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Linkage
- Zig-Zag paths and neusis constructions of a heptagon and a nonagon (\(\mathbb{M}\)), new article by Dave Richeson and Dan Lawson.
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Linkage
- Dollar-bill origami octahedron (\(\mathbb{M}\)), From @valentinabalance on Instagram.
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Voting with bitvector sums
As has happened in the past, there has been a recent scandal involving Hugo award voting. And as has happened in the past, this has caused me to completely ignore the particulars of the scandal and instead focus on the voting algorithms involved.
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The equilateral dimension of Riemannian manifolds
My recent preprint on integer distances in various metrics appears to have solved a 40-year-old open problem posed by Richard Guy in an interestingly-titled work, “An olla-podrida of open problems, often oddly posed” (Amer. Math. Monthly 1983, doi:10.1080/00029890.1983.11971188, jstor:2975549; both links are paywalled but maybe you can access one or the other). I’d read his paper before but forgotten it; after noticing it again I updated the preprint to make the connection explicit.
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Linkage
- The following lemma on distances between two out of three points around an obstacle is part of my new preprint on integer distances. I’m curious, has anyone seen it before?
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Integer distances in floppy metric spaces
Occasionally, the effort I put into editing Wikipedia articles pays off in research ideas. My latest preprint, “non-Euclidean Erdős–Anning theorems” (arXiv:2401.06328) is an example. It came out of my recent efforts to bring the Wikipedia article on the Erdős–Anning theorem (which states that a point set in the Euclidean plane with integer distances must either be finite or collinear) up to Wikipedia’s Good Article standard. One of the criteria for the standard is that the article should cover all “main aspects of the topic”, and one aspect that stood out to me as missing was: what about non-Euclidean geometries? I wasn’t able to find anything that had been published about this, and my Good Article reviewer didn’t notice the omission, so it wasn’t a problem for the Good Article review. But it left me wondering whether maybe there might be more to say about this topic. And there was!
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End-of-year linkage
- Harvard allegedly shut down a research project on disinformation in Facebook (\(\mathbb{M}\)) by noted social scientist Joan Donovan after getting a big donation from the Zuckerbergs (far outweighing the $12M Donovan had raised for the project) and planting a Facebook “fixer” on a dean’s advisory council. Donovan, a research scientist at Harvard since 2016, has since moved to a tenure-track faculty position at Boston University.
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Linkage
- Matt Blaze reminds all Ph.D. hopefuls that the way to get into a Ph.D. program is to apply to it. Emailing individual professors will not help: “No one gets in to a program that way. Just apply.”
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Correcting Harborth on rational distances
Heiko Harborth, in “Integral Distances in Point Sets” (at least, in the versions I can access online), claims without proof that the following construction produces an infinite dense set of points on the unit circle with rational-number distances: just take the points at all integer multiples of the angle of an integer right triangle. In my book and elsewhere I have uncritically repeated that claim. But this is just incorrect. The points \((1,0)\) and \((4/5,3/5)\) on the unit circle are separated from each other by the angle of a 3-4-5 right triangle, and are obviously at distance \(\sqrt{2/5}\) apart, not a rational number. You do get rational coordinates by this repeated rotation, but not rational distances.
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Linkage
- Lance Fortnow notices fewer faculty job ads in this year’s November CRA News and asks: “is there a real drop in hiring, or something else?”
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