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Triforce toroids
Recent edits to the Wikipedia article on toroidal polyhedra led me to a 1997 geometry.research discussion thread, “Polyhedra of positive genus”, in which John Conway describes a toroidal polyhedron with 36 equilateral-triangle sides, and suggests that this might be the fewest sides possible for a toroidal deltahedron.
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Linkage for mid-March
- Robin Houston wonders about cuboid terminology. The specific question is whether it should mean a shape with six rectangular sides (as commonly taught in school) or a shape with six quadrilateral sides (as used in some research communities). Let’s not even speak about Branko Grünbaum’s use of it to mean a shape formed by gluing together a power-of-two number of six-quad-side shapes.
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Spanners that don't change much
I have another preprint on the arXiv today with my student Hadi Khodabandeh: “Maintaining light spanners via minimal updates”, arXiv:2403.03290.
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Linkage for leap day
- The mysterious math of billiards tables (\(\mathbb{M}\)), Dave Richeson in Quanta.
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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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