Linkage with plum blossoms
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Remembering Joe Halpern (\(\mathbb{M}\)) focuses on Joe’s pivotal role in founding and guiding the CS section of the arXiv.
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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.
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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.
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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.
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Indonesian government blocks Wikipedia editors from logging in over lack of official registration of the site with the government (\(\mathbb{M}\)).
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It’s that time of the year when the plum blossoms in the alley behind my office catch the late afternoon sunlight (\(\mathbb{M}\)).
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Counting tetromino tilings of \(2\times n\) rectangles (\(\mathbb{M}\)). The answer turns out to be a squared Fibonacci number!
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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.
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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.
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3d-printable truncated octahedron 72-pencil pencilholder (\(\mathbb{M}\)), for all your hexastix pencil structure construction needs.
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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.
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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.