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Linkage

Jun 30, 2025

  • Robin Houston dissects the outer shell of a \(4\times 4\times 4\) polycube into seven interlocked pieces, picking up from a 2023 discussion where I found a six-piece solution. With bonus short video by George Miller showing that, with round enough cubes, it makes a nice snap-together puzzle.

  • There is some nice street art clustered in Santa Cruz’s Westside (\(\mathbb{M}\)). This one, “The Urchin Merchant”, is by Lauren Ys, part of PangeaSeed Foundation’s Sea Walls: Artists for Oceans project. From the project site you can find many more pieces of project art scattered around the world.

    "The Urchin Merchant", Street art in Westside Santa Cruz, California by Lauren Ys, 2021, part of PangeaSeed Foundation's Sea Walls: Artists for Oceans project. A mural decorates a keystone-shaped building, set against a dark blue sky. At center, a young woman with asian features representing an anthropomorphized starfish wears long flowing sleeves and a headdress decorated with sea anemones, sea urchins, starfish, and kelp. She holds a sea urchin, from which a bite appears to have been removed; a sea otter dances in an abalone shell in front of her. She is flanked by two floating mermaids with sharp teeth and fingernails, representing, left, an anthropomorphized sea urchin, and right, an anthropomorphized leopard shark. The urchin mermaid floats above a large sea urchin, has bright blue skin and braided hair, and wears red lipstick and punk clothing and chains with a spiky helmet. Safety pinned to her chest is a small poster reading "eat me". She is eating kelp with chopsticks, and holds another kelp stalk in her other hand. The leopard shark mermaid has mottled purple skin, large eyes at the ends of stalks on the side of her face, pinned-up dark blue hair, and lipstick matching her hair; she wears a fishnet top, a spiked bracelet on her right wrist, and a starfish on her right bicep. The background is pink, decorated with light purple silhouettes of sea urchins and kelp. A white van parked at lower right obscures the leopard shark mermaid's lower torso.

  • Your Brain on ChatGPT: Accumulation of Cognitive Debt when Using an AI Assistant for Essay Writing Task (\(\mathbb{M}\)), new preprint by Nataliya Kosmyna, Eugene Hauptmann, Ye Tong Yuan, Jessica Situ, Xian-Hao Liao, Ashly Vivian Beresnitzky, Iris Braunstein, and Pattie Maes. From the abstract: “EEG revealed significant differences in brain connectivity … LLM users displayed the weakest connectivity. … Self-reported ownership of essays was the lowest in the LLM group … LLM users also struggled to accurately quote their own work. … LLM users consistently underperformed at neural, linguistic, and behavioral levels. These results raise concerns about the long-term educational implications of LLM reliance and underscore the need for deeper inquiry into AI’s role in learning.”

  • Mystery of the quincunx’s missing quincunx (\(\mathbb{M}\)). After Mark-Jason Dominus asked whether there was ever actually a Roman five-uncia coin that had five spots in the pattern of the five-spot face of a die, @HydraPrever found one, a quincunx from Luceria ca 225–217.

  • Proceedings of the 41st International Symposium on Computational Geometry (\(\mathbb{M}\); open access).

  • @xgranade distinguishes valid uses of en-dashes from their use as an AI telltale.

  • Homework machine. Shel Silverstein on LLM-generated answers, from 1981.

  • Christian Lawson-Perfect on creating the Herschel Enneahedron (\(\mathbb{M}\)), on Numberphile.

  • The skewed, anamorphic sculptures and engineered illusions of Jonty Hurwitz (\(\mathbb{M}\)).

  • Yann Le Gall experiments with animating edge bundling, ending up with abstract animations that start dense and angular and slowly get simpler and more curvilinear.

  • Computer-vision research powers surveillance technology (\(\mathbb{M}\)), Kalluri, Agnew, Cheng, et al., in Nature suggest that this application does not come from “a few rogue entities” but rather that the whole research community is complicit, for instance in using “obfuscating language that allows documents to avoid direct mention of targeting humans”.

  • Protecting science: TIB builds dark archive for arXiv (\(\mathbb{M}\)).

  • Monostable tetrahedron (\(\mathbb{M}\)). Colin Wright still has the weighted bamboo-frame tetrahedron with only one stable side, constructed with Bob Dawson in the 1980s and described as lost in Dawson’s recent preprint “Building a monostable tetrahedron” with Gergő Almádi and Gábor Domokos. See also Quanta on the new preprint.

  • Alhazen’s problem (\(\mathbb{M}\)), another new Good Article on Wikipedia. The problem is: when you see an object reflected in a cylindrical or spherical mirror, where on the mirror does it appear? This was already notoriously hard by the late middle ages. It can be solved (after reducing to a circular mirror cross-section) by intersecting a hyperbola with the mirror, or with a quartic equation, but not by compass and straightedge.

    Huygens' solution of Alhazen's problem.by intersection with a hyperbola. Two blue points outside a circular mirror (black) are connected to each other by the blue path of a reflected light ray from one to the other. A red rectangular hyperbola crosses the mirror at the reflection point. Several points and lines used to construct the hyperbola are depicted in yellow: two rays from the center of the circle through the two blue points, the angle bisector of these rays and its perpendicular line through the center of the circle, two yellow points on the rays formed by circle inversion of the blue points through the mirror, the midpoint of these two inverse points, and two more yellow lines through the midpoint, parallel to the angle bisector and its perpendicular. The red hyperbola is asymptotic to these two lines through the midpoint and passes through the two inverse points.

    You know those curved patterns of light (caustics) that you see in the bottom of coffee cups outside in the sun? They are also very relevant for this problem. If you had a point light source inside your cup instead of far away, the caustic would separate points with two reflected images of the light source (on the dark side of the caustic) from points with four (on the bright side). With the sun outside the cup, you will instead get one or three reflection paths off the side of your cup onto its bottom (some of which may be blocked by the opposite side of the cup), but the underlying principle is the same.

    A minor technical problem I ran into when creating the hyperbola illustration above (source code): the svg format does not have a direct representation for hyperbolae so instead you have to approximate them with splines. But although calculating points on a hyperbola is very easy, choosing control points with splines is not. I ended up adding a polycurve routine to my Python SVG library to smooth a curve through given points automatically.

  • My talk slides from the annual Symposium on Computational Geometry in Kanazawa, Japan, where I presented my work on integer distances on non-Euclidean surfaces (\(\mathbb{M}\)). Next year the conference will be at Rutgers University in New Jersey (I know, US locations are problematic, but it is already too late to change). Two years from now it will be in Bangalore, the first time it has been in south Asia.

  • David Eppstein

Geometry, graphs, algorithms, and more