Linkage
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Constructing the Herschel enneahedron by truncating a triangular bipyramid (\(\mathbb{M}\), see also).
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Dagstuhl and its LIPIcs series of open-access conference proceedings add html versions of its papers (\(\mathbb{M}\)). Selected ones for now on a trial basis, likely all after a year, aiming at greater accessibility. Like arXiv they are using LaTeXML, but I suspect they are going to get more consistent results, first because LIPIcs already has a tightly controlled format that can be expected to work well with this conversion but second because they also check things manually. (LIPIcs is gold open access, not diamond: they charge a small processing fee per paper to cover their costs, usually rolled into conference registration fees.) An example: my paper “Non-Euclidean Erdős-Anning theorems”.
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University of California system-wide faculty senate unanimously “calls on the Office of the President and the UC Board of Regents to unequivocally reject governmental demands that compromise institutional autonomy and academic freedom” (\(\mathbb{M}\)). They write that “There can be no clearer attack on academic freedom than an attempt to dictate curricular content, research priorities, hiring decisions, and admissions standards. While the financial penalties threatened by the federal government would in effect radically curtail UC’s future, acquiescing in the idea that academic freedom, governance, and mission are negotiable would be equally ruinous”.
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On implementing Numberlink. Simon Tatham expresses his frustration at not being able to find and implement good generators and solvers for this puzzle, in which you fill a grid with spiraling paths between colored pairs of endpoints.
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Carefully engineered cuckoo hashing provides improved real-world performance for high-load hash tables (\(\mathbb{M}\), via). “Adding cuckoo hashing to a strong baseline is typically close to neutral for performance on low load factors, and greatly improves performance on high load factors”. They have benchmark code but unfortunately this is not (yet?) available as a library.
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The mathematics linking scattering amplitudes in particle physics to a geometric object called the amplituhedron can be better understood by also mapping the amplituhedron to the mathematics of origami (\(\mathbb{M}\)).
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Visual proof that unknotting number is not linear under connected sums (\(\mathbb{M}\), see also).
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The stamp folding problem from a mountain-valley perspective (\(\mathbb{M}\)). For a given mountain-valley assignment to a strip of stamps, how many foldings are there?
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Graph of the dependencies in Euclid’s Elements Book I, with data from Stephen Wolfram as implemented in Mathematica.
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Geometric Hermite curves with minimum strain energy (\(\mathbb{M}\)), Yong and Cheng, Computer Aided Geometric Design, 2004. I mentioned briefly last June that I had added to my Python SVG code a new “polycurve” routine to draw a smooth curve through a sequence of points. It did this by connecting each pair of points by a cubic Bézier spline, whose tangent directions at each point were chosen parallel to the line between the two neighboring points. So far, so natural. But with those choices, there are still two free parameters left for each spline, the lengths of the control segments in each tangent direction. The code I wrote last June chose these lengths in an ad-hoc way that worked ok in the examples I tried but had no theory to justify it. This problem (Hermite interpolation: finding a smooth curve to match two given endpoints with given tangent directions) is exactly what the linked paper solves, by choosing the control segment lengths to minimize a certain strain energy. So I rewrote my code to follow their method. Here’s an example; note that the first and last points are used only to set the direction of the curve and not as endpoints of the curve.
It’s somewhat less circular-looking now than before, for a test case I was using to tune the parameters of my earlier implementation: a smooth closed curve through the four corners of a square. And it can misbehave if consecutive triples of points form sharp angles, so that the tangent directions are far from the direction of the pair of points to be connected. But the linked paper proves that it does always give you a well-behaved curve when these angles are within certain ranges, and I’m happier to be using a principled and proven method than something ad hoc and only known to work on a small set of test cases.
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Gwen Fisher crafts a kirigami orthogonal-polyhedron Sierpiński triangle for mathober.
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The invited talks and tutorials from the International Symposium on Graph Drawing and Network Visualization are now online on YouTube (\(\mathbb{M}\)). They include talks by
- Huamin Qu on AI in graph visualization
- Hans Bodlaender on parameterized complexity in graph drawing
- Camilla Forsell on user evaluation
- Daniel Archimbault on dimensionality reduction and machine learning
- Markus Chimani on minimizing crossings
- Sara Di Bartolomeo on the “information visualization perspective”
(The first two are an hour and the others more like an hour and a half, so not for short attention spans. The shorter contributed talks were not recorded.)
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Nan Ma experiments with two 3d-printed coordinated-assembly puzzles, one with 12 pieces that slide together into a cube, and another with 30 pieces that slide together into a dodecahedron.
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A discussion on the relative accuracy of Wikipedia and nLab for type theory topics, concluding surprisingly that, although both are not good, and nLab has a stronger focus on contributions from experts, nLab’s content is worse than Wikipedia’s. According to one discussant, “there’s been an overwhelming amount of low-quality anonymous contributions that just put totally false things in the type theory pages.” My takeaways: Wikipedia’s sourcing requirements work, but we need to better encourage people who know what they’re talking about to contribute, and nLab should be avoided as a source for Wikipedia.