles are acute or rational multiples of $\pi$, but open in general.
Why do so many authors of enjoyable books end up being so problematic ($\mathbb{M}$)? J. K. Rowling, transphobe. Neil Gaiman, sexual predator. And now I’m saddened to learn that R. A. Lafferty was a Holocaust denier. (Via a Wikipedia discussion where another editor, who does believe the accusations against Lafferty, was falsely accused of white nationalism for enforcing Wikipedia’s strict sourcing standards in this case. The source above does not meet those standards, but it is good enough for me personally.)
The 2026 Michael and Sheila Held Prize goes to Irit Dinur, Subhash Khot, Guy Kindler, Dor Minzer and Muli Safra for their work on the 2-to-2 games theorem ($\mathbb{M}$).
Temporary outage of full-text search in Google Books ($\mathbb{M}$, via), only for books with preview rather than full view or snippet view. A few days later it started working again, as mysteriously as it stopped.
Calls for the IMU to publicly address the situation of the US hosting the International Congress of Mathematicians this year. Meanwhile the French Mathematical Society has announced that they will not be participating.

As I wrote in response: This is a big issue for academic conference planning in general, and not just for the ICM. My European friends are telling me that nobody they know there is currently willing to travel to the US and that for some of them there are institutional barriers in place (including issues of the security of institution-owned electronics) that prevent them from doing so. I have also talked to Canadians who have stopped attending conferences they would otherwise travel to in the US. Meanwhile there are also many people within the US who are unable to travel elsewhere for fear of not being allowed to return (which is not a strong argument for hosting international conferences in the US, but does raise problems for conference organizers elsewhere who might normally expect strong US attendance).
LICS breaks with ACM and IEEE, becoming an independently-run conference.
Domes made from rings of trapezoids topped by a pyramid of isosceles triangles have been seen in architecture since the construction of the Pantheon, and studied in geometry since della Francesca’s 15th-century De quinque corporibus regularibus. But what are they called? One possible answer: trapezium domes.
Generative AI and Wikipedia editing: What we learned in 2025 ($\mathbb{M}$, via), from the Wiki Education project, which mainly interfaces Wikipedia with academic course projects involving editing Wikipedia. Their takeaway message: “Wikipedia editors should never copy and paste the output from generative AI chatbots like ChatGPT into Wikipedia articles.”

In more detail, they used Pangram to detect AI-written content, then checked it by hand (finding very few false positives!). They found only a low rate of hallucinated sources. However, more concerningly, in more than 2/3 of the flagged articles, they found sourced sentences whose sources did not contain the information in the sentence. More strongly, “For most of the articles Pangram flagged as written by GenAI, nearly every cited sentence in the article failed verification.” This also led to a high rate of wasted WikiEdu staff time cleaning up after the student-made AI additions (“far more time attempting to verify facts in AI-generated articles than if we’d simply done the research and writing ourselves”).
The staircase paradox gets linked from BoingBoing ($\mathbb{M}$).