Linkage

Lithophanes, a 19th-century art medium involving backlit translucent engravings, revived via 3d printing as a single format for scientific images that blind people can read by feeling and sighted people can see (\(\mathbb{M}\), original research paper).
A quick “not a Reuleaux triangle” link: Lego triangles, incorrectly embellished as Lego Reuleaux triangles (\(\mathbb{M}\)). The second link even goes on to say “I don’t think [they] are quite convex enough to be proper Reuleaux triangles”. And in this it’s correct, if “convex” is interpreted to mean “bulgy”. You can tell because the corners are right angles. Proper Reuleaux triangles would have \(120^\circ\) corners.
As part of a search for real-world applications of combinatorial design theory, Jeremy Kun asks: “What is the authoritative text on combinatorial designs, anyway?” I suspect that the shakeup in the area caused by Peter Keevash’s use of the probabilistic method has caused what used to be the authoritative texts to become obsolete.
Squaring the circle illusion (\(\mathbb{M}\)). Henry Segerman 3d-prints a shape that, when rotated \(180^\circ\), changes appearance from a circle to a square. With the same area in the viewing frame, even.
Finding a vertex-to-vertex and edge-to-edge mapping (“homomorphism”) from an input directed graph to a fixed oriented tree or cycle can be either polynomial-time or \(\mathsf{NP}\)-complete, depending on the target. But the targets for which it is \(\mathsf{NP}\)-complete are surprisingly complicated! (\(\mathbb{M}\)) A recent search found the smallest hard tree to have \(20\) vertices, and the smallest hard cycle to have \(26\).
Tamal Dey and Yusu Wang have a new graduate-level textbook on computational topology (\(\mathbb{M}\)): Computational Topology for Data Analysis (Cambridge University Press, 2022). Ellen Gasparovic reviews it for the MAA.
In case anyone else is looking for a topical real-world example of a depth-first traversal of a tree (\(\mathbb{M}\)).
Chegg stops pretending not to have coursework-cheating as their main business; will no longer cooperate with university cheating investigations (\(\mathbb{M}\)).
Schwartz’s Pappus fractal (\(\mathbb{M}\), explanation). Start with two separate lines of three points \(abc\) and \(ABC\). By Pappus’s theorem the diagonal crossing points \(\alpha=aB\cdot Ab\), \(\beta=aC\cdot Ac\), and \(\gamma=bC\cdot Bc\) form another line of three points \(\alpha\beta\gamma\). Now recurse with \(abc\) and \(\alpha\beta\gamma\), and again with \(\alpha\beta\gamma\) and \(ABC\). You get a nice lightning-bolt fractal shape.
This year’s Graph Drawing conference proceedings are online (\(\mathbb{M}\)). As in past years, they’re using a system where the proceedings are published both on arXiv and in Springer LNCS.
An analysis of online exam-proctoring tool Proctorio finds it completely ineffective at catching cheaters (\(\mathbb{M}\), via). “The use of online proctoring is therefore best compared to taking a placebo: it has some positive influence, not because it works but because people believe that it works … policy makers would do well to balance the cost of deploying it (which can be considerable) against the marginal benefits of this placebo effect.”
Lang’s paper pentasia (\(\mathbb{M}\)). The kite and dart Penrose tiling lifts to a surface in 3d with two equilateral triangles per tile, overhanging for the darts. Robert Lang made these nice physical models. Explanations by Barry Cipra on the 1993 discovery of this surface with Conway and Lang and Barry Hayes, also on a related 3d surface for the rhombic Penrose tiling.
Three more new Wikipedia Good Articles (\(\mathbb{M}\)):
- Euclidean minimum spanning tree
- Doyle spiral, spiraling circle packings in which each circle is surrounded by a ring of six tangent circles
- Laves graph, a highly-symmetric 3-regular infinite graph in 3d
An algorithmic version of Mantel’s theorem (\(\mathbb{M}\)): If an \(n\)-vertex graph has more than \(\bigl\lfloor\tfrac{n^2}{4}\bigr\rfloor\) edges then we can find a triangle in it in linear time.

Proof: Delete min-degree vertices, maintaining the edge excess, until the remaining \(k\) vertices all have degree \(\ge\tfrac{k}{2}\). If \(k\) is odd, all have \(\ge \tfrac{k+1}{2}\); if \(k\) is even, at least one has \(\ge \tfrac{k}{2}+1\). Then a max-degree vertex and any neighbor have together \(\ge k+1\) incidences, so by the pigeonhole principle they have a common neighbor forming a triangle. \(\Box\)