Jekyll2023-03-16T05:35:37+00:00https://11011110.github.io/blog/feed.xml11011110Geometry, graphs, algorithms, and moreDavid EppsteinLinkage for the day after π day2023-03-15T21:38:00+00:002023-03-15T21:38:00+00:00https://11011110.github.io/blog/2023/03/15/linkage-day-after<p>…although I guess \(\pi+1\) day would be April 14? My site generator doesn’t appear to like putting formulas or markup in article titles; probably that’s a good thing.</p>
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<p><a href="https://ilaba.wordpress.com/2023/03/01/vanishing-sums-of-roots-of-unity/">Which sums of roots of unity vanish</a> <span style="white-space:nowrap">(<a href="https://mathstodon.xyz/@ilaba/109951705184301710">\(\mathbb{M}\)</a>)?</span> Blog post by Izabella Łaba.</p>
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<p>If, like me, you use LaTeX <code class="language-plaintext highlighter-rouge">\newif</code> to keep a single LaTeX source that can compile to either a full paper or its shortened proceedings version, in order to ensure that both versions stay in synch as you edit them, and you find yourself fighting with the LIPIcs proceedings server software, which demands that you avoid conditionals, especially for <code class="language-plaintext highlighter-rouge">documentclass</code>, then you might find helpful <a href="https://11011110.github.io/blog/assets/2023/stripif.py">a quick and dirty Python script that I wrote to strip all the conditionals from LaTeX sources</a> <span style="white-space:nowrap">(<a href="https://mathstodon.xyz/@11011110/109962076754520051">\(\mathbb{M}\)</a>).</span> Probably someone else has a better (more robust) version of this somewhere but I didn’t find it in some quick searches.</p>
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<p><a href="http://www.aleph.se/andart/archives/2014/04/tables_of_soyga_the_first_cellular_automaton.html">Tables of Soyga: the first cellular automaton</a> <span style="white-space:nowrap">(<a href="https://mathstodon.xyz/@11011110/109969429980701555">\(\mathbb{M}\)</a>,</span> <a href="https://news.ycombinator.com/item?id=35023440">via</a>)?
Somehow I hadn’t previously seen this 2014 post by Anders Sandberg arguing that the <a href="https://en.wikipedia.org/wiki/Book_of_Soyga">Book of Soyga</a>, a 16th-century mystic text owned by John Dee, used cellular automaton based cryptography long before the modern study of cellular automata or cryptography. The Jim Reeds paper deciphering the tables from the Book of Soyga is now a deadlink, but appears likely to be “<a href="https://doi.org/10.1007/1-4020-4246-9_10">John Dee and the magic tables in the book of Soyga</a>” (<a href="http://library.pyramidal-foundational-information.com/books/The%20Book%20Of%20Soyga.pdf">alt. link</a>).</p>
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<p><a href="https://en.wikipedia.org/wiki/Logic_of_graphs">Logic of graphs</a> <span style="white-space:nowrap">(<a href="https://mathstodon.xyz/@11011110/109973961208283484">\(\mathbb{M}\)</a>),</span> my latest Wikipedia Good Article. Probably too technical to be of any interest to a general audience, but I think it’s important enough to be worth the effort of covering well despite that.</p>
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<p><a href="https://madalenaparreira.com/SPAM">Cartographic Treasures of the von Willebrand Collection</a>. Apparently actually a contemporary art installation by Constança Arouca and Madalena Parreira from Portugal? The original post from which I got this has already been deleted.</p>
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<p><a href="http://matroidunion.org/?p=4782">Triangles, arcs, and ovals</a> <span style="white-space:nowrap">(<a href="https://mathstodon.xyz/@11011110/109986552544753107">\(\mathbb{M}\)</a>).</span> Jorn van der Pol on <a href="https://en.wikipedia.org/wiki/Tuza%27s_conjecture">Tuza’s conjecture</a> relating how many disjoint triangles you can find in a graph and how many edge removals it takes to eliminate them all, and on potential matroid generalizations of the conjecture.</p>
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<p><a href="https://mathstodon.xyz/@4sphere/109984969482502198">4sphere asks</a>: How many combinatorially-different Delaunay triangulations can we make with \(n\) points in \(\mathbb{R}^d\)?</p>
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<p><a href="https://www.wired.com/story/twitter-data-api-prices-out-nearly-everyone/"><em>Wired</em> on how Twitter’s new API-access pricing makes it inaccessible to social-network researchers</a> <span style="white-space:nowrap">(<a href="https://mathstodon.xyz/@11011110/110000531783680570">\(\mathbb{M}\)</a>),</span> who have in the past relied heavily on twitter’s easy access relative to other commercial social media providers.</p>
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<p><a href="https://www.polytechnique-insights.com/en/columns/science/origami-and-kirigami-in-the-service-of-science/">Origami and kirigami in the service of science</a> <span style="white-space:nowrap">(<a href="https://mathstodon.xyz/@11011110/110009041658302275">\(\mathbb{M}\)</a>,</span> <a href="https://math.koppernigk.net/origami-kirigami-and-science/">via</a>). Mostly about how inspiration from paperfolding can be used in the design of flexible devices such as flow-control valves which perform their function passively rather than requiring active control. By Sophie Ramananarivo at the Polytechnic Institute of Paris.</p>
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<p>I discovered today that some important emails sent last month to my campus email address never got there and never showed up in my spam filter <span style="white-space:nowrap">(<a href="https://mathstodon.xyz/@11011110/110012618166046747">\(\mathbb{M}\)</a>).</span> I strongly suspect overly aggressive campus-level spam filtering. I need to continue keeping that email separate from my gmail to preserve the privacy of student coursework, but I think this means that to the extent possible all mail from off-campus needs to be directed to my gmail. I have updated <a href="https://www.ics.uci.edu/~eppstein/contact.html">my contact information page</a> accordingly; if you need to contact me by email for some reason, please adjust your contact information.</p>
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<p><a href="https://mathstodon.xyz/@zbMATH/110016022739362318">zbMATH rejects a journal</a>, not merely because its most recent issue contained dubious proofs of both the twin prime conjecture and the Riemann hypothesis, but because too many earlier issues were packed with apparently automatically generated papers by apparently automatically generated authors.</p>
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<p><a href="https://mathstodon.xyz/@beeonaposy@recsys.social/110022283260307887">Caitlin Hudon pieces a \(\pi\) quilt</a> with color-coded squares for the first 143 decimal digits of \(\pi\). (The decimal point also occupies a square.)</p>
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<p><a href="https://mathstodon.xyz/@cstheory">TCS aggregator bot for Mastodon</a>.</p>
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</ul>David Eppstein…although I guess \(\pi+1\) day would be April 14? My site generator doesn’t appear to like putting formulas or markup in article titles; probably that’s a good thing.More mathematics books by women2023-03-08T17:40:00+00:002023-03-08T17:40:00+00:00https://11011110.github.io/blog/2023/03/08/more-mathematics-books<p>For the last several years, I’ve been celebrating International Women’s Day by posting lists of mathematics books written or coauthored by women: <a href="/blog/2020/03/08/mathematics-books-women.html">2020</a>, <a href="/blog/2021/03/08/more-mathematics-books.html">2021</a>, <a href="/blog/2022/03/08/mathematics-books-by-women.html">2022</a>. Here’s another set. The links go to Wikipedia articles on the books, where you can find more information about them collated from their published reviews. The level and selection is, as usual, random, based mainly on whether the book’s topic caught my interest and it had enough published reviews to justify a Wikipedia article.</p>
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<p><em><a href="https://en.wikipedia.org/wiki/The_Geometry_of_an_Art">The Geometry of an Art: The History of the Mathematical Theory of Perspective from Alberti to Monge</a></em> (2007), Kirsti Andersen. The development of the mathematics of perspective and descriptive geometry, and its applications by European artists, from the 15th to 18th centuries.</p>
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<p><em><a href="https://en.wikipedia.org/wiki/Extrinsic_Geometric_Flows">Extrinsic Geometric Flows</a></em> (2020), Ben Andrews, Bennett Chow, Christine Guenther, and Mat Langford. A geometric flow is a way of continuously moving a curve, surface, or other shape, with the speed and direction of motion depending on its shape. It is “extrinsic” when the flow depends on a higher-dimensional space in which the moving object is embedded, rather than just on the intrinsic geometry of the object itself. This is a graduate textbook on the subject.</p>
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<p><em><a href="https://en.wikipedia.org/wiki/Spatial_Mathematics:_Theory_and_Practice_through_Mapping">Spatial Mathematics: Theory and Practice through Mapping</a></em> (2013), Sandra Arlinghaus and Joseph Kerski. The mathematical background behind geodesy and spatial visualization in geographic information systems.</p>
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<p><em><a href="https://en.wikipedia.org/wiki/Problem_Solving_Through_Recreational_Mathematics">Problem Solving Through Recreational Mathematics</a></em> (1980), Bonnie Averbach and Orin Chein. Despite the title this is an undergraduate textbook, for general education courses aimed at non-mathematics students. Its premise is that the use of fun “recreational” problems can help motivate these students to learn mathematical problem-solving techniques.</p>
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<p><em><a href="https://en.wikipedia.org/wiki/Geometric_and_Topological_Inference">Geometric and Topological Inference</a></em> (2018), Jean-Daniel Boissonnat, Frédéric Chazal, and Mariette Yvinec. Computational geometry meets machine learning.</p>
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<p><em><a href="https://en.wikipedia.org/wiki/Independence_Theory_in_Combinatorics">Independence Theory in Combinatorics: An Introductory Account with Applications to Graphs and Transversals</a></em> (1980), Victor Bryant and Hazel Perfect. An undergraduate text on matroid theory, with a particular focus on graph-theoretic applications of matroids.</p>
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<p><em><a href="https://en.wikipedia.org/wiki/Beyond_Infinity_(mathematics_book)">Beyond Infinity: An Expedition to the Outer Limits of Mathematics</a></em> (2017), Eugenia Cheng. A general-audience book looking at the many ways mathematics has approached the infinite.</p>
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<p><em><a href="https://en.wikipedia.org/wiki/The_Symmetries_of_Things">The Symmetries of Things</a></em> (2008), John Horton Conway, Heidi Burgiel, and Chaim Goodman-Strauss. A bit annoying for its frequent use of neologism and revisionist history, but packed with detail about discrete symmetries of geometric objects.</p>
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<p><em><a href="https://en.wikipedia.org/wiki/A_Biography_of_Maria_Gaetana_Agnesi">A Biography of Maria Gaetana Agnesi</a></em> (2008), Antonella Cupillari. This mainly consists of a translation of Antonio Francesco Frisi’s Italian-language biography of Agnesi, augmented with many pages of notes and with translations of some of Agnesi’s mathematical works.</p>
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<p><em><a href="https://en.wikipedia.org/wiki/Finding_Ellipses">Finding Ellipses: What Blaschke Products, Poncelet’s Theorem, and the Numerical Range Know about Each Other</a></em> (2019), Ulrich Daepp, Pamela Gorkin, Andrew Shaffer, and Karl Voss. An undergraduate-level exposition of some deep connections between functional analysis (analytic functions with specified zeros), geometry (polygons simultaneously inscribed in and circumscribing conics), and linear algebra (convex sets containing the eigenvalues of a matrix).</p>
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<p><em><a href="https://en.wikipedia.org/wiki/Introduction_to_Lattices_and_Order">Introduction to Lattices and Order</a></em> (1990, 2002), Brian A. Davey and Hilary Priestley. A graduate textbook on order theory, also noteworthy for its tips on how to use LaTeX to make order-theoretic mathematical diagrams.</p>
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<p><em><a href="https://en.wikipedia.org/wiki/The_Geometry_of_the_Octonions">The Geometry of the Octonions</a></em> (2015), Tevian Dray and Corinne Manogue. Beyond the real numbers, complex numbers, and quaternions, the next step is the octonions, a division algebra but not a ring. This book surveys this topic at an advanced undergraduate level.</p>
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<p><em><a href="https://en.wikipedia.org/wiki/The_Cube_Made_Interesting">The Cube Made Interesting</a></em> (1960, 1964), Aniela Ehrenfeucht. Aimed at high school students, and originally written in Polish, on the rotational symmetries of a cube, its colorings, and on the ability to pass a cube through a hole in an equal-sized cube (“Prince Rupert’s cube”), illustrated with red-blue anaglyphic 3d visualizations.</p>
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<p><em><a href="https://en.wikipedia.org/wiki/The_Erd%C5%91s_Distance_Problem">The Erdős Distance Problem</a></em> (2011), Julia Garibaldi, Alex Iosevich, and Steven Senger, an advanced undergraduate monograph on the problem of arranging points to make as few distinct distances as possible, unfortunately made mostly obsolete soon after its publication by the polynomial method of Larry Guth and Nets Katz.</p>
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<p><em><a href="https://en.wikipedia.org/wiki/Lumen_Naturae">Lumen Naturae: Visions of the Abstract in Art and Mathematics</a></em> (2020), Matilde Marcolli. On inspirations and analogies connecting modern art, mathematics, and mathematical physics.</p>
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<p><em><a href="https://en.wikipedia.org/wiki/Black_Mathematicians_and_Their_Works">Black Mathematicians and Their Works</a></em> (1980), Virginia Newell, Joella Gipson, L. Waldo Rich, and Beauregard Stubblefield. Brief biographies of 62 black mathematicians, and reprints of 26 of their papers on mathematics and mathematics education, maybe the only book of its kind.</p>
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<p><em><a href="https://en.wikipedia.org/wiki/From_Zero_to_Infinity">From Zero to Infinity: What Makes Numbers Interesting</a></em> (1955, …, 2006), Constance Reid. A classic of general-audience mathematics exposition, on different kinds of numbers and topics in number theory.</p>
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<p><em><a href="https://en.wikipedia.org/wiki/Math_on_Trial">Math on Trial: How Numbers Get Used and Abused in the Courtroom</a></em> (2013), Leila Schneps and Coralie Colmez. A collection of case studies on mathematical fallacies occurring in famous court cases, aimed at a general audience.</p>
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<p><em><a href="https://en.wikipedia.org/wiki/Curvature_of_Space_and_Time,_with_an_Introduction_to_Geometric_Analysis">Curvature of Space and Time, with an Introduction to Geometric Analysis</a></em> (2020), Iva Stavrov. An undergraduate textbook on differential geometry and its applications in the theory of relativity.</p>
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<p><em><a href="https://en.wikipedia.org/wiki/The_History_of_Mathematics:_A_Very_Short_Introduction">The History of Mathematics: A Very Short Introduction</a></em> (2012), Jackie Stedall. This is less an overview of the history of mathematics itself (maybe too big a topic for a short book) and more an overview of the philosophy of the history of mathematics, as demonstrated through several case studies.</p>
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<p><em><a href="https://en.wikipedia.org/wiki/Ad_Quadratum:_The_Practical_Application_of_Geometry_in_Medieval_Architecture">Ad Quadratum: The Practical Application of Geometry in Medieval Architecture</a></em> (2002), Nancy Y. Wu. An edited volume of papers on geometry in medieval architecture, mostly of Gothic cathedrals.</p>
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<p><em><a href="https://en.wikipedia.org/wiki/Do_Not_Erase:_Mathematicians_and_their_Chalkboards">Do Not Erase: Mathematicians and their Chalkboards</a></em> (2021), Jessica Wynne. A photo-essay pairing photographs of mathematician’s chalkboards with reflections on their contents by the mathematicians. I listed this in last year’s collection of books for which I could not find enough reviews, but in this case I subsequently did find them.</p>
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<p>(<a href="https://mathstodon.xyz/@11011110/109990803163597638">Discuss on Mastodon</a>)</p>David EppsteinFor the last several years, I’ve been celebrating International Women’s Day by posting lists of mathematics books written or coauthored by women: 2020, 2021, 2022. Here’s another set. The links go to Wikipedia articles on the books, where you can find more information about them collated from their published reviews. The level and selection is, as usual, random, based mainly on whether the book’s topic caught my interest and it had enough published reviews to justify a Wikipedia article.Non-crossing Hamiltonian paths and cycles in output-polynomial time2023-03-01T17:51:00+00:002023-03-01T17:51:00+00:00https://11011110.github.io/blog/2023/03/01/non-crossing-hamiltonian<p>My paper “Non-crossing Hamiltonian paths and cycles in output-polynomial time”, to appear at SoCG, is now online as a preprint at <a href="https://arxiv.org/abs/2303.00147">arXiv:2303.00147</a>. This is the full version; the SoCG version will need to be cut down by omitting proofs to reach the 500-line proceedings limit. It’s about <a href="https://en.wikipedia.org/wiki/Polygonalization">polygonalization</a>, the problem of finding all ways of connecting dots in the plane into a simple polygon (allowing connections that pass straight through a dot, but not allowing missing a dot altogether). The main results are that we can list all of these in time polynomial in the output size, and in polynomial time get an approximate count of them that is bounded above and below the true count by a polynomial of its value. Previously, the best we knew were that there were at most exponentially many polygonalizations and that we could list them in exponential time.</p>
<p>I think of this as being in the vein of recent conferences like the <a href="https://www.siam.org/conferences/cm/conference/sosa23">Symposium on Simplicity in Algorithms</a> or the new “simplicity track” of the <a href="http://esa-symposium.org/">European Symposium on Algorithms</a>: simple algorithms whose analysis isn’t. In fact, the algorithm in my paper isn’t even new. It’s the same one that was already used to achieve exponential time, in a paper “Algorithmic enumeration of surrounding polygons” by Katsuhisa Yamanaka, David Avis, Takashi Horiyama, Yoshio Okamoto, Ryuhei Uehara, and Tanami Yamauchi, published in 2021 in <em>Discrete Applied Mathematics</em> (<a href="https://doi.org/10.1016/j.dam.2020.03.034">doi:10.1016/j.dam.2020.03.03</a>).</p>
<p>If we want to list all structures, from an exponentally large family of structures, in time polynomial per structure, then I think there’s really only one idea and a lot of elaboration on that idea. The idea is: describe your structures as the vertices of a large state space, with some sort of local operation for moving from state to state; prove that this local operation suffices to connect all the states together; and then apply a graph exploration algorithm like depth-first search to find all of the states from some starting state. The trouble is, for polygonalizations, we don’t know a good local operation. The obvious candidates, local moves that replace two or three edges of a polygon by a different set of edges, <a href="/blog/2020/01/29/unflippable-polygon.html">were proven not to work</a> in a 2002 paper by Carmen Fernando, Michael Houle, and Ferran Hurtado (<a href="https://doi.org/10.1016%2FS0304-3975%2801%2900409-1">doi:10.1016/S0304-3975(01)00409-1</a>). Instead, Yamanaka et al. propose to list all of the members of a larger family of structures, and then filter out the ones that are really polygonalizations. These more general structures are the “surrounding polygons” of their paper’s title.</p>
<p>A surrounding polygon is just a simple polygon that uses some of the given dots as vertices and contains the rest. The example below is taken from the last section of my paper. There I show that point sets like the one in the illustration, with one concave chain of dots inside a triangle, have \((n-1)2^{n-4}\) polygonalizations but a polynomially-larger number of surrounding polygons proportional to \(n(1+\varphi)^n\). Here \(\varphi\) is the golden ratio; this is <a href="/blog/2020/01/12/counting-grid-polygonalizations.html">not the first occurrence of the golden ratio in counting polygonalizations</a>. A reviewer told me that these point sets are called “party-hat sets” or “ice-cream cone sets” but I’m not sure I believe it; I couldn’t find those names in a Google Scholar search.</p>
<p style="text-align:center"><img src="/blog/assets/2023/pseudotriangle.svg" alt="A set of points in the form of a triangle with a concave chain of points replacing one of its edges, and a surrounding polygon of the points. The points that are vertices of the polygon are colored blue, and the other points surrounded by the polygon are colored red." /></p>
<p>The simplest surrounding polygon of any input is just its <a href="https://en.wikipedia.org/wiki/Convex_hull">convex hull</a>. You can get from any surrounding polygon that is not the convex hull to a simpler one by “<a href="https://en.wikipedia.org/wiki/Two_ears_theorem">ear-cutting</a>”: find two consecutive edges of the polygon that form two sides of an empty triangle outside the polygon, and replace them by a single shortcut edge. The shortcutted vertex becomes surrounded, and the area of the polygon grows, so repeated ear-cutting can only stop at the convex hull, implying that all surrounding polygons are connected through the convex hull. If you choose carefully which ear to cut, you give all surrounding polygons the structure of a tree, and the algorithm of Yamanaka et al. amounts to depth-first search of this tree. You can then find the polygonalizations just by running this algorithm and outputting only the surrounding polygons that use all the dots, at some tree leaves.</p>
<p>The idea of my new paper is to analyze these structures in the style of my book, <a href="https://www.ics.uci.edu/~eppstein/forbidden/"><em>Forbidden Configurations in Discrete Geometry</em></a>, in terms of simple parameters of point sets that are monotonic (they don’t go down when you add more points) and that depend only on the order-type of the point set and not its exact coordinates. The question I set out to answer is: which point sets have only a very small number of polygonalizations, and which have many? I quickly identified two ways in which a point set could only have a small number:</p>
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<p>Most of its points could belong to a single line. If a set of \(n\) points has \(n-k\) points on a line, and only a much smaller number \(k\) of points elsewhere, then most of the edges would have to connect paths of consecutive points along the line, and there aren’t very many ways of doing that. This number \(k\) is one of the parameters studied in my book. Working out the details of this argument showed more specifically that the number of polygonalizations is \(n^{O(k)}\): there are only \(O(k)\) points of any polygonalization where something interesting happens, and only \(O(n)\) choices for what happens there.</p>
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<p>Most of its points could belong to the convex hull. If all points belong to the convex hull, then that is the only polygonalization. And if there are \(n-k\) points on the hull, and only a much smaller number \(k\) of points elsewhere, then the only points where something interesting happens are the \(O(k)\) points that are either not on the hull, or adjacent to a non-hull point. All the rest of their points have to be connected to their two hull neighbors. So again the number of polygonalizations is \(n^{O(k)}\). The parameter used here, the number of points interior to the hull, was not from my book, but maybe it should have been.</p>
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<p>More strongly, upper bounds of the same form also apply to surrounding polygons. Allowing an interesting point to be skipped by the polygon doesn’t increase its number of choices much. Consecutive blocks of uninteresting points along a long line of points must either all be skipped or all be part of a surrounding polygon, again not increasing the number of choices by much. And a surrounding polygon cannot skip any point of the convex hull, because then it would not be surrounded. The part of the analysis that I found more difficult was proving that these are the only cases. If you have points that are mostly not on a line and mostly not on a hull, then there are exponentially many polygonalizations. And if you have one of the two situations with few polygonalizations described above, then the number of polygonalizations is accurately described by the upper bounds above. For details of these lower bounds, see the paper. The number of surrounding polygons can only be at least as large as the number of polygonalizations, because every polygonalization is a surrounding polygon.</p>
<p>Once that analysis was done, the algorithms for listing polygonalizations and for approximately counting them came for free. The lower bound and the upper bound on the number of polygonalizations have the same form as each other, so they give an accurate approximation without any more effort. And the bounds on the number of polygonalizations and on the number of surrounding polygons have the same form as each other, so the analysis of the algorithm for surrounding polygons (that it takes input-polynomial time per polygon) also shows that it generates all polygonalizations in output-polynomial time.</p>
<p>The “non-crossing Hamiltonian paths” of the new paper’s title are the same thing, but easier. The easier-to-generate structures are non-crossing paths, which you can form into a forest (rooted at the one-vertex paths) by a parent operation that removes the final edge of a path. And points in convex position still have many paths; the only point sets that have a small number of non-crossing Hamiltonian paths (or non-crossing paths) are the ones with most of the points on a single line.</p>
<p>(<a href="https://mathstodon.xyz/@11011110/109951209389425592">Discuss on Mastodon</a>)</p>David EppsteinMy paper “Non-crossing Hamiltonian paths and cycles in output-polynomial time”, to appear at SoCG, is now online as a preprint at arXiv:2303.00147. This is the full version; the SoCG version will need to be cut down by omitting proofs to reach the 500-line proceedings limit. It’s about polygonalization, the problem of finding all ways of connecting dots in the plane into a simple polygon (allowing connections that pass straight through a dot, but not allowing missing a dot altogether). The main results are that we can list all of these in time polynomial in the output size, and in polynomial time get an approximate count of them that is bounded above and below the true count by a polynomial of its value. Previously, the best we knew were that there were at most exponentially many polygonalizations and that we could list them in exponential time.Linkage for 200,000 edits to Wikipedia2023-02-28T17:40:00+00:002023-02-28T17:40:00+00:00https://11011110.github.io/blog/2023/02/28/linkage-200k-edits<ul>
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<p><a href="https://www.whitehouse.gov/briefing-room/presidential-actions/2023/02/16/executive-order-on-further-advancing-racial-equity-and-support-for-underserved-communities-through-the-federal-government/">Executive Order on Further Advancing Racial Equity and Support for Underserved Communities Through The Federal Government</a> <span style="white-space:nowrap">(<a href="https://mathstodon.xyz/@sorelle@mastodon.social/109875326202971711">\(\mathbb{M}\)</a>),</span> including guidelines for equitable use of AI and automated systems through a new <a href="https://www.whitehouse.gov/ostp/ai-bill-of-rights/">Blueprint for an AI Bill of Rights</a> (that is, rights for people to be protected against unfair uses of AI, not rights for artificial intelligences).</p>
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<p><a href="https://press.princeton.edu/ideas/why-prove-it">Why prove it?</a> <span style="white-space:nowrap">(<a href="https://mathstodon.xyz/@highergeometer/109854745668334423">\(\mathbb{M}\)</a>,</span> <a href="https://www.math.columbia.edu/~woit/wordpress/?p=13288">via</a>). John Stillwell on human-written vs machine-checkable proofs, with reference to the abc conjecture.</p>
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<p><a href="https://www.ics.uci.edu/~eppstein/pix/ltcc/index.html">Low tide at Crystal Cove State Park</a> <span style="white-space:nowrap">(<a href="https://mathstodon.xyz/@11011110/109894403539276698">\(\mathbb{M}\)</a>).</span></p>
<p style="text-align:center"><img src="https://www.ics.uci.edu/~eppstein/pix/ltcc/Seagrass2-m.jpg" alt="Low tide at Crystal Cove State Park, California" style="border-style:solid;border-color:black" /></p>
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<p>Another newly promoted Wikipedia Good Article: <a href="https://en.wikipedia.org/wiki/Polygonalization">Polygonalization</a> <span style="white-space:nowrap">(<a href="https://mathstodon.xyz/@11011110/109906715996859674">\(\mathbb{M}\)</a>),</span> about finding a polygon that uses all of a given set of points as vertices. The usual definitions allow it to go straight through some of the vertices, rather than always turning, though, and the illustration below shows why: for some point sets, including 3x3 grids, a polygon that turns everywhere might not exist.</p>
<p style="text-align:center"><img src="/blog/assets/2023/3x3_grid_polygonalizations.svg" alt="Eight ways of polygonalizing a 3x3 grid" /></p>
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<p><a href="https://terrytao.wordpress.com/2023/02/18/would-it-be-possible-to-create-a-tool-to-automatically-diagram-papers/">Would it be possible to create a tool to automatically diagram papers?</a> <span style="white-space:nowrap">(<a href="https://mathstodon.xyz/@tao/109887019658810502">\(\mathbb{M}\)</a>),</span> by Terry Tao, inspired by the diagrams the proof-assistant people have been using to guide their work.</p>
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<p>People who indulge in the fringe belief in the reality of certain folklore beasts are sad that <a href="https://boingboing.net/2023/02/22/the-cryptid-complications-of-wikipedias-editing-policies.html">Wikipedia now focuses on the folklore of these beasts without going into much detail about the fringe belief in their reality</a> <span style="white-space:nowrap">(<a href="https://mathstodon.xyz/@11011110/109918360457067075">\(\mathbb{M}\)</a>).</span> (Based on a both-sides-ist <em>Slate</em> article that I’m not going to link.)</p>
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<p><a href="https://www.youtube.com/watch?v=MDhT6-6Yr_I">Origami actuators</a> <span style="white-space:nowrap">(<a href="https://mathstodon.xyz/@logicalelegance@mastodon.online/109920746034435458">\(\mathbb{M}\)</a>),</span> for simple repetitive motions of origami models, by attaching flat-printed electromagnets to them.</p>
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<p>Gasarch writes: <a href="https://blog.computationalcomplexity.org/2023/02/it-is-more-important-than-ever-to-teach.html">It is more important than ever to teach your students probability (even non-stem students)</a> <span style="white-space:nowrap">(<a href="https://mathstodon.xyz/@11011110/109935523877122235">\(\mathbb{M}\)</a>).</span> Why: because your university may be making deals promoting online gambling to the same students, as the linked copy of a New York Times article details.</p>
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<p><a href="https://xtools.wmflabs.org/ec/en.wikipedia.org/David_Eppstein">Sometime in the last month (not exactly sure when) I passed the milestone of 200,000 edits (all non-automated) to Wikipedia</a> <span style="white-space:nowrap">(<a href="https://mathstodon.xyz/@11011110/109941265893054592">\(\mathbb{M}\)</a>).</span> That’s…a lot of edits. Although, as of earlier in the month when it was below 200,000, it only places me at 260 on the <a href="https://en.wikipedia.org/w/index.php?title=Wikipedia:List_of_Wikipedians_by_number_of_edits&oldid=1138516223">list of all-time prolific editors</a>. And a couple of the top ten are now blocked, so it’s not exactly always a place of pride.</p>
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<p><a href="https://www.thisiscolossal.com/2023/02/zai-divecha-phase-shift/">Mesmerizing paper sculptures and animations by Zai Divecha convey the subtlety of change</a> <span style="white-space:nowrap">(<a href="https://mathstodon.xyz/@colossal@mastodon.art/109937307601608046">\(\mathbb{M}\)</a>,</span> <a href="https://zaidivecha.com/">see also</a>). Basically a 3d papercraft zoetrope.</p>
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</ul>David EppsteinExecutive Order on Further Advancing Racial Equity and Support for Underserved Communities Through The Federal Government (\(\mathbb{M}\)), including guidelines for equitable use of AI and automated systems through a new Blueprint for an AI Bill of Rights (that is, rights for people to be protected against unfair uses of AI, not rights for artificial intelligences).Isohedral Delaunay complexes2023-02-25T09:19:00+00:002023-02-25T09:19:00+00:00https://11011110.github.io/blog/2023/02/25/isohedral-delaunay-complexes<p>The Delaunay complex of a set of points in the Euclidean plane partitions the convex hull of the points into polygonal cells. Each cell is the convex hull of a co-circular subset of the points whose circle does not contain any more points. It’s often called a <a href="https://en.wikipedia.org/wiki/Delaunay_triangulation">Delaunay triangulation</a>, because for points in <a href="General position">general position</a> the cells are all triangles, but I do not want to assume general position here. It is <a href="https://en.wikipedia.org/wiki/Isohedral_figure">isohedral</a> when all of the cells are symmetric to each other (maybe a little more strong than asking for them all to have the same shape). For example, the familiar tilings of the plane by squares or regular hexagons are both isohedral and Delaunay. Another example is a <a href="https://en.wikipedia.org/wiki/Deltoidal_trihexagonal_tiling">tiling of the plane by 60°–90°–120° kites</a>:</p>
<p style="text-align:center"><img src="/blog/assets/2023/tetrille-delaunay.svg" alt="Tiling of the plane by 60°–90°–120° kites, with shading showing that the circumcircles of each site are empty of other tiling vertices" style="width:100%;max-width:720px" /></p>
<p>Some other tilings, even very symmetric ones, might not be Delaunay. For instance, it is impossible to make a Delaunay version of the <a href="https://en.wikipedia.org/wiki/Cairo_pentagonal_tiling">Cairo pentagonal tiling</a> because its tiles have two complementary angles or two right angles, impossible for a co-circular pentagon.</p>
<p>In these cases, the symmetries are of the familiar kind, translations and rotations of the plane. But translation symmetry forces us to use infinitely many points. Can finite Delaunay complexes be isohedral? Sort of, maybe, but with a different kind of symmetry.
You can translate between Delaunay complexes on the plane and on a sphere by <a href="https://en.wikipedia.org/wiki/Stereographic_projection">stereographic projection</a>, and translations, rotations, and scaling in the plane become Möbius transformations on the sphere. So the projection onto the sphere of a square grid becomes a spherical Delaunay complex that is symmetric under Möbius transformations.</p>
<p style="text-align:center"><img src="/blog/assets/2023/stereographic-square-tiling.svg" alt="Stereographic projection of a square grid from the plane to a sphere" title="CC-BY-SA 4.0 image https://commons.wikimedia.org/wiki/File:Stereogr-proj-netz.svg by Ag2gaeh from Wikimedia commons" style="width:100%;max-width:720px" /></p>
<p>Rotations of the sphere are also a very special case of Möbius transformations, so we can look for Delaunay complexes with rotational symmetries. Suppose you have a polyhedron all of whose vertices lie on a sphere, and all of whose faces are symmetric to each other by rotations of the sphere. Then the intersection of the sphere with any face plane of the polyhedron is a circle through the vertices of a face that does not contain any other vertices, the defining property of a Delaunay cell. So these polyhedra are isohedral spherical Delaunay complexes. This is true, for instance, for the Platonic solids and for the two infinite families of <a href="https://en.wikipedia.org/wiki/Bipyramid">bipyramids</a> and the <a href="https://en.wikipedia.org/wiki/Trapezohedron">trapezohedra</a> but false for some other isohedral polyhedra like the <a href="https://en.wikipedia.org/wiki/Rhombic_dodecahedron">rhombic dodecahedron</a> and <a href="https://en.wikipedia.org/wiki/Triakis_tetrahedron">triakis tetrahedron</a> whose vertices cannot all be placed on a sphere.</p>
<p>You can map these spherical Delaunay complexes back onto the plane by stereographic projection again. You might think that the result is always a planar Delaunay complex in which all faces are symmetric to each other under Möbius transformation, but there’s a catch. The projection preserves circles, but it turns inside out the ones that contain the pole of the projection. If they were empty on the sphere, they instead turn into circles in the plane that contain every other point. These inside-out circles correspond to Delaunay cells on the sphere that do not map to Delaunay cells in the plane. For instance, projecting the cube vertices back down to the plane with the pole at the midpoint of a cube edge produces a Delaunay complex with only four quadrilaterals; the other two faces of the cube come from inside-out circles and do not become Delaunay cells.</p>
<p style="text-align:center"><img src="/blog/assets/2023/cube-edge-projection.svg" alt="Delaunay complex of a cube, stereographically projected onto the plane with its pole at an edge midpoint" style="width:100%;max-width:720px" /></p>
<p>All of this generalizes directly to 3d Delaunay triangulations, and to isohedral 4d polytopes with cospherical vertices, but less is known about what shapes are possible. The regular 4-polytopes, certainly, have symmetric facets and cospherical vertices, but there are other possibilities as well. The <a href="http://www.polytope.net/hedrondude/dice4.htm">isohedral 4-polytopes with up to 20 sides</a> have been classified, but I don’t know which of these can have cospherical vertices.</p>
<p>There are, at least, three different infinite families of isohedral 4d polytopes with cospherical vertices, analogous to the bipyramids and trapezohedra. To describe this, it helps to think of four-dimensional Euclidean space as having two complex numbers \(\alpha\) and \(\beta\) as coordinates, and the unit sphere as the points for which \(\vert\alpha\vert^2+\vert\beta\vert^2=1\). These are the state vectors of a <a href="https://en.wikipedia.org/wiki/Qubit">qubit</a>, so we can write these points on the sphere using <a href="https://en.wikipedia.org/wiki/Bra%E2%80%93ket_notation">quantum notation</a> as \(\alpha\,\vert0\rangle+\beta\,\vert1\rangle\), where \(\vert0\rangle\) and \(\vert1\rangle\) are just the two basis vectors for the two-complex-number coordinate system. In this notation, consider the following three sets of points, all on the unit sphere, for integer parameters \(n\) and \(m\):</p>
<ul>
<li>
<p>Let \(X\) be the set of \(n\) points \(e^{2\pi i/n}\,\vert0\rangle\), for the integers \(i\) with \(0\le i\lt n\). These form a regular \(n\)-gon in the plane \(\beta=0\).</p>
</li>
<li>
<p>Let \(Y\) be the set of \(m\) points \(e^{2\pi j/m}\,\vert1\rangle\), for the integers \(j\) with \(0\le j\lt n\). These form a regular \(m\)-gon, in the perpendicular plane \(\alpha=0\).</p>
</li>
<li>
<p>Let \(Z\) be the set of \(mn\) points</p>
\[\frac{1}{\sqrt 2}e^{2\pi i/n}\,\vert0\rangle + \frac{1}{\sqrt 2}e^{2\pi j/m}\,\vert1\rangle,\]
<p>for the same ranges of \(i\) and \(j\). These lie on a <a href="https://en.wikipedia.org/wiki/Flat_torus">flat torus</a>, the Cartesian product of two circles, and form the vertices of a tiling of the torus by rectangles.</p>
</li>
</ul>
<p>Then the convex hull of \(X\cup Y\) has as its facets \(mn\) congruent tetrahedra, each formed as the convex hull of an edge of the \(X\)-polygon and an edge of the \(Y\)-polygon. The convex hull of \(Z\) is a <a href="https://en.wikipedia.org/wiki/Duoprism">duoprism</a> whose facets are two kinds of prisms: the Cartesian product of an edge of the \(X\)-polygon with the whole \(Y\)-polygon, and vice versa. When \(n=m\) these two prisms are congruent and the resulting duoprism is isohedral, and dual to the convex hull of \(X\cup Y\). Here is a stereographic projection for \(n=m=18\), taken from the <a href="https://www.math.cmu.edu/~fho/jenn/polytopes/index.html">Jenn 3d website</a>:</p>
<p style="text-align:center"><img src="/blog/assets/2023/18x18-torus.png" alt="Stereographic projection into 3d of a 4-dimensional polytope, the (18,18)-duoprism, appearing as a torus tiled with squares" title="Public domain image https://www.math.cmu.edu/~fho/jenn/polytopes/18x18-torus.png" style="width:100%;max-width:720px" /></p>
<p>In this image, the most prominent feature is the tiling by squares of the torus containing \(Z\). If you follow sequences of edges of this square grid, through opposite edges at each vertex, you will also see many 18-gons. Some of the 18-gons slice the “inside” of the torus radially into distorted prisms; these are Delaunay cells. Many of the perpendicular 18-gons slice across the “donut hole” of the torus, forming more Delaunay cells. But some of the remaining 18-gons lie on the convex hull of the shape, and cannot be used as slices for the projected set. The missing slices cause the Delaunay triangulation of the stereographic projection to miss some cells, and that can only happen because the spheres for these cells were inverted by the projection.</p>
<p>You can also take the convex hull of \(X\cup Y\cup Z\). This has two triangular-prism facets for each tetrahedron of \(X\cup Y\), meeting at one of the squares of \(Z\). The reason I’m interested in this example comes from <a href="/blog/2023/02/20/geometric-graphs-unbounded.html">my most recent post, on flip-width of geometric graphs</a>. If you take an induced subgraph of this polytope, consisting only of the points in \(X\cup Y\cup Z\) whose coefficients \(i\) and \(j\) are both even, the result is a subdivided complete bipartite graph \(K_{n,n}\), where by “subdivided” I mean that each edge of \(K_{n,n}\) has been replaced by a two-edge path. This isn’t an interchange, in the sense of the previous post, but it has unbounded flip-width, because it is a sparse graph that does not have bounded expansion.</p>
<p>What I really want, though, is a 3d Euclidean Delaunay triangulation with unbounded flip-width, not a non-triangulation complex and not a 4-polytope (I already had one of those in my previous post). To get this, use a stereographic projection whose pole is on the central torus, in the middle of one of the squares (or really on the corresponding point of the unit sphere), and note that the Delaunay spheres of the polytope faces will intersect this torus in Delaunay circles of the squares. But for a square grid, the center of each square belongs only to the circumcircle of that square, not to any of the other circumcircles. So the pole of the projection will only belong to two of the Delaunay spheres, the two sharing the chosen square. The two prisms for these two spheres will be missing from the Delaunay complex (instead, their union, some sort of <a href="https://en.wikipedia.org/wiki/Gyrobifastigium">gyrobifastifium</a>, will form the convex hull of the points), but all the other prisms will still be present. They contain all the edges of the graph, so it still contains a large induced subdivided biclique. Perturbing the points slightly to get a triangulation rather than a complex doesn’t change this.</p>
<p>(<a href="https://mathstodon.xyz/@11011110/109926574982696332">Discuss on Mastodon</a>)</p>David EppsteinThe Delaunay complex of a set of points in the Euclidean plane partitions the convex hull of the points into polygonal cells. Each cell is the convex hull of a co-circular subset of the points whose circle does not contain any more points. It’s often called a Delaunay triangulation, because for points in general position the cells are all triangles, but I do not want to assume general position here. It is isohedral when all of the cells are symmetric to each other (maybe a little more strong than asking for them all to have the same shape). For example, the familiar tilings of the plane by squares or regular hexagons are both isohedral and Delaunay. Another example is a tiling of the plane by 60°–90°–120° kites:Geometric graphs with unbounded flip-width2023-02-20T21:20:00+00:002023-02-20T21:20:00+00:00https://11011110.github.io/blog/2023/02/20/geometric-graphs-unbounded<p>At the recent <a href="https://wogag.org/">Workshop on Geometry and Graphs</a> in Barbados, most of the technical activity involved working in small groups on research problems, but there was also a nice survey talk by <a href="https://web.math.princeton.edu/~rm1850/">Rose McCarty</a> on flip-width.<sup id="fnref:r" role="doc-noteref"><a href="#fn:r" class="footnote" rel="footnote">1</a></sup> This is a new and very general notion of width in graphs, introduced by <a href="https://www.mimuw.edu.pl/~szymtor/">Szymon Toruńczyk</a>.<sup id="fnref:t" role="doc-noteref"><a href="#fn:t" class="footnote" rel="footnote">2</a></sup> It is defined in terms of a certain cops-and-robbers game on graphs, and intended to capture the structure inherent in many types of graphs and to unify <a href="https://en.wikipedia.org/wiki/Bounded_expansion">bounded expansion</a> and <a href="https://en.wikipedia.org/wiki/Twin-width">twin-width</a>.</p>
<p>For instance, many algorithmic graph problems, such as searching for small patterns in larger graphs (“<a href="https://en.wikipedia.org/wiki/Subgraph_isomorphism_problem">subgraph isomorphism</a>”) can be formulated more abstractly in terms of of checking whether a graph models a given formula in the first-order <a href="https://en.wikipedia.org/wiki/Logic_of_graphs">logic of graphs</a>. Such problems are <a href="https://en.wikipedia.org/wiki/Parameterized_complexity">fixed-parameter tractable</a> when parameterized either by expansion or twin-width, and it is hoped that the same thing will extend to flip-width. Very recent partial results in this direction extend model checking algorithms from bounded expansion to “structurally nowhere dense classes”,<sup id="fnref:dms" role="doc-noteref"><a href="#fn:dms" class="footnote" rel="footnote">3</a></sup> but these classes do not even include everything with bounded twin-width, let alone flip-width.</p>
<p>For the purposes of this post, the only thing we need to know about bounded expansion is that graph families with this property must be sparse: in their graphs, the number of edges must be at most linear in the number of vertices.<sup id="fnref:no" role="doc-noteref"><a href="#fn:no" class="footnote" rel="footnote">4</a></sup> On the other hand, although graph families with bounded twin-width can be dense, they are limited in a different way: the number of graphs in the family, on a set of \(n\) unlabeled vertices, can only be singly exponential <span style="white-space:nowrap">in \(n\).<sup id="fnref:st" role="doc-noteref"><a href="#fn:st" class="footnote" rel="footnote">5</a></sup></span> One way to get a family of graphs that have bounded flip-width but not bounded expansion nor bounded twin-width is to take the union of two families, one dense with bounded twin-width and the other numerous with bounded expansion. For instance, take the graphs that are either <a href="https://en.wikipedia.org/wiki/Cograph">cographs</a> or <a href="https://en.wikipedia.org/wiki/Cubic_graph">3-regular</a>. But this is not a very natural family of graphs. Rose asked: is there a natural family of graphs with bounded flip-width but unbounded twin-width and expansion? For instance, there are many standard types of geometric graphs for which the twin-width and expansion are unbounded; could any of these have bounded flip-width?</p>
<h1 id="cops-and-robbers">Cops and robbers</h1>
<p>Like <a href="https://en.wikipedia.org/wiki/Treewidth">treewidth</a> and bounded expansion, flip-width can be defined using a certain <a href="https://en.wikipedia.org/wiki/Pursuit%E2%80%93evasion">cops-and-robbers game</a> on graphs.<sup id="fnref:t:1" role="doc-noteref"><a href="#fn:t" class="footnote" rel="footnote">2</a></sup></p>
<p>The games used for treewidth and expansion involve “cops with helicopters”, chasing a robber on a given graph. At each point in the game, the cops occupy a limited number of graph vertices. Then, at each move, the cops announce where they will move next, the robber moves to escape them along a path through the currently-unoccupied vertices, and then the cops fly directly to their new locations. The cops win if one of them lands on the robber’s current vertex, and the robber wins by evading the cops indefinitely. The treewidth of a graph is the maximum number of cops that a robber can evade, moving arbitrarily far on each move. A family of graphs has bounded expansion if and only if there is some function \(f\) such that only \(f(r)\) cops are needed to catch a robber who can move at most \(r\) steps per move.<sup id="fnref:t2" role="doc-noteref"><a href="#fn:t2" class="footnote" rel="footnote">6</a></sup></p>
<p>As Rose described, the same game can be described in a different way. Instead of occupying a vertex, the cops set up roadblocks on all the edges incident to it. On each move, the cops announce which vertices will be blockaded next. Then, the robber moves along un-blockaded edges. Finally, the cops remove their current blockades and put up new blockades at the vertices they announced. The cops win by leaving the robber at an isolated vertex, unable to move.</p>
<p style="text-align:center"><img src="/blog/assets/2023/roadblock.jpg" alt="Police roadblock in Washington, DC, January 15, 2021" title="CC-BY image by Mike Licht from Wikimedia commons, File:Inaugural preparation, January 15th Roadblock (50840138737).jpg" style="width:100%;max-width:540px" /></p>
<p>Flip-width is defined in the same way, but with more powerful cops. Instead of blockading a single vertex, they are allowed to perform a “flip” of a subset of vertices. This complements the subgraph within that subset: pairs of vertices that were connected become disconnected, and vice versa. So blockading a single vertex, for instance, can be accomplished by two flips: one flip of the vertex and its neighbors, and one flip of just the neighbors. The first flip disconnects the given vertex, and the second flip restores the original connectivity among the neighboring vertices. It doesn’t matter in which order these two flips (or any set of flips) is performed.</p>
<p style="text-align:center"><img src="/blog/assets/2023/flip-isolate.svg" alt="Isolating a vertex by two flips" style="width:100%;max-width:720px" /></p>
<p>In the flipping game used to define flip-width, at any point in the game, the cops will have performed some limited number of flips. Then in each move, the cops announce which sets of vertices they will flip next. The robber moves along a path in the current flipped graph, to evade these flips. Then, the cops undo their current flips and perform the new flips that they announced. The cops win if they leave the robber at an isolated vertex, unable to move, and the robber wins by avoiding this fate indefinitely. A family of graphs has bounded flip-width if there is some function \(f\) such that only \(f(r)\) flips per move are needed to catch a robber who can move at most \(r\) steps per move.</p>
<p>For the purposes of having bounded flip-width, two graphs that differ from each other only by a finite number \(\varphi\) of flips are essentially the same. If the cops can win on one, they can win on the other with only \(\varphi\) more flips per move. They only need to start by performing those \(\varphi\) flips to convert the second graph into the first one, and then leave those flips in place while they perform the winning strategy on the converted graph. So, for instance, the graphs that differ by a single flip from a 3-regular graph have bounded flip-width, but are again not a very natural class of graphs.</p>
<h1 id="interchanges">Interchanges</h1>
<p>Continuing the road network metaphor, and in the spirit of the <a href="https://en.wikipedia.org/wiki/Haven_(graph_theory)">havens</a> used to model escape strategies in the treewidth game, let’s define a structure I call an <em>interchange</em>, having pairwise connections between many points, which a robber can use to make a getaway from few enough cops.</p>
<p style="text-align:center"><img src="/blog/assets/2017/HighFive.jpg" alt="High Five Interchange at the intersection of I-635 and U.S. Route 75 in Dallas, Texas, looking towards the southwest" title="cropped from https://commons.wikimedia.org/wiki/File:High_Five.jpg by fatguyinalittlecoat on flickr, under a CC-BY 2.0 license" style="width:100%;max-width:540px" /></p>
<p>More precisely, define an interchange of order \(n\) to consist of the following components:</p>
<ul>
<li>
<p>Certain designated vertices, which we call <em>lanes</em>. The interchange should have \(n\) lanes, arranged into a sequence. These are colored blue in the following illustrations.</p>
</li>
<li>
<p>More designated vertices, called <em>ramps</em>. Each ramp is associated with a pair of lanes. When two lanes are \(n-3\) or fewer steps apart in the sequence, they have an associated ramp. (We don’t require ramps for pairs of the outermost lanes because they would not be helpful to the robber in the game.) The ramps are colored red in the following illustrations.</p>
</li>
<li>
<p>An edge between each ramp and its two associated lanes.</p>
</li>
<li>
<p>Optional edges between any two lanes or between any two ramps. These will be unused by the robber and do not affect the robber’s strategy. The optional edges mean that the class of all interchanges is huge, too large to have bounded twin-width. But more importantly for us, they allow us to construct geometric realizations of these graphs without worrying about whether or not the construction causes certain pairs of vertices to become adjacent.</p>
</li>
<li>
<p>For a ramp that connects lanes \(x\) and \(y\), optional edges to other lanes between \(x\) and \(y\) in the sequence (edges to lanes outside that range are not allowed).</p>
</li>
</ul>
<p>The image below shows an example, with the lanes blue, ramps red, optional edges yellow, and required edges black. The blue lanes are ordered in a sequence from left to right, but otherwise the placement of vertices is not meaningful; it’s the graph structure that matters.</p>
<p style="text-align:center"><img src="/blog/assets/2023/5-interchange.svg" alt="Interchange of order 5" style="width:100%;max-width:540px" /></p>
<p>As we show next, large-enough interchanges can be used by the robber to escape any fixed number of cops.</p>
<h1 id="escaping-through-junctions">Escaping through junctions</h1>
<p>Call a set of lanes <em>equivalent</em>, after certain flips have been made, if they are all treated the same by each flip: all included in the flipped set, or all excluded.
Define a <em>junction</em> to be a triple of equivalent lanes that are connected to each other by paths through one or two ramps, after the flips are made. Then:</p>
<ul>
<li>
<p>Every four equivalent lanes have at least one junction. For, if the lanes are \(a,b,c,d\) (in sequence order) then the <span style="white-space:nowrap">\(a\)–\(b\)</span> ramp either continues to connect \(a\) to \(b\), or it is flipped and instead connects <span style="white-space:nowrap">\(c\) to \(d\).</span> A third connection is provided either by the <span style="white-space:nowrap">\(b\)–\(c\)</span> ramp or its flip, which connects <span style="white-space:nowrap">\(a\) to \(d\).</span></p>
</li>
<li>
<p>In every six equivalent lanes, at least three of the lanes belong to two otherwise-disjoint junctions. I’ll skip the messy case analysis showing this.</p>
</li>
<li>
<p>Every two junctions are connected by at least one ramp between two of their lanes. If the junctions are \(a,b,c\) and \(d,e,f\), listed in the sequence order of all the lanes, then they have either \(a,b,e,f\) or \(d,e,b,c\) as a subsequence (depending on the ordering between <span style="white-space:nowrap">\(b\) and \(e\)).</span> In either case they are connected by the <span style="white-space:nowrap">\(b\)–\(e\)</span> ramp or its flip.</p>
</li>
<li>
<p>By the same argument, every two triples of equivalent lanes are connected by at least one ramp.</p>
</li>
</ul>
<p>These connections imply that, in an interchange that is big enough to guarantee the existence of junctions, the robber can win by always moving to a lane that will become part of a junction after the announced flips happen.</p>
<p>In more detail, suppose that the cops and a robber play the flipping game, with \(t\) flips per move and \(r\ge 6\), and that the graph includes an interchange of <span style="white-space:nowrap">order \(3\cdot 2^{2t}+1\).</span> This interchange is big enough to ensure that some four lanes are equivalent both in the current and announced set of flips. These four lanes include a junction under the announced flips. The robber can move to this new junction using at most two ramps within the current junction and then one more ramp to cross between the two triples of lanes. With an interchange that is a little larger, of <span style="white-space:nowrap">order \(5\cdot 2^{2t}+1\),</span> the robber can win with \(r\ge 4\), by moving to a lane that will become part of two otherwise-disjoint junctions, so that two other equivalent lanes will be reachable by only one ramp.</p>
<p>This strategy shows that, if a graph class contains arbitrarily large interchanges, it does not have bounded flip-width. We will use this idea to show that many natural classes of geometric graphs do not have bounded flip-width.</p>
<h1 id="geometric-graphs">Geometric graphs</h1>
<p>In each of the following types of geometric graph, it is possible to form arbitrarily large interchanges, as illustrated.</p>
<ul>
<li>
<p><a href="https://en.wikipedia.org/wiki/Interval_graph">Interval graphs</a> and <a href="https://en.wikipedia.org/wiki/Permutation_graph">permutation graphs</a>. Just make a left-to-right sequence of small disjoint blue intervals for the lanes, and connect them by longer red intervals for the ramps. Each red interval contains all of the blue intervals that it intersects, and permutation graphs are the same thing as <a href="https://www.graphclasses.org/classes/gc_288.html">interval containment graphs</a>. In contrast, the <a href="https://en.wikipedia.org/wiki/Indifference_graph">unit interval graphs</a> are known to have bounded twin-width,<sup id="fnref:tw3" role="doc-noteref"><a href="#fn:tw3" class="footnote" rel="footnote">7</a></sup> from which it follows that they also have bounded flip-width.</p>
<p style="text-align:center"><img src="/blog/assets/2023/interval-interchange.svg" alt="Interval graph forming an interchange of order 6" /></p>
</li>
<li>
<p>Circle graphs. These have the permutation graphs as a special case, but there’s also a direct construction.</p>
<p style="text-align:center"><img src="/blog/assets/2023/circle-interchange.svg" alt="Circle graph forming an interchange of order 6" style="width:100%;max-width:540px" /></p>
</li>
<li>
<p>Intersection graphs of axis-aligned line segments, no two collinear. Use long horizontal segments for the lanes, ordered vertically, and span them by vertical ramps.</p>
<p style="text-align:center"><img src="/blog/assets/2023/line-segment-interchange.svg" alt="Axis-aligned line segments forming an interchange of order 6" /></p>
</li>
<li>
<p>Intersection graphs of axis-parallel unit squares. Place the blue lane squares with their top right corners on a diagonal line, close enough together that any consecutive interval of them can be covered by a red ramp square.</p>
<p style="text-align:center"><img src="/blog/assets/2023/square-interchange.svg" alt="Squares forming an interchange of order 6" style="width:100%;max-width:540px" /></p>
</li>
<li>
<p>Unit disk graphs. This one is unfortunately difficult to see clearly because the details are tiny with respect to the overall form, even for the \(n=6\) example shown. Anyway, place \(n\) blue unit disks tangent to the outside of a circle of radius \(1+\varepsilon\) (yellow in the figure), so that their points of tangencies span an arc of diameter less than \(2\). Then place red unit disks with their centers inside the yellow circle, so that their intersections with the circle form arcs that look like the interval graph model above. Because their radius is smaller than the yellow circle, the red disks will bulge out of the yellow circle a little bit. They intersect the blue points of tangency in the pattern that we want, but the parts that bulge out may have some unwanted contacts with the other blue disks. To prevent this, make \(\varepsilon\) very small. As you decrease \(\varepsilon\), the red bulges will shrink towards the yellow circle, but the blue disks won’t change their positions or angles very much, so for sufficiently small values of \(\varepsilon\) there will be no unwanted contacts.</p>
<p style="text-align:center"><img src="/blog/assets/2023/disk-interchange.svg" alt="Unit disks forming an interchange of order 6" /></p>
</li>
<li>
<p>Unit distance graphs. Place the blue vertices equally spaced along an interval of length less than two and the red vertices that connect them on the points where unit circles centered on the blue vertices cross each other.</p>
<p style="text-align:center"><img src="/blog/assets/2023/unit-distance-interchange.svg" alt="Unit distance graph forming an interchange of order 6" /></p>
</li>
<li>
<p>Visibility graphs of simple polygons. Place the blue vertices in sequence order on a horizontal line, the red vertices that connect pairs of consecutive blue vertices in order on a parallel line above them, and the remaining red vertices in an arbitrary order on a parallel line below them. Draw a triangle between each red vertex and the two blue vertices it should connect, and take the union of the triangles. Fill any holes that might have been formed in taking the union. The resulting polygon has additional vertices but that doesn’t affect the existence of an interchange. (This construction is simplified from an earlier construction by Rose, of visibility graphs that can be flipped to contain subdivisions of complete graphs.)</p>
<p>Visibility graphs are <a href="https://en.wikipedia.org/wiki/Cop-win_graph">cop-win graphs</a>, meaning that a single cop wins a different cop-and-robber game in which both players can either move along a graph edge or stand still.<sup id="fnref:lsv" role="doc-noteref"><a href="#fn:lsv" class="footnote" rel="footnote">8</a></sup> But this doesn’t say anything about the flipping game: any graph can be made into a cop-win graph by adding a single <a href="https://en.wikipedia.org/wiki/Universal_vertex">universal vertex</a>, without changing whether it has bounded flip-width.</p>
<p style="text-align:center"><img src="/blog/assets/2023/visibility-interchange.svg" alt="Polygon whose visibility graph forms an interchange of order 6" /></p>
</li>
<li>
<p>Four-dimensional convex polytopes. I’m not even going to try to draw this one, but the construction is easy to describe in words. Just take the <a href="https://en.wikipedia.org/wiki/Barycentric_subdivision">barycentric subdivision</a> of a <a href="https://en.wikipedia.org/wiki/Neighborly_polytope">neighborly polytope</a>. Neighborly polytopes have edges and vertices forming complete graphs; the barycentric subdivision of any polytope is another polytope.<sup id="fnref:es" role="doc-noteref"><a href="#fn:es" class="footnote" rel="footnote">9</a></sup> It has a vertex for each face of the original polytope, and an edge for each incidence between faces of different dimensions. Arrange the vertices of the neighborly polytope into an arbitrary sequence as lanes; use the subdivision vertices coming from its edges as ramps. In this way the ramps will be connected only to their two associated lanes and to other subdivision points, but not to any other lanes.</p>
</li>
</ul>
<h1 id="where-now">Where now?</h1>
<p>We’re still missing a natural class of graphs with bounded flip-width, unbounded twin-width, and unbounded expansion. The known classes of geometric graphs looked promising as a direction to look for such classes, but these constructions rule that out in surprisingly many cases.</p>
<p>It still might be possible that the number of cops needed to catch a robber on these graphs could be low. The interchange construction only proves that it is at least logarithmic. But I don’t know of any useful algorithmic consequences of having a low but unbounded number of cops needed to catch a bounded-speed robber.</p>
<h1 id="notes-and-references">Notes and references</h1>
<div class="footnotes" role="doc-endnotes">
<ol>
<li id="fn:r" role="doc-endnote">
<p>Rose also helped me edit a preliminary version of this post. Thanks, Rose! Any remaining errors are my fault. <a href="#fnref:r" class="reversefootnote" role="doc-backlink">↩</a></p>
</li>
<li id="fn:t" role="doc-endnote">
<p>Szymon Toruńczyk (2023), “Flip-width: cops and robber on dense graphs”, <a href="https://arxiv.org/abs/2302.00352">arXiv:2302.00352</a> <a href="#fnref:t" class="reversefootnote" role="doc-backlink">↩</a> <a href="#fnref:t:1" class="reversefootnote" role="doc-backlink">↩<sup>2</sup></a></p>
</li>
<li id="fn:dms" role="doc-endnote">
<p>Jan Dreier, Nikolas Mählmann, and Sebastian Siebertz (2023), “First-order model checking on structurally sparse graph classes”, <a href="https://arxiv.org/abs/2302.03527">arXiv:2302.03527</a> <a href="#fnref:dms" class="reversefootnote" role="doc-backlink">↩</a></p>
</li>
<li id="fn:no" role="doc-endnote">
<p>Jaroslav Nešetřil and Patrice Ossona de Mendez (2012), “5.5: Classes with bounded expansion”, <em>Sparsity: Graphs, Structures, and Algorithms</em>, pp. 104–107, Springer, Algorithms and Combinatorics, vol. 28, <a href="https://doi.org/10.1007/978-3-642-27875-4">doi:10.1007/978-3-642-27875-4</a> <a href="#fnref:no" class="reversefootnote" role="doc-backlink">↩</a></p>
</li>
<li id="fn:st" role="doc-endnote">
<p>Pierre Simon and Szymon Toruńczyk (2021), “Ordered graphs of bounded twin-width”, <a href="https://arxiv.org/abs/2102.06881">arXiv:2102.06881</a> <a href="#fnref:st" class="reversefootnote" role="doc-backlink">↩</a></p>
</li>
<li id="fn:t2" role="doc-endnote">
<p>See Corollary 3.6 of Toruńczyk (2023)<sup id="fnref:t:2" role="doc-noteref"><a href="#fn:t" class="footnote" rel="footnote">2</a></sup> <a href="#fnref:t2" class="reversefootnote" role="doc-backlink">↩</a></p>
</li>
<li id="fn:tw3" role="doc-endnote">
<p>See Lemma 13 of Édouard Bonnet, Colin Geniet, Eun Jung Kim, Stéphan Thomassé, and Rémi Watrigant, “Twin-width III: Max Independent Set and Coloring”, <a href="https://arxiv.org/abs/2007.14161v2">arXiv:2007.14161v2</a> (this lemma is not in the <em>ICALP</em> 2021 version and numbered differently in other arXiv versions) <a href="#fnref:tw3" class="reversefootnote" role="doc-backlink">↩</a></p>
</li>
<li id="fn:lsv" role="doc-endnote">
<p>Anna Lubiw, Jack Snoeyink, and Hamideh Vosoughpour (2017), “Visibility graphs, dismantlability, and the cops and robbers game”, <em>CGTA</em> 66: 14–27, <a href="https://arxiv.org/abs/1601.01298">arXiv:1601.01298</a> <a href="#fnref:lsv" class="reversefootnote" role="doc-backlink">↩</a></p>
</li>
<li id="fn:es" role="doc-endnote">
<p>Günter Ewald and Geoffrey C. Shephard (1974), “Stellar subdivisions of boundary complexes of convex polytopes”, <em>Math. Ann.</em> 210: 7–16, <a href="https://doi.org/10.1007/BF01344542">doi:10.1007/BF01344542</a>. <a href="#fnref:es" class="reversefootnote" role="doc-backlink">↩</a></p>
</li>
</ol>
</div>
<p>(<a href="https://mathstodon.xyz/@11011110/109901138706218444">Discuss on Mastodon</a>)</p>David EppsteinAt the recent Workshop on Geometry and Graphs in Barbados, most of the technical activity involved working in small groups on research problems, but there was also a nice survey talk by Rose McCarty on flip-width.1 This is a new and very general notion of width in graphs, introduced by Szymon Toruńczyk.2 It is defined in terms of a certain cops-and-robbers game on graphs, and intended to capture the structure inherent in many types of graphs and to unify bounded expansion and twin-width. Rose also helped me edit a preliminary version of this post. Thanks, Rose! Any remaining errors are my fault. ↩ Szymon Toruńczyk (2023), “Flip-width: cops and robber on dense graphs”, arXiv:2302.00352 ↩Congratulations, Dr. Afshar!2023-02-17T16:06:00+00:002023-02-17T16:06:00+00:00https://11011110.github.io/blog/2023/02/17/congratulations-dr-afshar<p>Ramtin Afshar, a doctoral student in the UC Irvine Center for Algorithms and Theory of Computation advised by Mike Goodrich, passed his defense today, becoming Mike’s 25th completed doctoral student. Ramtin’s dissertation, <em>Exact Learning of Graphs from Queries</em>, was based on papers from <a href="https://doi.org/10.4230/LIPIcs.ESA.2020.3">ESA 2020</a>, <a href="https://doi.org/10.1007/978-3-031-20624-5_18">LATIN 2022</a>, and <a href="https://doi.org/10.4230/LIPIcs.STACS.2022.4">STACS 2022</a>, all of which involved asking questions to find out the structure of an unknown graph.</p>
<p>A possibly familiar example here is the <a href="https://en.wikipedia.org/wiki/Traceroute">traceroute</a> program, used to debug internet connections by finding a path from one networked computer to another. It uses a feature of internet protocols that allow packets to “time out” if they make too many hops, returning an error message back to the originating computer when they do. By setting the timeout to a parameter \(k\), you can force the timeout to happen at the <span style="white-space:nowrap">\(k\)th</span> step of a shortest path to another computer, and by doing so find out who is at that <span style="white-space:nowrap">\(k\)th</span> step. You might think that you would need to trace the routes between all pairs of computers on the network to find out where its edges are (and this does work, with a quadratic number of <span style="white-space:nowrap">\(k\)th-step</span> queries), but Ramtin and his coauthors (Goodrich and two other UCI students, Pedro Matias and Martha Osegueda) showed that with some natural assumptions on the network topology, only a near-linear number of queries is needed.</p>
<p>Beyond the papers used in his thesis, Ramtin is a coauthor on more papers in <a href="https://doi.org/10.1007/978-3-030-59212-7_12">SPIRE 2020</a> on related problems of string reconstruction, in <a href="https://doi.org/10.1145/3350755.3400229">SPAA 2020</a> on reconstructing evolutionary trees or other binary trees, and in <a href="https://doi.org/10.4230/LIPIcs.SEA.2022.9">SEA 2022</a> on learning road maps from shortest path hop counts. His traceroute work in STACS 2022 was also the subject of a brief announcement at <a href="https://doi.org/10.1145/3409964.3461822">SPAA 2021</a>.</p>
<p>His next step is to work for Google in the San Francisco Bay Area, involving a temporary two-body problem while his wife finishes her own studies at the University of Southern California.</p>
<p>Congratulations, Ramtin!</p>
<p>(<a href="https://mathstodon.xyz/@11011110/109882878328905995">Discuss on Mastodon</a>)</p>David EppsteinRamtin Afshar, a doctoral student in the UC Irvine Center for Algorithms and Theory of Computation advised by Mike Goodrich, passed his defense today, becoming Mike’s 25th completed doctoral student. Ramtin’s dissertation, Exact Learning of Graphs from Queries, was based on papers from ESA 2020, LATIN 2022, and STACS 2022, all of which involved asking questions to find out the structure of an unknown graph.Linkage partly from Barbados2023-02-15T18:37:00+00:002023-02-15T18:37:00+00:00https://11011110.github.io/blog/2023/02/15/linkage-partly-from<ul>
<li>
<p><a href="https://sforcey.github.io/sf34/hedra.htm">Hedra Zoo</a> <span style="white-space:nowrap">(<a href="https://mathstodon.xyz/@11011110/109794057137669490">\(\mathbb{M}\)</a>),</span> like the Online Encyclopedia of Integer Sequences, but for sequences of polyhedra rather than sequences of integers.</p>
</li>
<li>
<p><a href="https://mathstodon.xyz/@Danpiker/109773012543776299">Mathematical 3d prints by Dan Piker</a>: Morin’s surface, Boy’s surface, a Klein bottle, and a puzzle based on a 3d Pythagorean tiling.</p>
</li>
<li>
<p><a href="https://www.flickr.com/photos/tactom/8471902275/">Origami tessellation torus</a> <span style="white-space:nowrap">(<a href="https://mathstodon.xyz/@monsoon0/109800312114130835">\(\mathbb{M}\)</a>),</span> by Tomohiro Tachi.</p>
</li>
<li>
<p><a href="https://discrete-notes.github.io/crossing-number">Low crossing numbers</a> <span style="white-space:nowrap">(<a href="https://mathstodon.xyz/@11011110/109810071945968761">\(\mathbb{M}\)</a>),</span> blog post on <a href="https://doi.org/10.4230/LIPIcs.SoCG.2021.28">a SoCG 21 paper by Mónika Csikós and Nabil H. Mustafa</a>. Given points and a family of shapes containing subsets of them, the goal is to connect the points by a matching, path, or spanning tree so that no shape has its boundary crossed by many edges. Many geometric set families have these structures and this paper shows how to find it faster than was previously known.</p>
</li>
<li>
<p><a href="https://en.wikipedia.org/wiki/Quoridor">Quoridor</a> <span style="white-space:nowrap">(<a href="https://mathstodon.xyz/@11011110/109815696566120005">\(\mathbb{M}\)</a>),</span> a game with simple abstract rules that lead to an interesting mix of strategy and tactics, in which you often have to balance the future effects of a move on your own progress versus its effects on your opponent.</p>
</li>
<li>
<p><a href="https://www.youtube.com/watch?v=lthHMDXbP30">Knotty analog oscilloscope art</a> <span style="white-space:nowrap">(<a href="https://mathstodon.xyz/@henryseg/109809898659918995">\(\mathbb{M}\)</a>),</span> video by Henry Segerman with Matthias Goerner.</p>
</li>
<li>
<p>It’s not every day that I (and many other Mathstodon users) <a href="https://www.nytimes.com/2023/02/07/science/puzzles-rectangles-mathematics.html">get mentioned in the <em>New York Times</em></a> <span style="white-space:nowrap">(<a href="https://mathstodon.xyz/@phonner/109826175739500354">\(\mathbb{M}\)</a>)!</span> The context is an article by Siobhan Roberts on partitioning rectangles into similar rectangles.</p>
</li>
<li>
<p><a href="https://cs.utdallas.edu/SOCG23/socg.html">The list of accepted papers at SoCG’23</a> <span style="white-space:nowrap">(<a href="https://mathstodon.xyz/@11011110/109830669758019190">\(\mathbb{M}\)</a>),</span> the 39th International Symposium on Computational Geometry, next June in Dallas, is now online. I have one in the list; I’ll post in more detail on it once I have a preprint version ready. <a href="/blog/2022/09/21/counting-paths-convex.html">One of my earlier posts</a> is related and <a href="/blog/2022/08/22/permuted-points-interest.html">another post</a> was a lemma that I ended up not using.</p>
</li>
<li>
<p><a href="https://www.youtube.com/watch?v=F_43oTnTXiw">Beware the Runge spikes</a> <span style="white-space:nowrap">(<a href="https://mathstodon.xyz/@standupmaths/109819371159314006">\(\mathbb{M}\)</a>)!</span> Matt Parker on the Runge phenomenon, in which interpolating through more data points can make the quality of interpolation worse. Sadly, no mention of the <a href="https://en.wikipedia.org/wiki/Witch_of_Agnesi">Witch of Agnesi</a>.</p>
</li>
<li>
<p><a href="https://www.quantamagazine.org/mathematicians-complete-quest-to-build-spherical-cubes-20230210/"><em>Quanta</em> on low-surface-area convex polytopes that tile high-dimensional space by integer translations</a> <span style="white-space:nowrap">(<a href="https://mathstodon.xyz/@QuantaMagazine@mstdn.social/109841124938979652">\(\mathbb{M}\)</a>).</span> Based on “<a href="https://arxiv.org/abs/2301.02862">An integer parallelotope with small surface area</a>” by Assaf Naor and Oded Regev.</p>
</li>
<li>
<p>I just returned from a week-long workshop at the Bellairs Research Institute in Barbados, my first since the pandemic started <span style="white-space:nowrap">(<a href="https://mathstodon.xyz/@11011110/109849258853460072">\(\mathbb{M}\)</a>).</span> This time, Bellairs double-booked us with another workshop, so we met in Seabourne House and its ocean-view garden instead of the usual picnic table area. This worked surprisingly well, especially for the morning sessions. The photo below shows some of this area; <a href="https://www.ics.uci.edu/~eppstein/pix/seabourne/">the full gallery</a> has a few other shots of architectural details.</p>
<p style="text-align:center"><img src="https://www.ics.uci.edu/~eppstein/pix/seabourne/CommonRoom-m.jpg" alt="Seabourne House, Bellairs Research Institute, Holetown, Barbados" style="border-style:solid;border-color:black" /></p>
</li>
<li>
<p><a href="https://i.stack.imgur.com/k61In.png">Three unit regular tetrahedra pack neatly into a unit cube</a> <span style="white-space:nowrap">(<a href="https://mathstodon.xyz/@uzulim/109742191472459313">\(\mathbb{M}\)</a>).</span></p>
</li>
<li>
<p><a href="https://3quarksdaily.com/3quarksdaily/2023/02/some-comments-on-writing-popular-mathematics.html">John Allen Paulos on writing popular mathematics</a> <span style="white-space:nowrap">(<a href="https://mathstodon.xyz/@11011110/109858684384756666">\(\mathbb{M}\)</a>),</span> and Bertrand Russell’s other paradox, on the impossibility of combining intelligibility and precision.</p>
</li>
<li>
<p><a href="https://mathstodon.xyz/@aperiodical/109800517387277782">What can mathematicians do</a> <span style="white-space:nowrap">(<a href="https://mathstodon.xyz/@aperiodical/109800517387277782">\(\mathbb{M}\)</a>)?</span> Recordings of ten talks by disabled mathematicians.</p>
</li>
<li>
<p><a href="https://scottaaronson.blog/?p=7028">Scott Aaronson on the shortsightedness of xenophobia</a> <span style="white-space:nowrap">(<a href="https://mathstodon.xyz/@11011110/109872087293449471">\(\mathbb{M}\)</a>),</span> triggered by the denial of a visa to a would-have-been-incoming doctoral student from China.</p>
</li>
</ul>David EppsteinHedra Zoo (\(\mathbb{M}\)), like the Online Encyclopedia of Integer Sequences, but for sequences of polyhedra rather than sequences of integers.Linkage with glass viruses and a Cheeto sphere2023-01-31T23:04:00+00:002023-01-31T23:04:00+00:00https://11011110.github.io/blog/2023/01/31/linkage-glass-viruses<ul>
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<p><a href="https://windowsontheory.org/2023/01/16/new-in-focs-2023-a-conjectures-track/">FOCS 2023 introduces a new “conjectures track”</a> <span style="white-space:nowrap">(<a href="https://mathstodon.xyz/@11011110/109701298874363291">\(\mathbb{M}\)</a>),</span> seeking papers like the one by Khot that introduced the Unique Games Conjecture.</p>
</li>
<li>
<p><a href="https://arxiv.org/abs/2212.14200">Intersecting ellipses induced by a max-sum matching</a> <span style="white-space:nowrap">(<a href="https://mathstodon.xyz/@11011110/109708982979012608">\(\mathbb{M}\)</a>).</span> Any matching on \(2n\) plane points is shorter than any star with them as leaves, by the triangle inequality. But sometimes not much shorter: there is a matching and star center from which all matched edges have angles \(\ge 2\pi/3\), so the length ratio is \(\le 2/\sqrt3\). This preprint by Barabanshchikova & Polyanskii proves more: the max length matching itself has a star that approximates each edge individually with the same ratio.</p>
</li>
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<p><a href="https://www.quantamagazine.org/finally-a-fast-algorithm-for-shortest-paths-on-negative-graphs-20230118/">Three computer scientists have discovered a fast algorithm for finding shortest paths between points on graphs with negative weights</a> <span style="white-space:nowrap">(<a href="https://mathstodon.xyz/@QuantaMagazine@mstdn.social/109711314801457276">\(\mathbb{M}\)</a>),</span> <em>Quanta</em> summary of a recent breakthrough on a long-standing optimization problem. The original paper by Aaron Bernstein, Danupon Nanongkai, and Christian Wulff-Nilsen is in FOCS 2022 and online at <a href="https://arxiv.org/abs/2203.03456">arXiv:2203.03456</a>.</p>
</li>
<li>
<p><a href="https://blog.archive.org/2023/01/17/as-the-us-public-domain-expands-20-year-pause-for-the-canadian-public-domain-begins/">The release of works into the public domain in Canada goes into a 20-year hiatus</a> <span style="white-space:nowrap">(<a href="https://mathstodon.xyz/@11011110/109719958934723525">\(\mathbb{M}\)</a>)</span> after its North American partners forced Canada into using Life+70 instead of Life+50 protection terms.</p>
</li>
<li>
<p><a href="https://icml.cc/Conferences/2023/CallForPapers">ICML banned ChatGPT for writing submissions</a> <span style="white-space:nowrap">(<a href="https://mathstodon.xyz/@fortnow@fediscience.org/109677987334540502">\(\mathbb{M}\)</a>)</span> and then <a href="https://icml.cc/Conferences/2023/llm-policy">walked it back</a>.</p>
</li>
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<p><a href="https://www.thebulwark.com/the-economic-secret-hidden-in-a-tiny-discontinued-pasta/">The difficulty of of making machines to make machines to make the things you want</a> <span style="white-space:nowrap">(<a href="https://mathstodon.xyz/@Leahwrenn@mastodon.social/109701058818221148">\(\mathbb{M}\)</a>);</span> in this case, tiny star-shaped pasta.</p>
</li>
<li>
<p><a href="https://cp4space.hatsya.com/2023/01/18/tensor-rank-paper/">Bounds on the tensor rank of the determinant over various fields</a> <span style="white-space:nowrap">(<a href="https://mathstodon.xyz/@11011110/109737432277072901">\(\mathbb{M}\)</a>).</span> I’m intrigued by the connection between tensor ranks, tilings of space by orthogonal polyhedra, and flip graphs of ordered partitions of subsets alluded to in this blog post on <a href="https://arxiv.org/abs/2301.06586">a new preprint</a>.</p>
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<p><a href="https://futurism.com/cnet-ai-plagiarism">“CNET’s AI-written articles aren’t just riddled with errors. They also appear to be substantially plagiarized”</a> <span style="white-space:nowrap">(<a href="https://mathstodon.xyz/@11011110/109745677430217499">\(\mathbb{M}\)</a>,</span> <a href="https://news.ycombinator.com/item?id=34502939">via</a>). “The bot’s misbehavior ranges from verbatim copying to moderate edits to significant rephrasings, all without properly crediting the original. In at least some of its articles, it appears that virtually every sentence maps directly onto something previously published elsewhere.”</p>
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<p><a href="https://www.quantamagazine.org/the-computer-scientist-who-finds-life-lessons-in-board-games-20230125/"><em>Quanta</em> interview with Shang-Hua Teng</a> <span style="white-space:nowrap">(<a href="https://mathstodon.xyz/@QuantaMagazine@mstdn.social/109751056152008812">\(\mathbb{M}\)</a>).</span> “Teng discusses his upbringing in China and the math that good board games have in common.”</p>
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<p><a href="https://mathstodon.xyz/@kameryn/109756698881498419">Nice explainer on the continuum hypothesis</a> and how the idea that its independence makes the problem intractable “misses the actual nuance of the mathematical terrain”: for any “natural”-enough set, the dichotomy between the cardinalities of \(\mathbb{N}\) and \(\mathbb{R}\) is valid and provable.</p>
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<p>I have no mathematical point to share here <span style="white-space:nowrap">(<a href="https://mathstodon.xyz/@11011110/109765566564556642">\(\mathbb{M}\)</a>),</span> just a recoloring of an old image I made of the <a href="https://en.wikipedia.org/wiki/Folkman_graph">Folkman graph</a> that I thought came out interesting-looking. It is something like one of Escher’s impossible figures: Locally it looks like it could be made from overlapping translucent sheets of blue material, cut out in this pattern, but globally it doesn’t all fit together.</p>
<p style="text-align:center"><img src="/blog/assets/2023/Folkman-shaded.svg" alt="A figure with five-fold symmetry, made from arcs of circles, shaded in light and dark blue. There are 20 points where the arcs cross a central circle; if these points are are taken as the vertices of a graph, and the arcs between them taken as edges of a graph, the result is the Folkman graph, the smallest graph that is symmetric on its edges but not on its vertices." style="width:100%;max-width:600px" /></p>
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<p>You’ve probably heard about the glass flowers at Harvard, but have you seen <a href="https://www.lukejerram.com/glass/">Luke Jerram’s glass viruses and bacteria</a> <span style="white-space:nowrap">(<a href="https://mathstodon.xyz/@11011110/109771266602473224">\(\mathbb{M}\)</a>,</span> <a href="https://www.thisiscolossal.com/2023/01/luke-jerram-glass-microbes/">via</a>)?</p>
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<p><a href="https://www.si.edu/spotlight/geometric-models-jullien-models-for-descriptive-geometry">Jullien Models for Descriptive Geometry</a> <span style="white-space:nowrap">(<a href="https://mathstodon.xyz/@divbyzero/109751804697166396">\(\mathbb{M}\)</a>).</span> Fold-out paper-and-thread models of 3d geometric constructions, from the mid-1870s, in the collection of the <em>Smithsonian</em>.</p>
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<p><a href="https://www.latimes.com/california/story/2023-01-27/uc-scrambling-to-pay-big-wage-gains-for-academic-workers-grad-student-cuts-loom">“To afford historic labor contract, UC considers cutting TAs, graduate student admissions”</a> <span style="white-space:nowrap">(<a href="https://mathstodon.xyz/@11011110/109779527458911961">\(\mathbb{M}\)</a>).</span> Apparently the administration is treating this as a zero-sum game: they’ve agreed to pay teaching assistants significantly more per person, but they won’t give us any larger budget for them, apparently intending that there be fewer slots to make it all balance.</p>
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<p><a href="https://www.itsnicethat.com/articles/sam-keller-art-020322">Sam Keller’s <em>Cheetosphere</em></a> <span style="white-space:nowrap">(<a href="https://mathstodon.xyz/@daw15@universeodon.com/109767629934965993">\(\mathbb{M}\)</a>),</span> an artwork constructed much like a geodesic dome, but out of Cheetos.</p>
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</ul>David EppsteinFOCS 2023 introduces a new “conjectures track” (\(\mathbb{M}\)), seeking papers like the one by Khot that introduced the Unique Games Conjecture.Doubled planar drawings of doubled planar graphs2023-01-23T18:17:00+00:002023-01-23T18:17:00+00:00https://11011110.github.io/blog/2023/01/23/doubled-planar-drawings<p>If you start with a planar graph, and make two copies of each vertex, you should be able to draw the result as two planar graphs, right? But it’s more complicated than just copying a drawing of your starting graph, because you get four copies of each edge, and you have to put them all somewhere.</p>
<p>For instance, the graph \(K_{2,2,2}\) is planar (it’s the graph of an octahedron). Doubling it gives \(K_{4,4,4}\). And \(K_{4,4,4}\) can indeed be drawn as two planar graphs, but not as two octahedra. Here it is as two octagonal bipyramids:</p>
<p style="text-align:center"><img src="/blog/assets/2023/bipyramidal-K444.svg" alt="The complete tripartite graph K_{4,4,4} drawn as the union of two octagonal bipyramids" /></p>
<p>And here it is drawn as two planar graphs in a different way:</p>
<p style="text-align:center"><img src="/blog/assets/2023/nested-quad-K444.svg" alt="Another drawing of the complete tripartite graph K_{4,4,4} as the union of two planar graphs" /></p>
<p>Drawings like this, where a nonplanar graph is presented as the union of two planar graphs, are called <em>biplanar</em>. Another word for the same idea is <em>thickness</em>: the thickness of a graph is the number of planar subgraphs needed to cover all of its edges, and a graph is biplanar if it has thickness two. A famous unsolved problem in graph theory, <a href="https://en.wikipedia.org/wiki/Earth%E2%80%93Moon_problem">Ringel’s Earth–Moon problem</a>, asks how many colors are necessary to color biplanar graphs. The name comes from the idea that you might want to color a pair of maps of countries that all have space colonies on the Moon, using the same color for each country and its colony. The map of their adjacencies on the Earth gives you one planar graph, and the map of adjacencies between their colonies on the Moon gives you the other. \(K_{4,4,4}\) is not a very interesting example for this question, because it only needs three colors; some other biplanar graphs need as many as nine. On the other hand, we only know how to prove that twelve colors are always enough, so there’s a pretty big gap. Ellen Gethner published <a href="https://doi.org/10.1007%2F978-3-319-97686-0_11">a nice survey on the problem</a> in 2018, including also some other material on biplanar graphs.</p>
<p>One of Gethner’s conjectures from that survey is that doubled planar graphs (or as she calls them, 2-blowups) are always biplanar. The conjecture is plausible, because the number of edges is within the right range to be biplanar. If the starting planar graph has \(n\) vertices, it has at most \(3n-6\) edges, and its blowup has \(2n\) vertices and at most \(12n-24\) edges. This is fewer than \(12n-12\), the limit on the number of edges for biplanar graphs with \(2n\) vertices. And for a wide class of planar graphs, even those with the maximum possible number of edges, I can prove that their blowups are indeed biplanar. This works whenever the dual graph can be partitioned into two induced paths. In the original graph, these paths form strips of faces connected end-to-end, and in the biplanar drawing of the blowup they become sequences of nested quadrilaterals. For instance, \(K_{4,4,4}\) is the blowup of the octahedral graph \(K_{2,2,2}\), whose dual graph is a cube, and the cube can be partitioned into two induced paths:</p>
<p style="text-align:center"><img src="/blog/assets/2023/cube-path-partition.svg" alt="Partition of a cube into two induced paths" /></p>
<p>The second of the biplanar drawings of \(K_{4,4,4}\) above comes from this dual induced path partition.</p>
<p>Despite this positive evidence, the conjecture turns out to be false. My latest preprint, “On the biplanarity of blowups”, <a href="https://arxiv.org/abs/2301.09246">arXiv:2301.09246</a>, constructs counterexamples, planar graphs whose blowups are not biplanar. The general idea of the construction is very similar to one of my earlier papers, on <a href="/blog/2020/09/01/isosceles-polyhedra.html">polyhedral graphs that cannot be realized as convex polyhedra with isosceles-triangle faces</a>. Both papers are based on the construction of a <a href="https://en.wikipedia.org/wiki/Kleetope">Kleetope</a>, a polyhedron formed from another polyhedron by attaching a pyramid to each face.
If you repeat the Kleetope construction, you get a polyhedron enclosed by multiple layers of pyramids, and any drawing or geometric realization of this layered polyhedron also gives you a drawing or realization of the simpler polyhedra underneath those layers.
Each time you layer on more pyramids, any possible biplanar drawing of the result gets more and more constrained, until with enough layers it becomes completely impossible.</p>
<p>Because not all planar blowups are biplanar, the question arises: which ones are, and which ones aren’t? Is there an efficient algorithm that takes as input a planar graph, produces a biplanar drawing of its blowup if such a drawing exists, and tells you when such a drawing doesn’t exist? I don’t know.</p>
<p>(<a href="https://mathstodon.xyz/@11011110/109742463628336203">Discuss on Mastodon</a>)</p>David EppsteinIf you start with a planar graph, and make two copies of each vertex, you should be able to draw the result as two planar graphs, right? But it’s more complicated than just copying a drawing of your starting graph, because you get four copies of each edge, and you have to put them all somewhere.