Jekyll2020-11-18T22:25:28+00:00https://11011110.github.io/blog/feed.xml11011110Geometry, graphs, algorithms, and moreDavid EppsteinLinkage2020-11-15T21:05:00+00:002020-11-15T21:05:00+00:00https://11011110.github.io/blog/2020/11/15/linkage<ul>
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<p><a href="https://golem.ph.utexas.edu/category/2020/09/five_levels_of_continued_fract.html">The early history of continued fractions</a> (<a href="https://mathstodon.xyz/@11011110/105139147272940535">\(\mathbb{M}\)</a>). Includes the equivalences of infinitude with irrationality and periodicity with quadraticness, and the work of Euler and Gauss on continued fractions for values with nice series expansions.</p>
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<p>According to <a href="http://cams.ehess.fr/">EHESS</a>, <a href="https://en.wikipedia.org/wiki/Pierre_Rosenstiehl">Pierre Rosenstiehl</a> has died (<a href="https://mathstodon.xyz/@11011110/105147747900598849">\(\mathbb{M}\)</a>). I don’t think I met or interacted with Rosenstiehl, but I definitely interacted with his works: he was one of the founders of the International Symposium on Graph Drawing and co-editor in chief of the European Journal of Combinatorics, with important publications in topological and algorithmic graph theory on <a href="https://en.wikipedia.org/wiki/Bipolar_orientation">bipolar orientations</a> and the <a href="https://en.wikipedia.org/wiki/Left-right_planarity_test">left-right planarity test</a>.</p>
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<p><a href="https://www.aerialphotoawards.com/aerial-photos-of-the-year-2020-man-made-category">Aerial photos of the year, 2020</a> (<a href="https://mathstodon.xyz/@11011110/105155256595043126">\(\mathbb{M}\)</a>, <a href="https://www.metafilter.com/189287/A-Different-Perspective">via</a>). The link shows only the “man-made” category; be sure to click on all the others as well.</p>
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<p><a href="https://www.slac.stanford.edu/~kaehler/homepage/visualizations/dark-matter.html">Simulation and visualization of the large-scale structure of the universe</a> (<a href="https://mathstodon.xyz/@11011110/105161855079392754">\(\mathbb{M}\)</a>, <a href="https://apod.nasa.gov/apod/ap201025.html">via</a>), including webs of dark matter and the transition from an opaque early universe to its present mostly-transparent state.</p>
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<p><a href="https://arxiv.org/abs/2007.06838">The cocked hat</a> (<a href="https://mathstodon.xyz/@11011110/105167862476053808">\(\mathbb{M}\)</a>). This interestingly-titled recent preprint by Bárány, Steiger, and Toledo concerns locating yourself by measuring angles to three landmarks. Usually, random error causes rays from the landmarks to form a triangle rather than meeting at a point. Supposedly this triangle contains your position with probability 1/4, but the preprint shows that this can be proven only for distributions that force the rays to form a triangle. See also Toledo’s new book, <em><a href="http://www.cs.tau.ac.il/~stoledo/LocationEstimationFromTheGroundUp/">Location Estimation from the Ground Up</a></em>.</p>
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<p><a href="https://en.wikipedia.org/w/index.php?title=User_talk:DMacks&diff=prev&oldid=987454393">MDPI is asking its authors to spam Wikipedia with their work</a> (<a href="https://mathstodon.xyz/@11011110/105176559718608713">\(\mathbb{M}\)</a>), and <a href="https://www.elsevier.com/__data/assets/pdf_file/0013/201325/Get-Noticed_Brochure_2018.pdf">so is Elsevier</a>. I’m not surprised by MDPI doing this but, despite my low opinion of them, I thought Elsevier were better than that.</p>
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<p><a href="https://drawingmachines.org/">Drawing machines</a> (<a href="https://mathstodon.xyz/@11011110/105181738042763529">\(\mathbb{M}\)</a>, <a href="https://news.ycombinator.com/item?id=25033793">via</a>). An archive of optical/mechanical/automated drawing machines/devices/aids.</p>
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<p><a href="https://www.youtube.com/watch?v=LdnvxN4UUfs">Origami Fibonacci torus and knotted torus</a> (<a href="https://mathstodon.xyz/@11011110/105184956163150834">\(\mathbb{M}\)</a>). I have the impression that the Fibonacci part just gives it a nice organic look, <a href="http://www.starcage.org/englishindex.html">visible in much of Akio Hizume’s other architecture</a>, but what interests me is the way it rotates smoothly. That’s not something unfolded paper can do, because the inner parts of a torus have negative curvature, the outer parts are positive, and unfolded paper can’t change curvature.</p>
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<p>The floorplan below is of the <a href="https://en.wikipedia.org/wiki/K%C3%B6lntriangle">KölnTriangle</a>, a building in Cologne which has been claimed to be a Reuleaux triangle in cross-section (<a href="https://mathstodon.xyz/@11011110/105195769927037078">\(\mathbb{M}\)</a>). Obviously it’s not. The <a href="https://en.wikipedia.org/wiki/Torre_de_Collserola">Torre de Collserola in Barcelona</a> looks <a href="https://commons.wikimedia.org/wiki/File:Barcelona_Torre_de_Collserola_Planol_Evacuacio.jpg">much more promising</a> but a published description of its specific architecture would help make that more clear.</p>
<p style="text-align:center"><img src="/blog/assets/2020/KoelnTriangle-not-Reuleaux.jpg" alt="Kölntriangle floor plan from CC-BY-SA image https://commons.wikimedia.org/wiki/File:Fluchtwegeplan_K%C3%B6ln_Trangle.JPG by Bin im Garten, with overlaid Reuleaux triangle" /></p>
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<p><a href="https://www.scottaaronson.com/blog/?p=5094">The <em>Complexity Zoo</em> needs a new home</a> (<a href="https://mathstodon.xyz/@11011110/105199077818825073">\(\mathbb{M}\)</a>). Scott Aaronson’s collection of complexity classes and their relations had been at the U. Waterloo Institute for Quantum Computing, but their bureaucrats have decided that they cannot accept the liability of it possibly being out of compliance with accessibility rules, without allowing him to fix any issues or even tell him what issues it might have. As Scott writes, “Do you find it ironic that a central effect of these accessibility policies seems to be to make free academic content less accessible than before?”</p>
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<p>Two more recent deaths in discrete mathematics and theoretical computer science (<a href="https://mathstodon.xyz/@11011110/105205573414576460">\(\mathbb{M}\)</a>): <a href="http://emps.exeter.ac.uk/mathematics/news-events/news/articles/robinchapmanobituary.html">Robin J. Chapman, problem-solver extraordinaire</a> (and scarily almost exactly the same age as me), and
<a href="https://en.wikipedia.org/wiki/Chung_Laung_Liu">Dave (C. L.) Liu</a>, early researcher in scheduling algorithms and VLSI layout and later president of National Tsing Hua University (<a href="https://news.ltn.com.tw/amp/news/life/breakingnews/3346060">Chinese-language obituary</a>).</p>
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<p><a href="http://www.the-sandpit.com/contortions/contort.htm">Contortion engineering</a> (<a href="https://mathstodon.xyz/@11011110/105217523608810155">\(\mathbb{M}\)</a>). Engineering-style drawings of Escher-like impossible objects. An old link from my old Geometry Junkyard site — its old earthlink url went dead some time after 2016, when archive.org last captured it, but now it has a new home.</p>
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</ul>David EppsteinThe early history of continued fractions (\(\mathbb{M}\)). Includes the equivalences of infinitude with irrationality and periodicity with quadraticness, and the work of Euler and Gauss on continued fractions for values with nice series expansions.Pathbreaking for intervals2020-11-14T17:59:00+00:002020-11-14T17:59:00+00:00https://11011110.github.io/blog/2020/11/14/pathbreaking-for-intervals<p>The use of Nash’s Hex lemma in <a href="/blog/2020/11/10/hex-books-queues.html">my previous post</a>, according to which any 2-coloring of a triangular grid has a long monochromatic path, naturally raises the question: for which graphs other than the triangular grid is this true? In that post I mentioned that it is also true for certain outerplanar graphs, but false for bipartite graphs and for subcubic graphs. Here’s another class of graphs for which it is false: the graphs of bounded <a href="https://en.wikipedia.org/wiki/Pathwidth">pathwidth</a>. These graphs can always be 2-colored in a way that breaks up all long paths, leaving the remaining monochromatic paths of bounded size.</p>
<p>Rather than dealing with pathwidth directly, I want to formulate it geometrically in terms of interval graphs. Every graph can be completed (by adding extra edges) to form an <a href="https://en.wikipedia.org/wiki/Interval_graph">interval graph</a> of the same pathwidth; an interval graph has one-dimensional closed intervals as its vertices, connected by an edge whenever they intersect. The pathwidth of an interval graph is always one less than the maximum number of intervals that overlap at the same point (their ply). So rather than looking at graphs of bounded pathwidth I’ll be considering sets of intervals of bounded ply. For example in the illustration below which I drew a few years ago for Wikipedia, the intervals shown have ply 3, and their interval graph has pathwidth 2.</p>
<p style="text-align:center"><img src="/blog/assets/2020/interval-graph.svg" alt="A set of intervals and their associated interval graph" /></p>
<p>Intervals with low ply that are all either nested or disjoint can have no long paths in their intersection graph regardless of coloring. In any set of intervals forming a path, the path can pass through the outermost level of nesting at most once, subdividing it into two smaller paths with lower ply. It follows by induction the number of intervals in the path can be at most \(2^{\operatorname{ply}}-1\). (This is the standard argument behind an equivalence in graphs between longest path length and <a href="https://en.wikipedia.org/wiki/Tree-depth">tree-depth</a>.) So if we could partition an arbitrary set of intervals into two subsets within which all intervals are nested or disjoint, we’d be done. For instance, in the example above, the partition into \(\{A,B,F\}\) and \(\{C,D,E,G\}\) works. Not every set of intervals can be partitioned in this way, but I’ll show below that it’s always possible to lengthen some intervals, keeping the ply small, so that the lengthened intervals can be partitioned (colored) into two nested subsets.</p>
<p>As a subroutine in the construction, let’s find a set of points with the following properties:</p>
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<p>Each interval lies strictly between the leftmost and rightmost selected points.</p>
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<p>No interval covers more than one selected point.</p>
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<p>The union of the intervals that cover selected points equals the union of all the intervals.</p>
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<p>This can be done by a greedy algorithm that repeatedly selects an uncovered interval endpoint until the intervals that cover selected points also cover all other interval endpoints. Finish by selecting any two points to the left and right of all the intervals, and then number the selected points as \(x_0,x_1,\dots\) in left-to-right order. For example, one possible choice for our example would have \(x_0\) to the left of the intervals, \(x_1\) at the left endpoint of \(D\), \(x_2\) at the left endpoint of \(F\), and \(x_3\) to the right of the intervals. For this choice, \(B\) and \(E\) do not cover any selected points, but that’s not a problem. It’s not necessary to optimize how many points are selected.</p>
<p>Call an interval “odd” if it covers an odd-numbered point \(x_i\) and “even” if it covers an even-numbered point. In our example, \(A\), \(C\), and \(D\) are odd, and \(F\), and \(G\) are even, but \(B\) and \(E\) are neither odd nor even. We will color the odd intervals blue and the even intervals red. Because they are different colors, we don’t have to worry about nesting between odd and even intervals. However, the intervals that contain any single point \(x_i\) will have the same color and need to be lengthened to nesting intervals. To do so, we extend all these intervals to a single longer interval, from just after \(x_{i-1}\) to just before \(x_{i+1}\). Because of the way the points \(x_i\) were chosen, this does not create new intersections between intervals of the same color, and it makes all colored intervals nest with all uncolored ones (not just with each other). However, it can also increase the ply. If the ply was \(p\) previously, it can increase by as much as \(2p-1\), when a point that was previously covered by only one odd or even interval becomes covered by \(2p-1\) more of them.</p>
<p>Finally, we apply the same coloring and lengthening process recursively, to the remaining subsets of uncolored intervals between each pair of selected points.
To make sure the intervals stay nested after lengthening, we take care in each recursive subproblem to select its first and last points to be inside all intervals of the outer problem that contain the subproblem. Because every point in an interval of a recursive subproblem is also in at least one of the outer colored intervals, the recursive subproblem has smaller ply than the initial ply of the outer problem. The recursion stops at ply 2; for this ply the intersection graph is a tree (more precisely a caterpillar) and it is possible to 2-color the intervals without any lengthening so that no two intervals of the same color intersect, by 2-coloring the tree.</p>
<p>So if we start with intervals of ply 2, we produce a lengthened, 2-colored, and monochromatically nested set of intervals of ply 2 again (no change to the ply). If we start with intervals of ply 3, we produce a lengthened, 2-colored, and monochromatically nested set of intervals of ply at most 8, because the ply-3 interval-lengthening process added at most 5 to the existing ply. More generally for any starting ply \(p\) the new ply will be at most \(p^2+p-4\). And since the interval graph for the lengthened intervals has no monochromatic path with more than \(2^{p^2+p-4}-1\) intervals in it, neither does the interval graph for the original intervals, with the same 2-coloring.</p>
<p>(<a href="https://mathstodon.xyz/@11011110/105211974100055236">Discuss on Mastodon</a>)</p>David EppsteinThe use of Nash’s Hex lemma in my previous post, according to which any 2-coloring of a triangular grid has a long monochromatic path, naturally raises the question: for which graphs other than the triangular grid is this true? In that post I mentioned that it is also true for certain outerplanar graphs, but false for bipartite graphs and for subcubic graphs. Here’s another class of graphs for which it is false: the graphs of bounded pathwidth. These graphs can always be 2-colored in a way that breaks up all long paths, leaving the remaining monochromatic paths of bounded size.Hex, books, and queues2020-11-10T16:25:00+00:002020-11-10T16:25:00+00:00https://11011110.github.io/blog/2020/11/10/hex-books-queues<p>My latest preprint is “Stack-number is not bounded by queue-number”, <a href="https://arxiv.org/abs/2011.04195">arXiv:2011.04195</a>, with Vida Dujmović, Robert Hickingbotham, Pat Morin, and David R. Wood; Hickingbotham is a postgraduate student of Wood at Monash University. It solves a question implicit in the work of Heath, Leighton and Rosenberg (<a href="https://doi.org/10.1137/0405031">1992</a>) on whether graphs of bounded queue number have bounded stack number (they don’t), disproves <a href="https://en.wikipedia.org/wiki/Blankenship%E2%80%93Oporowski_conjecture">a conjecture of Blankenship and Oporowski</a> (1999) on whether subdividing a graph of unbounded stack number can reduce its stack number to bounded (it can), and answers a question of Bonnet, Geniet, Kim, Thomassé, and Watrigant (<a href="https://arxiv.org/abs/2006.09877">to appear at SODA 2021</a>) on whether graphs of unbounded stacknumber can have bounded sparse twin-width (they can). And it does so in part using an old trick from combinatorial game theory from a mathematician who won a Nobel prize for his work in game theory. But what are stack number and queue number? And what could combinatorial games possibly have to do with these sorts of questions in structural graph theory?</p>
<p>The game in question is <a href="https://en.wikipedia.org/wiki/Hex_(board_game)">Hex</a>, in which two players take turns placing their stones on hexagonal cells connected into a triangular grid and arranged in a rhombus. Each player tries to form a path of stones connecting two sides of the rhombus while simultaneously trying to block the other player from making a path. The rendering below from <a href="https://www.hexwiki.net/">HexWiki</a> shows a Hex board with a game in progress:</p>
<p style="text-align:center"><img src="/blog/assets/2020/HexWiki.jpg" alt="CC-BY-SA POV-Ray image of a Hex game, by Twixter on HexWiki, from https://commons.wikimedia.org/wiki/File:Hexposition02.jpg and https://www.hexwiki.net/index.php/File:Hexposition02.jpg" /></p>
<p>In tic-tac-toe, a similar game on a square grid, both players can end up blocking each other, producing a drawn game. But as John Nash famously proved in 1952, this can’t happen in Hex: every game ends with one or the other player forming a path. This inability to draw is usually explained as a topological property of disks in the plane, equivalent to the <a href="https://en.wikipedia.org/wiki/Brouwer_fixed-point_theorem">Brouwer fixed-point theorem</a>, although it also has a simple combinatorial proof involving walking along the boundary between the two colors of stones on a filled board, starting from a corner of the rhombus, and observing that the walk can only end at a non-opposite corner. But the lack of draws also implies the following statement about graphs: if you form a large <a href="https://en.wikipedia.org/wiki/Triangular_tiling">triangular grid</a> (the graph of positions and adjacencies of stones in Hex), and label its vertices with two colors (the colors of the stones of the two Hex players), then it must contain a long path: at least long enough to reach from one side of the Hex board to the other.</p>
<p>The same combinatorial proof for the Hex board works for any planar triangulation of a disk whose boundary can be divided into four paths with opposite pairs of paths far apart. Even some outerplanar graphs have the same long-path property. Suppose, for instance, that an outerplanar triangulated disk has a complete binary tree for its dual graph. Then we can form a long path by starting at a vertex of the root triangle and repeatedly walking down the tree, at each step moving to a neighbor of the same color whose triangle is closest in the dual tree. This walk takes a long step (such as the step on the top edge of the red path shown below) only when stepping from a vertex whose neighbors include a long path of the other color (the blue path shown below), and otherwise it takes many steps.</p>
<p style="text-align:center"><img src="/blog/assets/2020/outerplanar-path-property.svg" alt="An outerplanar graph in which every 2-coloring has a long monochromatic path" /></p>
<p>On the other hand, the square grid or other bipartite graphs don’t have the long path property, because if you color them bipartitely then there are no nontrivial monochromatic paths. Neither do subcubic graphs, because their maximum cuts always partition them into two subsets whose longest path has length at most one. The Hex board graph turns out to be the right choice for our purposes, because as well as the long path property it has bounded degree (unlike the outerplanar example) and a nice regular planar structure making stack and queue layouts easy.</p>
<p>So what are these layouts? Stack layout is another name for <a href="https://en.wikipedia.org/wiki/Book_embedding">book embedding</a>, in which the vertices of a graph are arranged in a line and its edges are placed without crossings into “pages”, half-planes bounded by the line. If you traverse the vertices of the graph in the order of the line, add an edge to its page when you traverse its first endpoint, and remove an edge from the page when you traverse the second endpoint, then the order of additions and removals is last-in-first-out, the same as a stack. The stack number or book thickness is the smallest number of pages you need to construct a layout like this. Below is an <a href="/blog/2015/10/03/why-shallow-minors.html">example I’ve used before</a>, a 3-page book embedding of the complete graph \(K_5\), whose stack number is three:</p>
<p style="text-align:center"><img src="/blog/assets/2015/3page-K5.svg" alt="Book embedding" /></p>
<p>If you replace the last-in-first-out ordering of additions and removals in a stack by first-in-first-out ordering, you get a queue. So a <a href="https://en.wikipedia.org/wiki/Queue_number">queue layout</a> is just an ordering of the vertices into a line, and a partition of the edges into “pages”, so that the traversal of the vertices by their line order produces a queue ordering of additions and removals of edges within each page. As <a href="https://doi.org/10.1007%2F978-3-642-18469-7_7">Auer et al described at GD 2010</a>, these layouts can also be described topologically, by representing each page as a cylinder with the line going longitudinally along it, and requiring each edge to be placed in such a way that it loops all the way around the cylinder. The queue number is the minimum number of these queues, or cylindrical pages, that you need to organize the graph in this way. It’s also closely related to the compact layout of graphs in 3d.</p>
<p>Below is an example I recently drew for a new Wikipedia article on <a href="https://en.wikipedia.org/wiki/Shuffle-exchange_network">shuffle-exchange networks</a>. These are very nonplanar graphs, but the layout shown can almost be drawn without crossings on two cylinders (one for the edges that bend around the left side of the vertices, another for the edges on the right) or on a single surface with a figure-eight cross-section, formed by gluing two cylinders together on a line. However, this only takes care of the curved edges. If you look closely, there are also short horizontal edges between consecutive vertices, which in a queue layout would need to wrap around another third cylinder. So these graphs have queue number at most three.</p>
<p style="text-align:center"><img src="/blog/assets/2020/Order-4_shuffle-exchange.svg" alt="Shuffle-exchange network" /></p>
<p>With all that as background, the central example in our new preprint is the <a href="https://en.wikipedia.org/wiki/Cartesian_product_of_graphs">Cartesian product</a> of a triangular grid with a <a href="https://en.wikipedia.org/wiki/Star_(graph_theory)">star</a>. It looks like this, but possibly with a bigger grid or star:</p>
<p style="text-align:center"><img src="/blog/assets/2020/star-times-hex.svg" alt="Cartesian product of a star with a triangular grid" /></p>
<p>The reason Cartesian products are useful in this context is that they behave nicely for queue layouts but not quite as nicely for stack layouts. If two graphs \(G\) and \(H\) both have queue layouts with a bounded number of queues, and in addition the layout of \(H\) is <em>strict</em>, meaning that within each page, each vertex has at most one earlier and one later neighbor, then we can lay out their product by grouping the vertices of the product into copies of \(G\) and placing each copy separately, with the order of the copies determined by the layout of \(H\). The resulting layout will still have a bounded queue number. The star has bounded queue number, but its layouts are not strict, because it has high degree at the center of the star. As a planar graph the triangular grid has bounded queue number, and because it also has bounded degree its layouts can be made strict. Therefore, the product graph again has bounded queue number.</p>
<p>For stack layouts, a similar product layout works, but only when \(H\) is bipartite. We can reverse the ordering within the copies of \(G\) coming from one side of the bipartition, and the edges going from one copy to another will still be stack-ordered. But when \(H\) is not bipartite, there is no way to consistently choose which copies of \(G\) to reverse. The long path property of Hex boards, in some sense, provides a quantitative measure for the fact that their graphs are far from being bipartite.</p>
<p>Of course, this only shows that a certain stack layout doesn’t work, not that all layouts fail. Our proof that all stack layouts of our product graph fail proceeds in a sequence of reductions in which we assume we are given a layout, and successively reduce it to more-constrained layouts on smaller graphs until reaching a contradiction. The first reductions use the pigeonhole principle to find products of the grid with a smaller star, in which all leaf copies of the grid have the same layout. Next we use the <a href="https://en.wikipedia.org/wiki/Erd%C5%91s%E2%80%93Szekeres_theorem">Erdős–Szekeres theorem</a> to find a product with an even smaller star, in which each copy of the star is consistently ordered by the layout in one of two ways. Finally, we use the result about long paths in Hex to find a product of this small star with a path, in which all copies of the star are consistently ordered in the same way, and prove that this ordering of this graph cannot have a bounded stack number.</p>
<p>Heath, Leighton and Rosenberg introduced both stack number and queue number in their 1992 work, and provided an example for which the stack number is exponentially bigger than the queue number, naturally raising the question whether even bigger separations are possible. Our work settles this question by showing that there is no function (exponential or otherwise) that can be used to bound stack number as a function of queue number. In the opposite direction, Heath, Leighton & Rosenberg conjectured that queue number cannot be made to grow significantly more quickly than stack number, but this remains open.</p>
<p>In 1999, Blankenship and Oporowski observed that subdividing the edges of a graph can lead to better stack layouts: for instance adding a single subdivision point to each edge of a complete graph reduces the stack number from linear to the square root of the number of vertices. They conjectured that this improvement cannot be too extreme: reducing the stack number from non-constant to constant should require a non-constant number of subdivision points. Our example disproves the Blankenship–Oporowski conjecture. The star-triangular grid product graph has non-constant stack number, but because it has bounded queue number its subdivisions with three subdivision points per edge have bounded stack number, according to a result of Dujmović and Wood (<a href="https://dmtcs.episciences.org/346">2005</a>). The same paper of Dujmović and Wood also stated the question of whether stack number can be bounded by a function of queue number more explicitly.</p>
<p>The definition of twin-width is too technical to summarize here; see the <a href="https://arxiv.org/abs/2006.09877">forthcoming SODA paper by Bonnet et al.</a>, which shows among other results that it is bounded for graphs of bounded stack number. Our results show that this is not an equivalence: there exist graphs (including the star-triangular grid products) that have unbounded stack number, but still have bounded sparse twin-width.</p>
<p>(<a href="https://mathstodon.xyz/@11011110/105188872486464972">Discuss on Mastodon</a>)</p>David EppsteinMy latest preprint is “Stack-number is not bounded by queue-number”, arXiv:2011.04195, with Vida Dujmović, Robert Hickingbotham, Pat Morin, and David R. Wood; Hickingbotham is a postgraduate student of Wood at Monash University. It solves a question implicit in the work of Heath, Leighton and Rosenberg (1992) on whether graphs of bounded queue number have bounded stack number (they don’t), disproves a conjecture of Blankenship and Oporowski (1999) on whether subdividing a graph of unbounded stack number can reduce its stack number to bounded (it can), and answers a question of Bonnet, Geniet, Kim, Thomassé, and Watrigant (to appear at SODA 2021) on whether graphs of unbounded stacknumber can have bounded sparse twin-width (they can). And it does so in part using an old trick from combinatorial game theory from a mathematician who won a Nobel prize for his work in game theory. But what are stack number and queue number? And what could combinatorial games possibly have to do with these sorts of questions in structural graph theory?Constant width from involutes of pseudotriangles2020-11-02T17:26:00+00:002020-11-02T17:26:00+00:00https://11011110.github.io/blog/2020/11/02/constant-width-involutes<p>In his <a href="https://www.eecs.yorku.ca/~jeff/courses/fun/">online collection of fun stuff</a>, Jeff Edmonds recently posted <a href="https://www.eecs.yorku.ca/~jeff/courses/fun/Equal_Distance.docx">a method of constructing curves of constant width</a> by spinning a pencil on a flat surface, with a varying axis, and tracking the movement of its ends. It is pretty similar to the classical method of crossed lines described by Martin Gardner in <em>The Unexpected Hanging</em>, in which one constructs an arrangement of lines in the plane, sorts them in circular order by slope, and builds a curve out of circular arcs centered at the crossing points of consecutive lines in this sorted order. However, it grows the curve at both ends simultaneously, rather than only at one end, and chooses the lines dynamically rather than in advance. Regardless, the result is the same: a piecewise-circular constant-width curve.</p>
<p>This got me wondering how we might go about constructing curves of constant width that are not piecewise-circular. Instead of a finite set of lines, we could use a continuous family of lines, one of each slope, but that’s a little difficult to visualize. Instead, there’s a simpler method that works more like Jeff’s spinning pencil, which Robinson (<a href="https://doi.org/10.1112/blms/16.3.264">“Smooth curves of constant width and transnormality”, <em>Bull. LMS</em> 1984</a>) attributes to Euler (<a href="https://scholarlycommons.pacific.edu/euler-works/513/">“De curvis triangularibus”, 1778</a>):</p>
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<p>Draw a closed curve in the plane that has only one tangent line of each slope.</p>
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<p>Rotate a “long enough” tangent line segment of some fixed length around this curve without sliding it.</p>
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<p>Trace the paths of the endpoints of the line segment.</p>
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<p>The first step may already seem a little confusing. Don’t curves usually have at least two tangent lines of each slope, their support lines from opposite sides? Well, yes, for convex curves. But for a <a href="https://en.wikipedia.org/wiki/Pseudotriangle">pseudotriangle</a>, a curve whose boundary is concave everywhere except at three extreme points, there might only be one tangent line of each slope. A standard example of such a curve is the <a href="https://en.wikipedia.org/wiki/Deltoid_curve">deltoid</a>, and the following animation stolen from the <a href="https://mathcurve.com/courbes2d.gb/deltoid/deltoid.shtml">mathcurve.com page on deltoids</a> shows a tangent line segment (black) rotating around a deltoid (blue) and tracing out a curve of constant width (red).</p>
<p style="text-align:center"><img src="/blog/assets/2020/involute-deltoid.gif" alt="Animation of a tangent line segment rolling around a deltoid, with its endpoints tracing out a curve of constant width, from https://mathcurve.com/courbes2d.gb/deltoid/deltoid.shtml" /></p>
<p>The traced curve is always perpendicular to the rotating line segment, so locally at least this segment behaves like the width of the curve in each given direction. And since we don’t change the length of the segment while it rotates, the width stays constant. The same deltoid, the same curve of constant width, and two positions of its tangent line segment can also be seen in the illustration below from a 1954 mathematics paper, <a href="https://doi.org/10.2307/2307215">“Rotors within rotors” by Michael Goldberg in the <em>Amer. Math. Monthly</em></a>. I’ve overlaid a red Reuleaux triangle to show that, like <a href="/blog/2020/08/30/linkage.html">so</a> <a href="/blog/2020/07/05/shape-wankel-rotor.html">many</a> <a href="/blog/2020/06/30/linkage.html">other</a> <a href="/blog/2018/06/24/la-maddalena-non-reuleaux.html">curvy</a> <a href="/blog/2018/04/17/mythical-reuleaux-manhole.html">triangles</a>, this is not a Reuleaux triangle, even though it has constant width. Although its corners are drawn to look kind of pointy, they should actually be smooth, and the rest of the curve bulges farther out from its sides than a Reuleaux triangle would.</p>
<p style="text-align:center"><img src="/blog/assets/2020/rotors-within-rotors.png" alt="The involute of deltoid, as depicted by Goldberg in "Rotors within rotors", with an overlaid Reuleaux triangle" /></p>
<p>More formally, this process of rotating and tracing tangent line segments produces a curve called the <a href="https://en.wikipedia.org/wiki/Involute">involute</a> of the deltoid. An involute of a curve is more typically described as what you get when you fix one end of a length of string at a point on the curve, wrap it tightly around the curve, and then unwrap it while keeping it taut, tracing a curve with the other end of the string as you do. The two ends of the rotating tangent line segment can both be thought of as being formed in the same way from two strings, with one of them unwrapping the deltoid from one direction while the other wraps it back up in the other direction. In the deltoid example the segment was the same length as the sides of the deltoid, which were all equal, but it also works with unequal sides or longer segments, as long as the rotating segment is long enough to reach all three cusps.</p>
<p>You might worry whether the segment always comes back to its starting position after each rotation, and this does require a little care in the initial choice of length and placement of the line segment. If the length of the rotating segment and eventual width of the traced curve are \(w\), the pseudotriangle sides have lengths \(a\), \(b\), and \(c\), and the segment starts with \(x\) units of extra length extending past the cusp prior to side \(a\) in its rotation, then it will have \(w-a-x\) units of extra length at the next cusp, \(w-b-(w-a-x)=x+a-b\) units at the third cusp, and \(w-c-(x+a-b)=w-x-a+b-c\) units at the last cusp. To make a curve of constant width we need the amount of extra length at the start and end to be equal, which happens when we set this length to be \(x=(w-a+b-c)/2\).</p>
<p>If the pseudotriangle that the tangent segment rotates around includes a line segment, there will be a discontinuity in its rotating movement as the axis of rotation shifts from one end of the segment to the other, much like the changes of axis described by Edmonds, but the same process still works. If the pseudotriangle has a point where its slope changes discontinuously (for instance, if it is a polygon rather than a smooth curve), then the rotating segment will rotate around this point, with its ends tracing circular arcs, as it continuously moves between the same slopes; this can happen either at the three convex points of the pseudotriangle or along the concave curves between them. In particular, if your pseudotriangle is actually an equilateral triangle and the rotating segment has the same length as its sides, you get a Reuleaux triangle.</p>
<p>It’s also possible to form closed curves with only one tangent line of each slope that are not pseudotriangles. An example is the standard pentagram (whose involute is the Reuleaux pentagon), or a cuspy and irregular pentagram like the one below (whose involute is another curve of constant width without circular arcs). The same process works for these, with a slightly more complicated calculation of how to place a rotating segment of a given length for a given starting curve, involving alternating sums of side lengths.</p>
<p style="text-align:center"><img src="/blog/assets/2020/cuspy-star.svg" alt="A curved five-point star with one tangent line in each direction" /></p>
<p>What happens when the rotating line segment is too short, so that it doesn’t reach one or more of the cusps? I’m not sure in general, but for the deltoid the result can be the same deltoid (for which the rotating line segment is one possibility for Goldberg’s “rotor within a rotor”, although the rotor he describes is larger) or another similar but smaller deltoid inside it. See the mathcurve link for details.</p>
<p>(<a href="https://mathstodon.xyz/@11011110/105144037558097550">Discuss on Mastodon</a>)</p>David EppsteinIn his online collection of fun stuff, Jeff Edmonds recently posted a method of constructing curves of constant width by spinning a pencil on a flat surface, with a varying axis, and tracking the movement of its ends. It is pretty similar to the classical method of crossed lines described by Martin Gardner in The Unexpected Hanging, in which one constructs an arrangement of lines in the plane, sorts them in circular order by slope, and builds a curve out of circular arcs centered at the crossing points of consecutive lines in this sorted order. However, it grows the curve at both ends simultaneously, rather than only at one end, and chooses the lines dynamically rather than in advance. Regardless, the result is the same: a piecewise-circular constant-width curve.Linkage for a trick-or-treat-less Halloween2020-10-31T22:59:00+00:002020-10-31T22:59:00+00:00https://11011110.github.io/blog/2020/10/31/linkage<ul>
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<p><a href="http://www.math.uwaterloo.ca/tsp/star/gaia1.html">3d flythrough of a near-optimal TSP tour through a dataset of nearly 221 stars</a> (<a href="https://mathstodon.xyz/@11011110/105048388019149239">\(\mathbb{M}\)</a>, <a href="https://news.ycombinator.com/item?id=24807080">via</a>). I found the “full view of tour” a lot easier to navigate than the mini-view on the main page.</p>
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<p><a href="http://lightherder.blogspot.com/">The light herder</a> (<a href="https://mathstodon.xyz/@11011110/105056854067105557">\(\mathbb{M}\)</a>, <a href="https://boingboing.net/2020/10/18/amazing-in-camera-patterns-with-a-video-feedback-kinetic-sculpture.html">via</a>). Dave Blair makes dynamic fractals from old-school video feedback.</p>
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<p><a href="https://www.siam.org/conferences/cm/program/accepted-papers/soda21-accepted-papers">Symposium on Discrete Algorithms (SODA 2021) accepted papers</a> (<a href="https://mathstodon.xyz/@11011110/105062503320962361">\(\mathbb{M}\)</a>). Lots of interesting looking titles there, but you’ll have to search online for links to the corresponding papers.</p>
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<p><a href="https://tomlehrersongs.com/">Tom Lehrer has made his song lyrics public domain, or as close to it as one can legally get</a> (<a href="https://mathstodon.xyz/@11011110/105076781151866254">\(\mathbb{M}\)</a>, <a href="https://news.ycombinator.com/item?id=24833683">via</a>, <a href="https://boingboing.net/2020/10/20/brilliant-satirist-tom-lehrers-catalog-now-in-the-public-domain.html">via2</a>). But you have to download them within four years because his domain may go away after that. In honor of which, here’s a link to an (audio-only) version of <a href="https://www.youtube.com/watch?v=IL4vWJbwmqM">a little ditty about plagiarism</a> (only be sure always to call it please research).</p>
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<p><a href="https://dev.scottdarby.com/chaos-ink/">Chaos ink</a> (<a href="https://mathstodon.xyz/@11011110/105082966399876773">\(\mathbb{M}\)</a>, <a href="https://boingboing.net/2020/10/18/chaos-ink-disturb-a-tank-of-virtual-liquid-metal.html">via</a>). It’s rendered to look like waves in liquid metal, but I think it’s actually some kind of reaction-diffusion equation, in which you can move your mouse around to try to control where the reactions are centered.</p>
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<p><a href="https://nebusresearch.wordpress.com/2020/10/21/my-all-2020-mathematics-a-to-z-statistics/">2020 Mathematics A to Z: Statistics</a> (<a href="https://mathstodon.xyz/@nebusj/105085756046257623">\(\mathbb{M}\)</a>). On the differences between statistics and mathematics, historical connections between statistics and eugenics, and new connections to algorithmic fairness.</p>
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<p><a href="https://sites.google.com/view/bad-math-day-spring-2020/home">Bay Area Discrete Math Day, November 21</a> (<a href="https://mathstodon.xyz/@11011110/105091339250415859">\(\mathbb{M}\)</a>). This year, it’s a day of online discrete math talks, so you don’t actually need to be in the SF Bay Area to participate.</p>
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<p><a href="https://liorpachter.wordpress.com/2020/09/10/sexual-harassment-case-number-1052/">Lior Pachter reports on confirmed sexual harassment within the computational geometry community, at SoCG 2016 in Boston, by Adrian Dumitrescu</a> (<a href="https://mathstodon.xyz/@11011110/105097516664950105">\(\mathbb{M}\)</a>). According to the victim, <a href="https://twitter.com/RupeiXu/status/1302069912286957571">SoCG organizers told her they would try to bar Dumitrescu from future events, but told Dumitrescu he could not be barred</a> (see also <a href="https://twitter.com/RupeiXu/status/1310211818716049409">this update</a>). She says <a href="https://twitter.com/RupeiXu/status/1303309158427615234">there was also a second victim, whose academic career was “ruined” as a result</a>.</p>
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<p><a href="https://sites.google.com/view/wepa2020">Fourth International Workshop on Enumeration Problems and Applications</a> (<a href="https://mathstodon.xyz/@11011110/105102101211362136">\(\mathbb{M}\)</a>). I haven’t participated in previous instances but this year I’m on the program committee. Submission deadline November 8; online workshop December 7–10.</p>
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<p><a href="https://igorpak.wordpress.com/2020/10/26/the-guest-publishing-scam/">Igor Pak hates journal special issues</a> (<a href="https://mathstodon.xyz/@11011110/105116591625994062">\(\mathbb{M}\)</a>). The underlying problem appears to be loss of quality control compared to regular papers. He suggests handling festschrifts as books instead, and publishing surveys and reminiscences instead of research papers in them.</p>
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<p><a href="http://atlas.gregas.eu/graphs/31">The 84-vertex cubic symmetric graph</a> (<a href="https://mathstodon.xyz/@11011110/105121991976224491">\(\mathbb{M}\)</a>) is drawn nicely on <a href="http://www.mathpuzzle.com/">mathpuzzle.com</a> (update of June 27) using its structure as a <a href="https://www.abstract-polytopes.com/atlas/504/156/3.html">36-heptagon symmetric tiling of a non-orientable surface of Euler characteristic \(-6\)</a>. The Petrie dual of this tiling is <a href="https://www.abstract-polytopes.com/atlas/504/156/9.html">another symmetric tiling of the same graph with 28 nonagons on a higher-genus surface</a>. Does anyone know of other sources on these tilings? Or nice 3d embeddings of their surfaces?</p>
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<p><a href="https://plus.maths.org/content/prize-young-mathematicians"><em>Plus</em> magazine on Cambridge’s Whitehead Prize winners</a> (<a href="https://mathstodon.xyz/@11011110/105128113002070698">\(\mathbb{M}\)</a>): A nice general-audience explainer of</p>
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<p>Maria Bruna’s derivation of macro-level models from micro-level behavior, applied to vacuum cleaner design</p>
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<p>Holly Krieger’s connections between prime factors in integer sequences and special points on the Mandelbrot set</p>
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<p>Henry Wilton on the impossibility of determining whether infinite symmetry groups have finite quotients</p>
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<p><a href="https://www.iflscience.com/technology/ai-camera-ruins-soccar-game-for-fans-after-mistaking-referees-bald-head-for-ball/">Silly computer news of the day: automatically aimed soccer game video camera follows bald referee’s head instead of the ball, causing fans to miss key plays</a> (<a href="https://mathstodon.xyz/@11011110/105131999819480420">\(\mathbb{M}\)</a>, <a href="https://news.ycombinator.com/item?id=24955651">via</a>).</p>
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</ul>David Eppstein3d flythrough of a near-optimal TSP tour through a dataset of nearly 221 stars (\(\mathbb{M}\), via). I found the “full view of tour” a lot easier to navigate than the mini-view on the main page.Graphs whose cycles all touch2020-10-26T16:16:00+00:002020-10-26T16:16:00+00:00https://11011110.github.io/blog/2020/10/26/graphs-whose-cycles<p>An interesting recent question on MathOverflow asks about <a href="https://mathoverflow.net/q/374793/440">graphs in which all cycles touch</a>. Here, touching is meant in the same sense as a <a href="https://en.wikipedia.org/wiki/Bramble_(graph_theory)">bramble</a> in graph structure theory: every two cycles either share a vertex or contain the two endpoints of an edge from one cycle to the other. The graphs with this property include all the complete graphs (girth 3), complete bipartite graphs (girth 4), and theta graphs (arbitrarily high girth but very simple structure). As originally phrased, it asked whether there exists \(g\) such that graphs of girth \(\ge g\) with all cycles touching have bounded treewidth. Partial results given there by Tony Huynh and me show that the condition of bounded treewidth can be replaced by bounded vertex cover number or a bounded number of vertex-disjoint cycles without changing the answer.</p>
<p>This led me to look for graphs that have high girth, all cycles touching, and as many vertex-disjoint cycles as I could construct. So far, the best I have found is four vertex-disjoint cycles, as shown in graphs of the following form:</p>
<p style="text-align:center"><img src="/blog/assets/2020/4-disjoint-touching-cycles.svg" alt="A graph with four vertex-disjoint long cycles, and all cycles touching" /></p>
<p>It consists of four theta graphs (the pairs of blue vertices connected by multiple long paths of yellow vertices, with the eight blue pole vertices of the theta graphs connected into two four-vertex paths. I’ve drawn it with yellow paths of length 16, and three paths per theta, but these numbers are arbitrary. One can easily find four vertex-disjoint cycles, within each of the four thetas, ignoring the edges between the pole vertices.</p>
<p>There is no cycle using only the blue pole vertices, so every cycle in the overall graph must include at least one complete yellow path connecting its two poles. Therefore, every cycle is at least as long as this yellow path length. These paths can be made arbitrarily long, so the graphs constructed in this way can have arbitrarily large girth.</p>
<p>The six edges of the two four-vertex paths between the pole vertices include an edge between each of the six pairs of pole vertices. But each cycle uses at least one pair of pole vertices, so this implies that every two cycles touch, either by sharing a pole vertex or by each containing one endpoint of one of these path edges.</p>
<p>Therefore the graphs constructed in this way have arbitrarily large girth, have all cycles touching, and contain four vertex-disjoint cycles. It also has feedback vertex number four. The MathOverflow question asks whether the four vertex-disjoint cycles can be replaced by an arbitrarily large number of cycles, or equivalently whether the feedback vertex number can be increased, but at this point I don’t even know whether either number can be replaced by five.</p>
<p>(<a href="https://mathstodon.xyz/@11011110/105103937279022043">Discuss on Mastodon</a>)</p>David EppsteinAn interesting recent question on MathOverflow asks about graphs in which all cycles touch. Here, touching is meant in the same sense as a bramble in graph structure theory: every two cycles either share a vertex or contain the two endpoints of an edge from one cycle to the other. The graphs with this property include all the complete graphs (girth 3), complete bipartite graphs (girth 4), and theta graphs (arbitrarily high girth but very simple structure). As originally phrased, it asked whether there exists \(g\) such that graphs of girth \(\ge g\) with all cycles touching have bounded treewidth. Partial results given there by Tony Huynh and me show that the condition of bounded treewidth can be replaced by bounded vertex cover number or a bounded number of vertex-disjoint cycles without changing the answer.The graphs of stably matchable pairs2020-10-19T20:29:00+00:002020-10-19T20:29:00+00:00https://11011110.github.io/blog/2020/10/19/graphs-stably-matchable<p>The <a href="https://en.wikipedia.org/wiki/Stable_marriage_problem">stable matching problem</a> takes as input the preferences from two groups of agents (most famously medical students and supervisors of internships), and pairs up agents from each group in a way that encourages everyone to play along: no pair of agents would rather go their own way together than take the pairings they were both given. A solution can always be found by the <a href="https://en.wikipedia.org/wiki/Gale%E2%80%93Shapley_algorithm">Gale–Shapley algorithm</a>, but there are generally many solutions, described by the <a href="https://en.wikipedia.org/wiki/Lattice_of_stable_matchings">lattice of stable matchings</a>. Some pairs of agents are included in at least one stable matching, while some other pairs are never matched. In this way, each instance of stable matchings gives rise to a graph, the <em>graph of stably matchable pairs</em>. This graph is the subject and title of my latest preprint, <a href="https://arxiv.org/abs/2010.09230">arXiv:2010.09230</a>, which asks: Which graphs can arise this way? How hard is it to recognize these graphs, and infer a stable matching instance that might have generated them? How does the graph structure relate to the lattice structure?</p>
<p>For some answers, see the preprint. One detail is connected to <a href="/blog/2020/10/18/polyhedra-without-disjoint.html">my previous post, on polyhedra with no two disjoint faces</a> (even though there are no polyhedra in the new preprint): the (prism,\(K_{3,3}\))-minor-free graphs discussed there come up in proving an equivalence between outerplanar graphs of stably matchable pairs and lattices of <a href="https://en.wikipedia.org/wiki/Closure_problem">closures</a> of <a href="https://en.wikipedia.org/wiki/Polytree">oriented trees</a>. Instead of providing any technical details of any the other results in the paper, though, I thought it would be more fun to show a few visual highlights.</p>
<p>The following figure shows a cute mirror-inversion trick (probably already known, although I don’t know where or by whom) for embedding an arbitrary bipartite graph as an induced subgraph of a regular bipartite graph. I use it to show that graphs of stably matchable pairs have no forbidden induced subgraphs:</p>
<p style="text-align:center"><img src="/blog/assets/2020/regularize.svg" alt="Embedding a bipartite graph as an induced subgraph of a regular bipartite graph" width="60%" /></p>
<p>This next one depicts a combinatorial description of a stable matching instance having a \(6\times 5\) grid as its graph, in terms of the top and bottom matchings in the lattice of matchings, the “rotations” that can be used to move between matchings in this lattice, and a partial order on the rotations. For what I was doing in this paper, these rotation systems were much more convenient to work with than preferences.</p>
<p style="text-align:center"><img src="/blog/assets/2020/5x6.svg" alt="Rotation system describing a system of stable matchings having a 6x5 grid as its graph" width="80%" /></p>
<p>All the main ideas for a proof of NP-completeness of recognizing these graphs, by reduction from <a href="https://en.wikipedia.org/wiki/Not-all-equal_3-satisfiability">not-all-equal 3-satisfiability</a>, are visible in the next picture. The proof now in the paper is significantly more complicated, though, because the construction in this image produces nonplanar graphs but I wanted a proof that would also apply in the planar case.</p>
<p style="text-align:center"><img src="/blog/assets/2020/nae3sat-to-matching.svg" alt="NP-completeness reduction from NAE3SAT to recognizing graphs of stably matchable pairs" /></p>
<p>The last one shows a sparse graph that can be represented as a graph of stably-matching pairs (because it’s outerplanar, bipartite, and biconnected) but has a high-degree vertex. If we tried to test whether it could be realized by doing a brute-force search over preference systems, the time would be factorial in the degree, but my preprint provides faster algorithms that are only singly exponential in the number of edges.</p>
<p style="text-align:center"><img src="/blog/assets/2020/factorial.svg" alt="Outerplanar graph of stably matchable pairs with a factorial number of potential preference systems" /></p>
<p>(<a href="https://mathstodon.xyz/@11011110/105065476283424319">Discuss on Mastodon</a>)</p>David EppsteinThe stable matching problem takes as input the preferences from two groups of agents (most famously medical students and supervisors of internships), and pairs up agents from each group in a way that encourages everyone to play along: no pair of agents would rather go their own way together than take the pairings they were both given. A solution can always be found by the Gale–Shapley algorithm, but there are generally many solutions, described by the lattice of stable matchings. Some pairs of agents are included in at least one stable matching, while some other pairs are never matched. In this way, each instance of stable matchings gives rise to a graph, the graph of stably matchable pairs. This graph is the subject and title of my latest preprint, arXiv:2010.09230, which asks: Which graphs can arise this way? How hard is it to recognize these graphs, and infer a stable matching instance that might have generated them? How does the graph structure relate to the lattice structure?Polyhedra without disjoint faces2020-10-18T17:06:00+00:002020-10-18T17:06:00+00:00https://11011110.github.io/blog/2020/10/18/polyhedra-without-disjoint<p>Some research I’ve been doing led me to consider the (prism,\(K_{3,3}\))-minor-free graphs. It’s not always easy to go from <a href="https://en.wikipedia.org/wiki/Forbidden_graph_characterization">forbidden minors</a> to the graphs that forbid them, or vice versa, but in this case I think there’s a nice characterization, which I’m posting here because it doesn’t fit into the research writeup: these are the graphs whose nontrivial triconnected components are \(K_5\), <a href="https://en.wikipedia.org/wiki/Wheel_graph">wheel graphs</a>, or the graph \(K_5-e\) of the <a href="https://en.wikipedia.org/wiki/Triangular_bipyramid">triangular bipyramid</a>. The illustration below shows an example of a graph with this structure, with its nontrivial triconnected components colored red and yellow. There’s a simpler and more geometric way to say almost the same thing: the only convex polyhedra that do not have two vertex-disjoint faces are the pyramids and the triangular bipyramid.</p>
<p style="text-align:center"><img src="/blog/assets/2020/prism-k33-free.svg" alt="A (prism, K_{3,3})-minor-free graph, with its nontrivial triconnected components colored red and yellow" /></p>
<p>Some definitions:</p>
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<p>Here by the prism graph I mean the graph of the triangular prism. Any other prism has this one as a minor, and so is irrelevant as a forbidden minor. However, the pyramids in this structure can have any polygon as their base, corresponding to wheel graphs with arbitrarily many vertices.</p>
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<p>\(K_{3,3}\) is a complete bipartite graph with three vertices on each side of its bipartition, famous as the <a href="https://en.wikipedia.org/wiki/Three_utilities_problem">utility graph</a>, one of the two forbidden minors for planar graphs. The triangular prism graph and \(K_{3,3}\) are the only two <a href="https://en.wikipedia.org/wiki/Cubic_graph">3-regular graphs</a> with six vertices.</p>
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<p style="text-align:center"><img src="/blog/assets/2020/prism-k33.svg" alt="The prism graph and K_{3,3}" /></p>
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<p>The triconnected components of a graph are the graphs associated with the nodes of its <a href="https://en.wikipedia.org/wiki/SPQR_tree">SPQR tree</a>, or of the SPQR trees of its biconnected components. These are cycle graphs, dipole multigraphs, or 3-connected graphs, and by “nontrivial” I mean the ones that are not cycles or dipoles. A triconnected component might not be a subgraph of the given graph, because it can have additional edges that correspond to paths in the given graph. For instance, subdividing the edges of any graph into paths, or more generally replacing edges by arbitrary series-parallel graphs, does not change its set of nontrivial triconnected components.</p>
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<p>I’m using “face” in the usual three-dimensional meaning, a two-dimensional subset of the boundary of the polyhedron. For higher-dimensional polytopes, “face” has a different meaning that also includes vertices and edges, and “facet” would be used to refer to the \((d-1)\)-dimensional faces, but using that terminology seems overly pedantic here.</p>
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<p>Sketch of proof of the characterization of polyhedra without two disjoint faces: Consider any polyhedron without disjoint faces. If one face shares an edge with all the others, it’s a <a href="https://en.wikipedia.org/wiki/Halin_graph">Halin graph</a>, a graph formed by linking the leaves of a tree into a cycle; if the tree is a star, it’s a pyramid, and otherwise contracting all but one of the interior edges of the tree, and then all but four of the cycle edges, will produce a prism minor. In the remaining case, some two faces share only a vertex \(v\), which must have degree four or more. Each face that is disjoint from \(v\) must touch all that faces incident to \(v\), which can only happen when there is one face disjoint from \(v\) (a pyramid) or two faces disjoint from \(v\), neither of which has an edge disjoint from the other one (a bipyramid).</p>
<p>Sketch of a lemma that every convex polyhedron with two disjoint faces has a prism minor: glue a pyramidal cap into each of the two faces, producing a larger convex polyhedron which by either <a href="https://en.wikipedia.org/wiki/Steinitz%27s_theorem">Steinitz’s theorem</a> or <a href="https://en.wikipedia.org/wiki/Balinski%27s_theorem">Balinski’s theorem</a> is necessarily 3-connected, and find three vertex-disjoint paths between the apexes of the attached pyramids. The parts of these paths outside the two glued pyramids, together with the boundaries of the two faces, form a subdivision of a prism.</p>
<p>Sketch of proof of the characterization of (prism,\(K_{3,3}\))-minor-free graphs: The nontrivial triconnected components are exactly the maximal triconnected minors of the given graph, so if either of the two triconnected forbidden minors is to be found in the given graph, it will be found in one of the triconnected components. \(K_5\) and the triangular bipyramid are too small to have one of the forbidden minors. The only 3-connected minors of the pyramid graphs are smaller pyramids, obtained by contracting one of the cycle edges of the pyramid, so these also do not have a forbidden minor. Therefore the graphs of the stated form are all (prism,\(K_{3,3}\))-minor-free.</p>
<p>In the other direction, suppose that a graph is (prism,\(K_{3,3}\))-minor-free.
Each triconnected component is a minor, so it must also be (prism,\(K_{3,3}\))-minor-free. What can these components look like? Forbidding \(K_{3,3}\) as a minor rules out nonplanar components other than \(K_5\), by a theorem of Wagner<sup id="fnref:wagner" role="doc-noteref"><a href="#fn:wagner" class="footnote">1</a></sup> and Hall.<sup id="fnref:hall" role="doc-noteref"><a href="#fn:hall" class="footnote">2</a></sup> So the remaining components that we need to consider are triconnected planar graphs with no prism minor. These cannot have two disjoint faces by the lemma, and so they can only be pyramids or the triangular bipyramid.</p>
<div class="footnotes" role="doc-endnotes">
<ol>
<li id="fn:wagner" role="doc-endnote">
<p>K. Wagner. Über eine Erweiterung des Satzes von Kuratowski. <em>Deutsche Mathematik</em>, 2:280–285, 1937. <a href="#fnref:wagner" class="reversefootnote" role="doc-backlink">↩</a></p>
</li>
<li id="fn:hall" role="doc-endnote">
<p>D. W. Hall. A note on primitive skew curves. <em>Bulletin of the American Mathematical Society</em>, 49(12):935–936, 1943. <a href="https://doi.org/10.1090/ S0002-9904-1943-08065-2">doi:10.1090/ S0002-9904-1943-08065-2</a>. <a href="#fnref:hall" class="reversefootnote" role="doc-backlink">↩</a></p>
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<p>(<a href="https://mathstodon.xyz/@11011110/105058649830809584">Discuss on Mastodon</a>)</p>David EppsteinSome research I’ve been doing led me to consider the (prism,\(K_{3,3}\))-minor-free graphs. It’s not always easy to go from forbidden minors to the graphs that forbid them, or vice versa, but in this case I think there’s a nice characterization, which I’m posting here because it doesn’t fit into the research writeup: these are the graphs whose nontrivial triconnected components are \(K_5\), wheel graphs, or the graph \(K_5-e\) of the triangular bipyramid. The illustration below shows an example of a graph with this structure, with its nontrivial triconnected components colored red and yellow. There’s a simpler and more geometric way to say almost the same thing: the only convex polyhedra that do not have two vertex-disjoint faces are the pyramids and the triangular bipyramid.Linkage2020-10-15T22:15:00+00:002020-10-15T22:15:00+00:00https://11011110.github.io/blog/2020/10/15/linkage<ul>
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<p><a href="https://scilogs.spektrum.de/hlf/mirzakhani-and-meanders/">Mirzakhani and meanders</a> (<a href="https://mathstodon.xyz/@11011110/104963847400612388">\(\mathbb{M}\)</a>). On some more-than-coincidental similarities in formulas found by Mirzakhani for numbers of geodesics on hyperbolic surfaces and by Vincent Delecroix, Elise Goujard, Peter Zograf, and Anton Zorich <a href="https://arxiv.org/abs/1705.05190">in a new preprint</a> for numbers of <a href="https://en.wikipedia.org/wiki/Meander_(mathematics)">meanders</a>, closed curves with a given number of intersections with a line.</p>
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<p><a href="https://mycqstate.wordpress.com/2020/09/29/it-happens-to-everyonebut-its-not-fun/">Retraction of “a proof of soundness of the Raz-Safra low-degree test against entangled-player strategies, a key ingredient in the proof of the quantum low-degree test, itself a key ingredient in the \(\mathsf{MIP}^*=\mathsf{RE}\) paper”</a> (<a href="https://mathstodon.xyz/@11011110/104969573344196233">\(\mathbb{M}\)</a>). \(\mathsf{MIP}^*=\mathsf{RE}\) is patched and remains believed true but not fully refereed. This post provides a lot more than the standard we-found-a-bug notice: a good description of what happened, what it implies technically, and how it affects the authors and community.</p>
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<p><a href="https://blogs.lse.ac.uk/impactofsocialsciences/2020/09/30/for-academic-publishing-to-be-trans-inclusive-authors-must-be-allowed-to-retroactively-change-their-names/">For academic publishing to be trans-inclusive, authors must be allowed to retroactively change their names</a> (<a href="https://mathstodon.xyz/@11011110/104972193066839079">\(\mathbb{M}\)</a>, <a href="https://retractionwatch.com/2020/10/03/weekend-reads-unicorn-poo-and-other-fraudulent-covid-19-treatments-disgraced-researchers-and-drug-company-payouts-a-fictional-account-of-real-fraud/">via</a>). I agree — more than once in researching Wikipedia bios I found past publications under deadnames. If the authors prefer this to be better hidden, while continuing to be credited for their past work, we should try to honor that preference.</p>
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<p>It’s easy to point and laugh at the <a href="https://tex.stackexchange.com/questions/565387/mathbb-r-is-not-showing-in-reference-bibtex">researcher who thought bibtex from Google scholar was usable</a> (<a href="https://mathstodon.xyz/@11011110/104980666583964923">\(\mathbb{M}\)</a>), but their question brings up a more serious question: why is Google’s bibtex so bad? Even the junk I get from <code class="language-plaintext highlighter-rouge">curl -LH "Accept: application/x-bibtex" http://doi.org/...</code> is mostly usable in comparison. I’m tempted to suggest that they go to MathSciNet for the good stuff but I’m worried they won’t have access.</p>
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<p><a href="https://boingboing.net/2020/09/30/ten-kinetic-sculptures-by-anne-lilly.html">Ten kinetic sculptures by Anne Lilly</a> (<a href="https://mathstodon.xyz/@11011110/104988795486796768">\(\mathbb{M}\)</a>).</p>
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<p><a href="https://jix.one/the-assembly-language-of-satisfiability/">The assembly language of satisfiability</a> (<a href="https://mathstodon.xyz/@jix/104971574457861322">\(\mathbb{M}\)</a>). Why Boolean satisfiability is too low-level to work well as a way to express the kind of problems satisfiability-solvers can solve, and how <a href="https://en.wikipedia.org/wiki/Satisfiability_modulo_theories">satisfiability modulo theories</a> can help.</p>
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<p><a href="https://cp4space.hatsya.com/2020/10/01/subsumptions-of-regular-polytopes/">Which regular polytopes have their vertices a subset of other regular polytopes in the same dimension</a> (<a href="https://mathstodon.xyz/@11011110/104998010300898992">\(\mathbb{M}\)</a>)? We don’t know! The answer is closely connected to the existence of <a href="https://en.wikipedia.org/wiki/Hadamard_matrix">Hadamard matrices</a>, which are famously conjectured to exist in dimensions divisible by four. A solution to the Hadamard matrix existence problem would also solve the polytope problem.</p>
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<p><a href="https://www.quantamagazine.org/computer-scientists-break-traveling-salesperson-record-20201008/">Computer scientists break traveling salesperson record</a> (<a href="https://mathstodon.xyz/@11011110/105006269895209659">\(\mathbb{M}\)</a>). I <a href="/blog/2020/07/15/linkage.html">linked to this back in July</a> when <a href="https://arxiv.org/abs/2007.01409">Karlin, Klein, and Gharan’s preprint</a> giving a \((1/2-\varepsilon)\)-approximation to TSP first came out, but now it’s getting wider publicity in <em>Quanta</em>. See also <a href="https://www.sciencenews.org/article/shayan-oveis-gharan-theoretical-computer-scientist-sn-10-scientists-watch">an earlier (paywalled) piece on the same story in <em>ScienceNews</em></a>.</p>
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<p>Symmetry, quasisymmetry, and kite-rhomb tessellations in the mathematical modeling of virus surface structures: <a href="https://ima.org.uk/721/fighting-infections-with-symmetry/">IMA</a>,
<a href="https://inference-review.com/article/mathematical-virology"><em>Inference</em></a>,
<a href="https://archive.bridgesmathart.org/2018/bridges2018-237.pdf">Bridges</a> (<a href="https://mathstodon.xyz/@11011110/105009372623320055">\(\mathbb{M}\)</a>).</p>
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<p><a href="https://shop.deutschepost.de/freies-quadrat-briefmarke-zu-1-70-eur-10er-bogen">New German postage stamp features the missing square puzzle</a> (<a href="https://muensterland.social/@rgx/105007333917605810">\(\mathbb{M}\)</a>, <a href="https://en.wikipedia.org/wiki/Missing_square_puzzle">see also</a>).</p>
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<p><a href="https://www.newstatesman.com/international/science-tech/2020/07/ra-fisher-and-science-hatred">R. A. Fisher and the science of hatred</a> (<a href="https://mathstodon.xyz/@11011110/105020588148970072">\(\mathbb{M}\)</a>). If you’ve been wondering why noted academics of yesteryear like <a href="https://en.wikipedia.org/wiki/Ronald_Fisher">R. A. Fisher</a> (a major figure in statistics) and <a href="https://en.wikipedia.org/wiki/David_Starr_Jordan">David Starr Jordan</a> (founding president of Stanford University) have been having their names taken off things lately, the link looks like a good explainer of their views on eugenics, and why those views are now regarded as deeply racist, even for their times.</p>
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<p><a href="http://hardmath123.github.io/minimal-surface.html">Sol LeWitt and the soapy pit</a> (<a href="https://mathstodon.xyz/@11011110/105023612862469185">\(\mathbb{M}\)</a>, <a href="https://abhikjain360.github.io/2020/08/01/The-186th-Carnival-of-Mathematics.html">via</a>, <a href="https://aperiodical.com/2020/10/carnival-of-mathematics-186/">via2</a>). LeWitt was an artist who in 1974 made a piece exhibiting all of the possible subsets of edges of the cube. The comfortably numbered blog examines what you get if you use these as frames for making soap films.</p>
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<p><a href="http://landezine.com/index.php/2013/02/funenpark-by-landlab/">Funenpark</a> (<a href="https://mathstodon.xyz/@11011110/105029637909838642">\(\mathbb{M}\)</a>). To be clear, Funenpark is not a fun-park. It is a high-density residential development on former industrial land near Amsterdam. What interests me is their <a href="https://www.flickr.com/photos/shiratski/2242870712/">pentagonal tiles</a>. It’s not one of the <a href="https://en.wikipedia.org/wiki/Pentagonal_tiling">15 monohedral pentagon tilings</a>: the tiles have two shapes, one forming half of a regular hexagon (all angles \(> 60^\circ\)) and another surrounding the hexagons (sharp angle \(= 60^\circ\)). Still, a nice pattern.</p>
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<p>Sometimes when I’ve been doing big literature searches on jstor (manually clicking on dozens of links because jstor’s search results don’t tell me which book is being reviewed, delayed by maybe a second or so per click so that I don’t get stopped by jstor’s anti-bot filters) I then get locked out of Google Scholar for a day or so on the same IP address because Google thinks I’m a bot. It doesn’t happen when I search Scholar directly. Has anyone else noticed this? Any idea how to avoid it? (<a href="https://mathstodon.xyz/@11011110/105037500352288970">\(\mathbb{M}\)</a>)</p>
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<p>While I’m linking Dutch pentagonal tiling architecture, here’s <a href="https://www.19hetatelier.nl/nieuws/wiskundige-vijfhoek-op-gevel-basisschool-de-garve-lochem/">an elementary school in Lochem decorated with the Mann–McLoud–Von Derau tile</a> (<a href="https://mathstodon.xyz/@11011110/105042753206122004">\(\mathbb{M}\)</a>, <a href="https://twitter.com/alexvdbrandhof/status/1004661466149085184">via</a>), which in 2015 became the 15th and final Euclidean monohedral pentagonal tile to be found. The link is in Dutch but Google translate works well except at one point: the school’s name, “De Garve”, means “the sheaf”, and the article remarks that this is appropriate for a pattern that looks like ears of corn.</p>
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</ul>David EppsteinMirzakhani and meanders (\(\mathbb{M}\)). On some more-than-coincidental similarities in formulas found by Mirzakhani for numbers of geodesics on hyperbolic surfaces and by Vincent Delecroix, Elise Goujard, Peter Zograf, and Anton Zorich in a new preprint for numbers of meanders, closed curves with a given number of intersections with a line.Linkage2020-09-30T17:15:00+00:002020-09-30T17:15:00+00:00https://11011110.github.io/blog/2020/09/30/linkage<ul>
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<p><a href="https://www.quantamagazine.org/a-new-algorithm-for-graph-crossings-hiding-in-plain-sight-20200915/">A new algorithm for graph crossings, hiding in plain sight</a> (<a href="https://mathstodon.xyz/@11011110/104876209471911167">\(\mathbb{M}\)</a>). Dynamic graph planarity testing, in <em>Quanta</em>. The original papers are <a href="https://arxiv.org/abs/1910.09005">arXiv:1910.09005, in SODA 2020</a> and <a href="https://arxiv.org/abs/1911.03449">arXiv:1911.03449, in STOC 2020</a>, by Jacob Holm and Eva Rotenberg.</p>
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<p><a href="https://drops.dagstuhl.de/opus/portals/lipics/index.php?semnr=16159">Fun with Algorithms proceedings, now online</a> (<a href="https://mathstodon.xyz/@11011110/104884749945714543">\(\mathbb{M}\)</a>). So if you want to read about robot bamboo trimmers, phase transitions in the mine density of minesweeper, applications of the Blaschke–Lebesgue inequality to the game of battleship, multiplication of base-Fibonacci numbers, or trains that can jump gaps in their tracks, you know where to go. The conference itself has been rescheduled to next May. Maybe by then we can actually get a trip to an Italian resort island out of it.</p>
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<p><a href="https://fractalkitty.com/2020/07/02/week-37-cantor-set-kirigami/">Cantor set kirigami</a> (<a href="https://mathstodon.xyz/@11011110/104889995571500186">\(\mathbb{M}\)</a>). One of many many mathy-craft blog posts at Fractal Kitty, which I found via <a href="https://blogs.ams.org/blogonmathblogs/2020/08/24/fractal-kitty-blog-a-tour/">the AMS math blog tour</a>.</p>
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<p>Probability theorist <a href="https://en.wikipedia.org/wiki/Nina_Holden">Nina Holden</a>, quantum complexity theorist <a href="https://en.wikipedia.org/wiki/Urmila_Mahadev">Urmila Mahadev</a>, and knot theorist <a href="https://en.wikipedia.org/wiki/Lisa_Piccirillo">Lisa Piccirillo</a> win the <a href="https://breakthroughprize.org/News/60">2021 Maryam Mirzakhani New Frontiers Prizes</a> (<a href="https://mathstodon.xyz/@11011110/104894368463336926">\(\mathbb{M}\)</a>). For more on them their work, see <a href="https://johncarlosbaez.wordpress.com/2020/09/19/the-brownian-map/">Baez on the Brownian map</a>, <a href="https://www.quantamagazine.org/graduate-student-solves-quantum-verification-problem-20181008/"><em>Quanta</em> on quantum verification</a>, and <a href="https://www.quantamagazine.org/graduate-student-solves-decades-old-conway-knot-problem-20200519/"><em>Quanta</em> on Conway’s knot problem</a>.</p>
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<p><a href="https://www.youtube.com/channel/UCIyDqfi_cbkp-RU20aBF-MQ/videos">Richard Borcherd’s YouTube channel</a> (<a href="https://mathstodon.xyz/@jsiehler/104870496696544903">\(\mathbb{M}\)</a>), “a trove of mathematical lectures at various levels”.</p>
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<p><em><a href="https://archive.org/details/gri_33125012889602">Perspectiva corporum regularium</a></em> (1568), by Wenzel Jamnitzer (<a href="https://mathstodon.xyz/@11011110/104907440380152299">\(\mathbb{M}\)</a>, <a href="https://en.wikipedia.org/wiki/Perspectiva_corporum_regularium">see also</a>). I don’t know how readable the brief medieval German text connecting the regular polyhedra to Plato’s theory of the four elements is, but the pictures of elaborated variations of the regular polyhedra can be understood in any language.</p>
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<p>Two new Wikipedia articles inspired by papers at Graph Drawing 2020 (<a href="https://mathstodon.xyz/@11011110/104911250878365185">\(\mathbb{M}\)</a>):</p>
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<p><a href="https://en.wikipedia.org/wiki/Geodetic_graph">Geodetic graph</a>, a graph in which all shortest paths are unique, inspired by “<a href="https://arxiv.org/abs/2008.07637">Drawing shortest paths in geodetic graphs</a>”</p>
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<p><a href="https://en.wikipedia.org/wiki/Kirchberger%27s_theorem">Kirchberger’s theorem</a>, that if every points in a red-blue point set are linearly separable then all of them are, inspired by “<a href="https://arxiv.org/abs/2005.12568">Topological drawings meet classical theorems from convex geometry</a>”</p>
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</ul>
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<p><a href="https://www.europeanwomeninmaths.org/ewm-open-letter-on-the-covid-19-pandemic/">European Women in Mathematics have written an open letter advocating proactive support of temporary employees, applicants, women, and parents in academia</a> (<a href="https://mathstodon.xyz/@11011110/104918402781146229">\(\mathbb{M}\)</a>, <a href="https://twitter.com/hollykrieger/status/1308375574285606913">via</a>), to forestall disproportionate losses in diversity in the wake of the covid pandemic. It’s addressed to European authorities but most of the same concerns apply more globally.</p>
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<p><a href="https://wikimediafoundation.org/news/2020/09/24/china-blocks-wikimedia-foundations-accreditation/">China blocks Wikimedia Foundation from being an observer to the World Intellectual Property Organization</a> (<a href="https://mathstodon.xyz/@11011110/104926568710157787">\(\mathbb{M}\)</a>, <a href="https://news.ycombinator.com/item?id=24588913">via</a>), apparently because it has a chapter in Taiwan.</p>
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<p><a href="https://www.iqoqi-vienna.at/blog/article/dishonesty-in-academia-the-deafening-silence-of-the-royal-society-open-science-journal-on-an-accept/">Royal Society Open Science journal publishes crank quantum paper despite negative referee reports, and has not responded to two-year-old open letter from two of the referees and several other quantum heavy hitters requesting its retraction</a> (<a href="https://mathstodon.xyz/@11011110/104927678937136375">\(\mathbb{M}\)</a>, <a href="https://news.ycombinator.com/item?id=24593465">via</a>).</p>
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<p>Michael Wehar posted a nice algorithms / fine grained complexity question on the CS theory stack exchange: <a href="https://cstheory.stackexchange.com/q/47588/95">how quickly can we test whether a 2d matrix has a square of four non-zero entries</a> (<a href="https://mathstodon.xyz/@11011110/104934034382351929">\(\mathbb{M}\)</a>)? The obvious method, looping over nonzeros and testing the squares each might be part of, is cubic when there are many nonzeros. And there can be many nonzeros without forcing a square to exist. Is there a standard hardness assumption under which strongly subcubic is impossible?</p>
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<p><a href="https://cacm.acm.org/magazines/2020/10/247584-bouncing-balls-and-quantum-computing/fulltext">The connection between the unexpected appearance of \(\pi\) in counting the bounces of billiard balls of different sizes and Grover’s algorithm for quantum search</a> (<a href="https://mathstodon.xyz/@11011110/104941135715864349">\(\mathbb{M}\)</a>): hidden constraints that keep things on a unit circle. Based on <a href="https://arxiv.org/abs/1912.02207">a preprint by Adam Brown</a>.</p>
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<p><a href="https://rjlipton.wordpress.com/2020/09/22/puzzle-reviews-by-a-puzzle-writer/">Puzzle reviews by a puzzle writer</a> (<a href="https://mathstodon.xyz/@11011110/104949202213901273">\(\mathbb{M}\)</a>). Lipton and Regan look at a few puzzles from the book <em>Bicycles or Unicycles: A Collection of Intriguing Mathematical Puzzles</em>, by Velleman and Wagon, concentrating on one that places a pebble at the origin of the positive quadrant and asks to clear a \(3\times 3\) square by moves that replace a pebble by one above and one to its left. The puzzle writer is Jason Rosenhouse, who <a href="https://www.ams.org/journals/notices/202009/rnoti-p1382.pdf">reviewed <em>Bicycles or Unicycles</em> in the <em>Notices</em></a>.</p>
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<p><a href="https://www.ams.org/journals/notices/202009/rnoti-p1397.pdf">Otto Neugebauer, famous as a historian of mathematics, also championed internationalism and diversity during Nazi times</a> (<a href="https://mathstodon.xyz/@11011110/104952629031738190">\(\mathbb{M}\)</a>, <a href="https://blogs.ams.org/beyondreviews/2020/09/28/otto-neugebauer-redux/">via</a>).</p>
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<p><a href="https://www.nature.com/articles/d41586-020-02746-y"><em>Nature</em> covers the stories of five international students and postdocs whose plans to join US academia were disrupted by Trumpist visa restrictions</a> (<a href="https://mathstodon.xyz/@11011110/104955593693842251">\(\mathbb{M}\)</a>, <a href="https://news.ycombinator.com/item?id=24634486">via</a>).</p>
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</ul>David EppsteinA new algorithm for graph crossings, hiding in plain sight (\(\mathbb{M}\)). Dynamic graph planarity testing, in Quanta. The original papers are arXiv:1910.09005, in SODA 2020 and arXiv:1911.03449, in STOC 2020, by Jacob Holm and Eva Rotenberg.