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Linkage
- The exciting new world of AI prompt injection (\(\mathbb{M}\), via). Promote your business by running a bot that uses other people’s social media post text to prompt a text-writing AI that generates customized responses to those posts. What could go wrong?
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Counting paths in convex polygons
Let’s count non-crossing paths through the all points of a convex polygon. There is a very simple formula for this, \(n2^{n-3}\) undirected paths through an \(n\)-gon, but why? Here’s a simple coloring-based argument that immediately gives this formula.
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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).
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Comparing distances along lines
I’ve written here several times about Gilbert tessellations, most recently in last year’s post about cellular automata that naturally generate them. These are polygonal subdivisions of the plane, generated from the tracks of particles moving at the same speed, where the particles start as oppositely-moving pairs with random locations and directions, and continue moving until they crash into the track of another particle. Here’s Wikipedia’s illustration of these things, generated in 2012 by Claudio Rocchini:
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Linkage
- You’re probably familiar with machine-learning-based translation between natural languages, based on finding patterns in large datasets of known translations, for instance as used by Google translate. Now the Xena people are trying to use the same methods to convert LaTeX-formatted natural-language descriptions of mathematical propositions into the formal language used by the Lean theorem prover (\(\mathbb{M}\)).
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Permuted points of interest
Suppose you have a map, with certain points of interest marked. To avoid cluttering the map with the labels of these points, you want to list the labels in a line down the side of the map, with each point of interest connected to its label by a line segment. Preferably, the line segments should not cross each other, as they did when I tried to match the sidebar to the points in this example from Google Maps:
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Linkage
- The logic of definite descriptions (\(\mathbb{M}\)). Joel Hamkins wrestles with logical formulations of “the”, indicating that a description uniquely identifies something. If you define a logical operator whose value is the thing identified by its argument, and use it in a context that doesn’t uniquely identify something, does your overall formula still have a truth value? Fortunately(?) the stakes are low: several plausible choices produce logics with the same power as classical logic.
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Linkage
- Ars Mathemalchemica: From Math to Art and Back Again (\(\mathbb{M}\)), article in the Notices by Susan Goldstine, Elizabeth Paley, and Henry Segerman about the Mathemalchemy traveling multimedia mathematical art installation.
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Midsphere facts and fallacies
Some three-dimensional convex polyhedra have a midsphere, a sphere tangent to all of their edges. More strongly, every three-dimensional convex polyhedron has a combinatorially equivalent form, called a “canonical polyhedron”, that has a midsphere. The literature on midspheres is a little sparse and missing some key facts and counterexamples, so I thought I would collect some of them here.
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Flipping until you are lost
Start with any triangulation of a convex polygon, and then repeatedly choose a random diagonal and flip it, replacing the two triangles it borders with two different triangles. Eventually, these random flips will cause your triangulation to be nearly equally likely to be any of the possible triangulations of the polygon. But how long is “eventually”? My student Daniel Frishberg has a new answer, in our preprint “Improved mixing for the convex polygon triangulation flip walk” (arXiv:2207.09972).
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