Jekyll2019-05-16T05:01:03+00:00https://11011110.github.io/blog/feed.xml11011110Geometry, graphs, algorithms, and moreDavid EppsteinLinkage2019-05-15T22:00:00+00:002019-05-15T22:00:00+00:00https://11011110.github.io/blog/2019/05/15/linkage<ul>
<li>
<p><a href="https://culturacientifica.com/2019/05/01/el-poema-de-los-numeros-primos/">El poema de los números primos</a> (<a href="https://mastodon.social/@victorhck/102020247143426975"><script type="math/tex">\mathbb{M}</script></a>). Exhibit of the mathematically-inspired artworks of <a href="https://en.wikipedia.org/wiki/Esther_Ferrer">Esther Ferrer</a>, at <a href="https://en.wikipedia.org/wiki/Tabakalera">Tabakalera</a> in San Sebastián, Spain.</p>
</li>
<li>
<p><a href="https://igorpak.wordpress.com/2019/04/26/how-combinatorics-became-legitimate-according-to-laszlo-lovasz-and-endre-szemeredi/">How combinatorics became legitimate</a> (<a href="https://mathstodon.xyz/@11011110/102036466457669447"><script type="math/tex">\mathbb{M}</script></a>). Igor Pak recommends two interesting video interviews with László Lovász and Endre Szemerédi. The whole interviews are quite long but they’re broken into 10-minute clips and Igor has picked out the ones relevant to the title.</p>
</li>
<li>
<p><a href="https://www.quantamagazine.org/how-twisted-graphene-became-the-big-thing-in-physics-20190430/">Magic angles and superconductivity in twisted graphene</a> (<a href="https://mathstodon.xyz/@11011110/102044547042991323"><script type="math/tex">\mathbb{M}</script></a>). If you twist two sheets of hexagonally tiled carbon relative to each other you can get a superconductor, but only for certain very specific twist angles.</p>
</li>
<li>
<p><a href="https://practicaltypography.com/ligatures-in-programming-fonts-hell-no.html">Matthew Butterick says no to ligatures in programming fonts</a> (<a href="https://mathstodon.xyz/@11011110/102047783718443248"><script type="math/tex">\mathbb{M}</script></a>, <a href="https://news.ycombinator.com/item?id=19805053">via</a>). I tend to agree. They make some things cuter but more things inconsistent. The lack of a short double back arrow in the Fira example is telling. And anyone who expects to see individual characters has to know what font they’re displayed in and how it mangles them to understand what they’re reading. But if you like these for your own text editing, whatever. Just show me the ASCII when I have to view it in my browser.</p>
</li>
<li>
<p><a href="https://arxiv.org/abs/1905.01325">Breaking the Bellman–Ford shortest-path bound</a> (<a href="https://mathstodon.xyz/@11011110/102051970319136719"><script type="math/tex">\mathbb{M}</script></a>). Amr Elmasry claims a time bound of <script type="math/tex">O(m\sqrt{n})</script> for single-source shortest paths in graphs that may have cycles and negative edge weights, but no negative cycles. If correct, this would be a big improvement over the <script type="math/tex">O(mn)</script> time for Bellman–Ford. However, I got stuck somewhere around Lemma 3 when trying to understand it. Anyone else have better progress?</p>
</li>
<li>
<p><a href="http://www.generalist.org.uk/blog/2019/gender-and-deletion-on-wikipedia/">Some actual data on how the subject’s gender influences biography creation and deletion on Wikipedia</a> (<a href="https://mathstodon.xyz/@11011110/102058999289661319"><script type="math/tex">\mathbb{M}</script></a>). Still-existing older articles on women are more likely to have gone through a deletion discussion than men, but we don’t know whether more were nominated or equally many nominated but women survived better, and whether the inequality of nominations has lessened recently or the greater nomination rate for women takes longer to kick in and is still prevalent.</p>
</li>
<li>
<p><a href="https://arxiv.org/abs/1904.12761">The graphs behind Reuleaux polyhedra</a> (<a href="https://mathstodon.xyz/@11011110/102064338449854355"><script type="math/tex">\mathbb{M}</script></a>), by
Luis Montejano, Eric Pauli, Miguel Raggi, and Edgardo Roldán-Pensado.
These shapes are the intersections of equal-radius balls centered at their vertices; smoothing some edges gives them constant width. Their vertices are the finite point sets with the most diameters. Their vertex-edge graphs are self-dual, unlike other polyhedral graphs. And their vertex-diameter graphs are 4-colorable. Examples include pyramids over odd polygons.</p>
</li>
<li>
<p>It’s not like it’s difficult to make your own out of, you know, paper, but if you want a colorful kit to teach yourself about the Miura-ori and three other folds, <a href="https://www.thisiscolossal.com/2019/05/paper-folding-kit-by-kelli-anderson/">this one looks pretty if a little overpriced at $20 for eight sheets of paper</a> (<a href="https://mathstodon.xyz/@11011110/102070078247023735"><script type="math/tex">\mathbb{M}</script></a>).</p>
</li>
<li>
<p><a href="https://arxiv.org/abs/1905.02167">Tensor products of graphs can require fewer colors than their factors</a> (<a href="https://mathstodon.xyz/@11011110/102072759747197000"><script type="math/tex">\mathbb{M}</script></a>). This short new preprint by Yaroslav Shitov gives counterexamples to <a href="https://en.wikipedia.org/wiki/Hedetniemi%27s_conjecture">Hedetniemi’s conjecture</a> from 1966. In a new blog post <a href="https://gilkalai.wordpress.com/2019/05/10/sansation-in-the-morning-news-yaroslav-shitov-counterexamples-to-hedetniemis-conjecture/">Gil Kalai explains the construction</a>.</p>
</li>
<li>
<p><a href="https://discrete-notes.github.io/natural-history">Algorithms and natural history</a> (<a href="https://mathstodon.xyz/@11011110/102080291126103628"><script type="math/tex">\mathbb{M}</script></a>). In a new blog, Laurent Feuilloley writes about some algorithmic problems on polyhedra coming from the measurement of skulls, diamond cutting, and the use of symmetry to undo deformations of fossils.</p>
</li>
<li>
<p>Did you know that Swiss mathematician <a href="https://en.wikipedia.org/wiki/Alice_Roth">Alice Roth</a> invented <a href="https://en.wikipedia.org/wiki/Swiss_cheese_(mathematics)">Swiss cheese</a>? (<a href="https://mathstodon.xyz/@11011110/102087095765063877"><script type="math/tex">\mathbb{M}</script></a>). A Swiss cheese is a disk with smaller disks removed, leaving no interior. <a href="https://blogs.scientificamerican.com/roots-of-unity/the-serendipity-of-swiss-cheese/"><em>Scientific American</em> alerted me to this amusing terminology</a> but I got a clearer idea what they’re good for from <a href="http://www.math.tamu.edu/~boas/courses/618-2015a/roth.pdf">an exercise using them to show complex conjugation to be well-behaved on a compact domain but hard to approximate by rational functions</a>.</p>
</li>
<li>
<p><a href="https://ooni.torproject.org/post/2019-china-wikipedia-blocking/">China is now blocking all language editions of Wikipedia</a> (<a href="https://mathstodon.xyz/@11011110/102089517303119369"><script type="math/tex">\mathbb{M}</script></a>, <a href="https://boingboing.net/2019/05/13/report-china-now-blocks-wikip.html">via</a>), expanding its previous block which applied only to the Mandarin edition.</p>
<p>Of course their internet blockage is hardly the biggest problem with China these days. I was surprised to find that some of my usually-well-informed friends hadn’t even heard of “<a href="https://www.france24.com/en/20190510-reporters-plus-surviving-china-uighur-camps-repression">the largest mass incarceration of the 21st century</a>” and “<a href="https://www.theguardian.com/world/2018/dec/07/uighur-leaders-warn-chinas-actions-could-be-precursors-to-genocide">precursors to genocide</a>”, <a href="https://www.washingtonpost.com/opinions/global-opinions/china-cant-prettify-the-human-rights-catastrophe-in-xinjiang/2019/03/24/4c844f62-45ca-11e9-90f0-0ccfeec87a61_story.html">China’s concentration camps</a> for <a href="https://www.amnesty.org/en/latest/news/2018/09/china-up-to-one-million-detained/">up to a million Uighur people</a>. So read and learn.</p>
</li>
<li>
<p><a href="https://en.wikipedia.org/wiki/Hazel_Perfect">Hazel Perfect</a> (<a href="https://mathstodon.xyz/@11011110/102097362417443508"><script type="math/tex">\mathbb{M}</script></a>). A new Wikipedia article on the inventor of <a href="https://en.wikipedia.org/wiki/Gammoid">gammoids</a> (how I came across her name this time) and <a href="https://aperiodical.com/2013/03/much-ado-about-noether/">Christian Lawson-Perfect’s mathematical hero</a> (despite or because of the unexplained similarity of names).</p>
</li>
<li>
<p><a href="https://www.dailykos.com/stories/2019/5/13/1857360/-Poll-says-that-56-of-Americans-don-t-want-kids-taught-Arabic-numerals-We-have-some-bad-news">In a recent poll, “56% of Americans said Arabic numerals should not be taught in American schools”</a> (<a href="https://mathstodon.xyz/@11011110/102100871760570133"><script type="math/tex">\mathbb{M}</script></a>).</p>
</li>
</ul>David EppsteinEl poema de los números primos (). Exhibit of the mathematically-inspired artworks of Esther Ferrer, at Tabakalera in San Sebastián, Spain.Playing with model trains and calling it graph theory2019-05-02T19:03:00+00:002019-05-02T19:03:00+00:00https://11011110.github.io/blog/2019/05/02/playing-model-trains<p>You’ve probably played with model trains, for instance with something like the <a href="https://en.wikipedia.org/wiki/Brio_(company)">Brio</a> set shown below.<sup id="fnref:fn"><a href="#fn:fn" class="footnote">1</a></sup> And if you’ve built a layout with a model train set, you may well have wondered: is it possible for my train to use all the parts of my track?</p>
<p style="text-align:center"><img src="/blog/assets/2019/brio-33133.jpg" alt="Brio train set" /></p>
<p>For instance, in the layout shown in this image, if your train starts on the far right, moving downward, it will be stuck in a loop that it can never escape. There are no choice points where the train can switch to another track until it returns to the Y at the right, moving in the same direction. On the other hand, if you allow yourself to reverse the train, it can reverse back through the other entrance to the Y and reach the rest of the track. It’s also possible for a long-enough train to block itself, preventing it from escaping certain parts of the track that a short train could negotiate more easily.</p>
<p>My newest preprint, “Reconfiguring Undirected Paths” (with Demaine, Hesterberg, Jain, Lubiw, Uehara, and Uno, <a href="https://arxiv.org/abs/1905.00518">arXiv:1905.00518</a>), considers an abstract model for such problems, in which the train track is modeled as an undirected graph and the train is a simple path in the graph. You can slide the train by adding an edge to one end of the path and removing an edge from the other end; we don’t distinguish which end of the train is which, so it can slide in both directions. The vertices of the graph model points where you can choose which of several directions to slide the train. Because it’s an undirected graph, these are like the three-way and four-way junctions in the middle of the image (allowing the train to enter and exit along any pair of track segments) rather than the Y junctions at the far right (where a train that enters at one of the two top edges of the Y has to exit the bottom).</p>
<p>For instance, in a <script type="math/tex">2\times 3</script> grid graph, the different positions of a length-<script type="math/tex">3</script> path and the ways that one position can shift into another can be visualized as the state space shown below.</p>
<p style="text-align:center"><img src="/blog/assets/2019/path-reconfig-states.svg" alt="PSPACE-hardness reduction for path reconfiguration" /></p>
<p>Testing whether a long train can slide from one position to another turns out to be PSPACE-complete, even on graphs of bounded bandwidth, by a reduction from <a href="https://en.wikipedia.org/wiki/Nondeterministic_constraint_logic">nondeterministic constraint logic</a>. Here’s an example of an NCL problem transformed by our reduction into a path-sliding problem:</p>
<p style="text-align:center"><img src="/blog/assets/2019/path-reconfig-redux.svg" alt="PSPACE-hardness reduction for path reconfiguration" width="80%" /></p>
<p>Our main results are a <a href="Parameterized complexity">fixed-parameter tractable algorithm</a> parameterized by train length (so it’s fast for short trains) and a linear time algorithm when the graph is a tree. Both cases are based on the same intuition, that the problem becomes easier if we can maneuver the train onto a long enough path. For the parameterized version, if the graph has a path twice as long as the train that can be reached from the starting position of the train, and another long path that can reach the ending position, then we can maneuver the train onto the first long path, send it on an express route directly from the first long path to the second one, and then maneuver it from there into its final position. On the other hand, until we find these long paths, we can restrict our attention to a subgraph with no long paths; this implies that it has bounded <a href="Tree-depth">tree-depth</a> and makes searching within the subgraph easy. The linear time tree algorithm similarly involves a lot of back-and-forth maneuvering of the train to free up longer and longer segments of it until the whole train is freed to move from the start to the goal.</p>
<p>A shorter version of our paper will appear at <a href="http://wads.org/">WADS</a> this summer.
While it was in submission to WADS, a related preprint appeared on arXiv: “The Parameterized Complexity of Motion Planning for Snake-Like Robots”, by Gupta, Sa’ar, and Zehavi (<a href="https://arxiv.org/abs/1903.02445">arXiv:1903.02445</a>). They show that for a graph-theoretic model of the <a href="https://en.wikipedia.org/wiki/Snake_(video_game_genre)">Snake video game</a>, getting the snake from one position to another is fixed-parameter tractable in the length of the snake. For this problem, snakes are again paths in graphs, but they can move only in one direction, and the techniques they use to prove fixed-parameter tractability involve sparsifying the state space instead of maneuvering into long paths. <a href="/blog/2018/08/06/congratulations-dr-gupta.html">Sid Gupta was my student</a> at UCI before taking his current postdoc in Israel, but I haven’t talked to him about this, so I think their work must be independent and its appearance at about the same time a coincidence.</p>
<div class="footnotes">
<ol>
<li id="fn:fn">
<p>Searching on tineye finds that this image was on Amazon in 2008. Presumably it was supplied to them by Brio? <a href="#fnref:fn" class="reversefootnote">↩</a></p>
</li>
</ol>
</div>
<p>(<a href="https://mathstodon.xyz/@11011110/102029697142872437">Discuss on Mastodon</a>)</p>David EppsteinYou’ve probably played with model trains, for instance with something like the Brio set shown below.1 And if you’ve built a layout with a model train set, you may well have wondered: is it possible for my train to use all the parts of my track? Searching on tineye finds that this image was on Amazon in 2008. Presumably it was supplied to them by Brio? ↩Linkage2019-04-30T23:15:00+00:002019-04-30T23:15:00+00:00https://11011110.github.io/blog/2019/04/30/linkage<ul>
<li>
<p><a href="https://blog.computationalcomplexity.org/2019/04/good-article-terrible-headline.html">Good article, terrible headline</a> (<a href="https://mathstodon.xyz/@11011110/101938798669973189"><script type="math/tex">\mathbb{M}</script></a>). Bill Gasarch rants about several recent instances of clickbaity, inaccurate, and overhyped media coverage of theoretical computer science topics. I suspect the answer to his question “is it just our field?” is no.</p>
</li>
<li>
<p><a href="https://www.vox.com/science-and-health/2019/4/16/18311194/black-hole-katie-bouman-trolls">Vox on the sexist backlash against astronomer Katie Bouman</a> (<a href="https://mathstodon.xyz/@11011110/101942756338391262"><script type="math/tex">\mathbb{M}</script></a>, <a href="https://www.cnn.com/2019/04/12/us/andrew-chael-katie-bouman-black-hole-image-trnd/index.html">see also</a>), of black hole image fame, after she was cast by the media in the “lone genius” role typically reserved for men and untypical of how science actually happens.</p>
</li>
<li>
<p><a href="https://aperiodical.com/2019/04/mathematical-sign-language-interview-with-dr-jess-boland/">Mathematical sign language</a> (<a href="https://mathstodon.xyz/@11011110/101950143529837988"><script type="math/tex">\mathbb{M}</script></a>). Hearing-impaired eletrical engineering researcher Jess Boland discovered that weren’t enough technical terms in British Sign Language to cover the mathematics she uses in her work, so she’s been creating new ones as well as promoting the ones BSL already had. Katie Steckles interviews her for <em>The Aperiodical</em>.</p>
</li>
<li>
<p><a href="https://arxiv.org/abs/1804.05452">Regular polygon surfaces</a> (<a href="https://mathstodon.xyz/@11011110/101955536664219652"><script type="math/tex">\mathbb{M}</script></a>). Ian Alevy answers <a href="http://cs.smith.edu/~jorourke/TOPP/P72.html#Problem.72">Problem 72 of The Open Problems Project</a>: every topological sphere made of regular pentagons can be constructed by gluing regular dodecahedra together. You can also <a href="https://momath.org/mathmonday/the-paragons-system/">glue dodecahedra to get higher-genus surfaces</a>, but Alevy’s theorem doesn’t apply, so we don’t know whether all higher-genus regular-pentagon surfaces are formed that way.</p>
</li>
<li>
<p><a href="https://www.insidehighered.com/news/2019/04/12/czech-president-blocks-professorships-academic-critics">Czech president Miloš Zeman “has repeatedly used presidential powers to block the professorships of political opponents”</a> (<a href="https://mathstodon.xyz/@11011110/101965701030220573"><script type="math/tex">\mathbb{M}</script></a>). Charles University is now suing to allow their promotions to go through.</p>
</li>
<li>
<p><a href="https://arxiv.org/abs/1904.08845">Planar point sets determine many pairwise crossing segments</a> (<a href="https://mathstodon.xyz/@11011110/101968467896290245"><script type="math/tex">\mathbb{M}</script></a>). János Pach, Natan Rubin, and Gábor Tardos make significant progress on whether every<br />
<script type="math/tex">n</script> points in the plane have a large matching where all edges cross each other. A 1994 paper by Paul Erdős and half a dozen others only managed to prove this for “large” meaning <script type="math/tex">\Omega(\sqrt{n})</script>. The new paper proves a much stronger bound, <script type="math/tex">\Omega(n/2^{O(\sqrt{\log n})})</script> (Ryan Williams’ favorite function).</p>
</li>
<li>
<p><a href="https://randomascii.wordpress.com/2019/04/21/on2-in-createprocess/">Why asymptotics matters</a> (<a href="https://mathstodon.xyz/@11011110/101970781407484011"><script type="math/tex">\mathbb{M}</script></a>, <a href="https://news.ycombinator.com/item?id=19716673">via</a>): because if you don’t pay attention to it you get problems like this slow quadratic-time process creation bug in Windows 10.</p>
</li>
<li>
<p><a href="https://mathstodon.xyz/@jsiehler/101982200745112808">Snap cube puzzle</a>. The cubes have one peg and five holes; how many ways can you snap them into a connected <script type="math/tex">2\times 2</script> block with no pegs showing? See link in discussion thread for spoilers.</p>
</li>
<li>
<p>There’s lots of reasons to be unenthusiastic about newly-official-presidential-candidate Biden involving multiple instances of poor treatment of African-Americans and women, but here’s another more techy reason: <a href="https://www.pastemagazine.com/articles/2019/04/biden-to-attend-fundraiser-hosted-by-comcast-blue.html">his first major fundraiser as a candidate closely involves anti-net-neutrality lobbyists from Comcast</a> (<a href="https://mathstodon.xyz/@11011110/101987719804605064"><script type="math/tex">\mathbb{M}</script></a>).</p>
</li>
<li>
<p><a href="https://adsabs.github.io/blog/transition-reminder">SAO/NASA Astrophysics Data System updates its user interface</a> (<a href="https://mathstodon.xyz/@11011110/101996739440443058"><script type="math/tex">\mathbb{M}</script></a>). <a href="http://adsabs.harvard.edu/">The ADS</a> is a useful database of papers in astronomy and related fields. From comments on their post, the new UI is very slow. It is <a href="http://adsabs.github.io/help/faq/">unusable without JavaScript</a>. And it <a href="https://en.wikipedia.org/wiki/Special:Diff/892128592">“sends the users’ personal identifying information to at least 5 third-party companies”</a>. This is progress?</p>
</li>
<li>
<p><a href="https://mathoverflow.net/q/329910/440">I ask for a reference for an easy fact about divisibility representations of partial orders</a> (<a href="https://mathstodon.xyz/@11011110/102002516978139958"><script type="math/tex">\mathbb{M}</script></a>). The MathOverflow community isn’t very helpful, preferring instead to simultaneously complain that it’s too trivial and explain why it’s true to me as if I didn’t already say in my question that I thought it was trivial.`</p>
</li>
<li>
<p><a href="https://mathstodon.xyz/@henryseg/101975738950740643">Cannon-Thurston maps for veering triangulations</a>, whatever those are. Henry Segerman posts some pretty pictures from his joint work with David Bachman and Saul Schleimer.</p>
</li>
<li>
<p><a href="https://www.mathunion.org/fileadmin/CWM/Initiatives/CWMNewsletter1.pdf">Newsletter of the IMU Committee for Women</a> (<a href="https://aperiodical.com/2019/04/imu-committee-for-women-in-mathematics-now-has-a-newsletter/">via</a>). Includes an interview with Marie-Francoise Roy and the announcement of the book <em>World Women in Mathematics 2018</em>.</p>
</li>
<li>
<p><a href="https://rjlipton.wordpress.com/2019/04/30/network-coding-yields-lower-bounds/">Network coding yields lower bounds</a> (<a href="https://mathstodon.xyz/@11011110/102018096543192991"><script type="math/tex">\mathbb{M}</script></a>). Lipton and Regan report on <a href="https://arxiv.org/abs/1902.10935">a new paper by Afshani, Freksen, Kamma, and Larsen</a> on lower bounds for multiplication. If algorithmically opening and recombining network messages never improves fractional flow, then <script type="math/tex">O(n\log n)</script> circuit size for multiplication is optimal. But the same lower bound holds for simpler bit-shifting operations, so it’s not clear how it could extend from circuits to bignum algorithms.</p>
</li>
</ul>David EppsteinGood article, terrible headline (). Bill Gasarch rants about several recent instances of clickbaity, inaccurate, and overhyped media coverage of theoretical computer science topics. I suspect the answer to his question “is it just our field?” is no.Euler characteristics of non-manifold polycubes2019-04-23T16:37:00+00:002019-04-23T16:37:00+00:00https://11011110.github.io/blog/2019/04/23/euler-characteristics-nonmanifold<p>From a <script type="math/tex">2\times 2</script> block of cubes, remove two non-adjacent and non-opposite cubes. The resulting polycube has a boundary that is not a <a href="https://en.wikipedia.org/wiki/Manifold">manifold</a>: between the two removed cubes, there is an edge shared by four squares, but a two-dimensional manifold can only have two faces per edge. Nevertheless, we can compute its Euler characteristic as the number of vertices (<script type="math/tex">25</script>) minus the number of edges (<script type="math/tex">47</script>) plus the number of square faces (<script type="math/tex">24</script>). <script type="math/tex">25-47+24=2</script>, the same number we would expect for the Euler characteristic of a topological sphere! What does it mean?</p>
<p style="text-align:center"><img src="/blog/assets/2019/nonmanifold-polycube.svg" alt="Removing two non-adjacent and non-opposite cubes from a 2x2 block of cubes" /></p>
<p>Any finite union of cubes of the integer lattice (not even necessarily connected) has as its boundary a set of vertices, edges, and squares, with each edge incident to an even number of squares. We can define the Euler characteristic to be the number of vertices minus edges plus squares, in the usual way. But we can also compute it in a different, more intrinsic and topological, way. For any <script type="math/tex">\varepsilon</script> in the range <script type="math/tex">% <![CDATA[
0<\varepsilon<1/2 %]]></script>, define the “shrunken interior” of the polycube to be the set of points of the interior farther than <script type="math/tex">\varepsilon</script> from the boundary, and define the “shrunken exterior” in the same way. Then the shrunken interior and shrunken exterior both have (possibly disconnected) 2-manifolds as boundaries. We can define their Euler characteristics in the standard way from any cell decomposition of these boundaries (it doesn’t matter which cell decomposition we choose). Then the Euler characteristic of the polycube is the average of the Euler characteristics of the shrunken interior and shrunken exterior!</p>
<p>In the case of the mutilated <script type="math/tex">2\times 2</script> block, the shrunken interior and shrunken exterior are both topological balls (ignoring the puncture at infinity as it doesn’t have a boundary), so the average of their Euler characteristics is the Euler characteristic of a sphere, as we calculated.</p>
<p>There’s probably a simpler and more conceptual way of doing it, but here’s an explanation for why the Euler characteristic of the polycube boundary is the average of the Euler characteristics of the interior and exterior. Form a cell complex on the boundary of the interior and exterior, together, in the following way: expand each square of the polycube boundary to a cuboid with thickness 0.1, expand each edge into a cylinder with diameter 0.2 (big enough to enclose all the intersections of two expanded squares), and expand each vertex into a sphere with diameter 0.3 (big enough to enclose all the intersections of two cylinders but small enough that no two of these spheres touch). Remove the union of these expanded shapes from the space, and consider what’s left. It has the same topology as the union of the shrunken interior and exterior, and its boundary is now naturally divided up into cells: offset squares patches on the sides of each expanded square face, cylindrical patches on each expanded edge, and spherical patches on each expanded vertex, with curves where two patches meet.</p>
<p>Let’s calculate the Euler characteristic of this cell complex. Each square of the polycube leads to two offset square patches, so the <script type="math/tex">F</script> squares contribute <script type="math/tex">2F</script> to the Euler characteristic. Each edge <script type="math/tex">e</script> of the polycube might be adjacent to two or four squares; call this number <script type="math/tex">d(e)</script>. Then the cylinder around <script type="math/tex">e</script> includes <script type="math/tex">d(e)</script> surface patches between pairs of squares and <script type="math/tex">2d(e)</script> curves connecting them to the square patches. The patches count <script type="math/tex">+1</script> and the curves count <script type="math/tex">-1</script> for a total contribution to the Euler characteristic of <script type="math/tex">-d(e)</script>.</p>
<p>Finally, each vertex of the polycube becomes a sphere, subdivided by the patches and curves of the complex. These spheres also contain all the vertices of the complex. The Euler characteristic of a subdivided sphere would be <script type="math/tex">2</script>, but the vertex spheres have some parts of their subdivision removed. Where each edge cylinder or expanded square comes into the sphere, a patch of surfaces is removed, and the curves between these removed patches are also removed. An edge <script type="math/tex">e</script> with degree <script type="math/tex">d(e))</script> contributes to the removal of <script type="math/tex">d(e)</script> curves and <script type="math/tex">1+d(e)/2</script> patches (one for itself and <script type="math/tex">1/2</script> for each adjacent square). So if there are <script type="math/tex">V</script> vertices in the polycube, the <script type="math/tex">2V</script> contribution from the Euler characteristics of the subdivided spheres is modified by subtracting <script type="math/tex">1-d(e)/2</script> for each incident edge. The total modification at both endpoints of each edge is <script type="math/tex">-2+d(e)</script>. The <script type="math/tex">d(e)</script> that we calculated here is cancelled by the <script type="math/tex">-d(e)</script> on the cylinder for <script type="math/tex">e</script>, and we are left with a total modification of <script type="math/tex">-2E</script> where <script type="math/tex">E</script> is the number of polycube edges.</p>
<p>Putting all the pieces of this calculation together, the complex we have constructed on the union of the shrunken interior and exterior has Euler characteristic <script type="math/tex">2V-2E+2F</script>. Therefore, the Euler characteristic of the polycube boundary itself, <script type="math/tex">V-E+F</script>, equals the average of the characteristics of the interior and exterior. The same reasoning shows more generally that whenever you have a finite cell complex embedded into <script type="math/tex">\mathbb{R}^3</script>, dividing space up into chambers, the Euler characteristic of the complex equals half the sum of Euler characteristics of the manifolds bounding shrunken chambers.</p>
<p>Although Euler characteristics of 2-manifolds embedded without boundary in <script type="math/tex">\mathbb{R}^3</script> are always even, this averaging method can produce non-manifold surfaces with odd Euler characteristic. For instance, consider mutilating the <script type="math/tex">2\times 2</script> block in a different way, by removing two opposite cubes. The interior and exterior of the resulting polycube are both connected, but the interior is a solid torus and the exterior is a ball. So the Euler characteristic of the polycube should be the average of the torus and sphere, <script type="math/tex">1</script>. And if we actually calculate it we get <script type="math/tex">25-48+24=1</script>.</p>
<p>As this example shows, it’s possible for a polycube to have different topologies of surface on the interior and exterior, and it’s also possible to have different numbers of surfaces: for instance, two cubes attached vertex-to-vertex produce two interior surfaces but only one exterior. For cell complexes in <script type="math/tex">\mathbb{R}^3</script>, there appears to be no restriction on which combinations of surfaces are possible. But for cell complexes in other spaces (other 3-manifolds than Euclidean space) it may be possible to embed 2-manifolds with odd Euler characteristic. When this happens, the number of odd chambers of a cell complex must always be even. For, the parity of the sum of the Euler characteristics of the chambers must be even, in order to be able to divide by two and get an integer as the Euler characteristic of the cell complex.</p>
<p>(<a href="https://mathstodon.xyz/@11011110/101978143052398446">Discuss on Mastodon</a>)</p>David EppsteinFrom a block of cubes, remove two non-adjacent and non-opposite cubes. The resulting polycube has a boundary that is not a manifold: between the two removed cubes, there is an edge shared by four squares, but a two-dimensional manifold can only have two faces per edge. Nevertheless, we can compute its Euler characteristic as the number of vertices () minus the number of edges () plus the number of square faces (). , the same number we would expect for the Euler characteristic of a topological sphere! What does it mean?Linkage2019-04-15T17:43:00+00:002019-04-15T17:43:00+00:00https://11011110.github.io/blog/2019/04/15/linkage<ul>
<li>
<p><a href="https://aperiodical.com/2019/03/realhats-writing-a-latex-package/">“You know how the \hat command in LaTeΧ puts a caret above a letter? … Well I was thinking it would be funny if someone made a package that made the \hat command put a picture of an actual hat on the symbol instead?”</a>
And then Matthew Scroggs and Adam Townsend went ahead and <a href="https://ctan.org/pkg/realhats">did it</a> (<a href="https://mathstodon.xyz/@11011110/101849504150959463"><script type="math/tex">\mathbb{M}</script></a>).</p>
</li>
<li>
<p><a href="https://syntopia.github.io/Polytopia/polytopes.html">Generating 4d polyhedra from their symmetries</a> (<a href="https://mathstodon.xyz/@11011110/101860652773207990"><script type="math/tex">\mathbb{M}</script></a>, <a href="https://web.archive.org/web/20190306075446/https://plus.google.com/+RoiceNelson/posts/13EEovjAjh3">via</a>), by Mikael Hvidtfeldt Christensen.</p>
</li>
<li>
<p><a href="https://windowsontheory.org/2019/04/03/focs-2019-real-website-and-submission-server/">Windows on Theory</a> and <a href="https://www.scottaaronson.com/blog/?p=4154">Scott Aaronson</a> both warn about a fake web site for <a href="http://focs2019.cs.jhu.edu/">FOCS 2019</a>, whose submission deadline just passed (<a href="https://mathstodon.xyz/@11011110/101864693573223236"><script type="math/tex">\mathbb{M}</script></a>).</p>
</li>
<li>
<p><a href="http://service.ifam.uni-hannover.de/~geometriewerkstatt/gallery/index.html">GeometrieWerkstatt Gallery</a> (<a href="https://mathstodon.xyz/@11011110/101872198999374479"><script type="math/tex">\mathbb{M}</script></a>). A collection of weirdly-shaped mathematical surfaces, mostly of constant mean curvature.</p>
</li>
<li>
<p><a href="http://focs2019.cs.jhu.edu/awards/">Sandi Irani wins IEEE TCMF Distinguished Service Award</a> (<a href="https://mathstodon.xyz/@11011110/101877216457233962"><script type="math/tex">\mathbb{M}</script></a>). The award recognizes her work chairing the <a href="https://www.ics.uci.edu/~irani/safetoc.html">ad hoc committee to combat harassment and discrimination in the theory of computing community</a>, and then getting many theory conferences to follow its recommendations.</p>
</li>
<li>
<p><a href="http://web.colby.edu/thegeometricviewpoint/2014/04/25/periodic-billiard-paths/">Periodic billiard paths</a> (<a href="https://mathstodon.xyz/@11011110/101883476235740072"><script type="math/tex">\mathbb{M}</script></a>). If the boundary of a given polygon is made of mirrors, these are paths that a laser beam could take that would eventually reflect back to the starting point and angle and then repeat infinitely. It remains a heavily-studied open question whether such paths exist in every triangle. This blog post from 2014 provides a proof that they do exist in polygons whose vertex angles are all rational multiples of <script type="math/tex">\pi</script>.</p>
</li>
<li>
<p><a href="https://retractionwatch.com/2019/04/08/with-a-badly-handled-tweet-plos-angers-scientists-after-a-blog-disappears/">PLOS disappears one (or maybe more) of its hosted blogs</a> (<a href="https://mathstodon.xyz/@11011110/101894568161561631"><script type="math/tex">\mathbb{M}</script></a>) without any warning to the blog author, without any attempt at keeping old blog links still working, and with only a belated apology.</p>
</li>
<li>
<p><a href="https://slate.com/news-and-politics/2019/03/scotus-gerrymandering-case-mathematicians-brief-elena-kagan.html">The Supreme Court’s math problem</a> (<a href="https://mathstodon.xyz/@11011110/101900050555580515"><script type="math/tex">\mathbb{M}</script></a>, <a href="https://www.metafilter.com/180163/The-Supreme-Courts-Math-Problem">via</a>). Jordan Ellenberg explains why, in testing for gerrymandering, asking about deviation from proportional representation is the wrong question. Democratic systems naturally concentrate power to the majority rather than being proportional. The right question is whether that concentration is at the natural level, or is artificially accelerated in one direction or another.</p>
</li>
<li>
<p><a href="https://blog.archive.org/2019/04/10/official-eu-agencies-falsely-report-more-than-550-archive-org-urls-as-terrorist-content/">EU falsely calls Internet Archive’s major collection pages, scholarly articles, and copies of US government publications “terrorism” and demands they be taken down from the internet</a> (<a href="https://mathstodon.xyz/@11011110/101908397856087187"><script type="math/tex">\mathbb{M}</script></a>, <a href="https://boingboing.net/2019/04/11/one-hour-service.html">see also</a>). The EU is about to vote to require terrorism takedowns to happen within an hour, and these requests are coming on European times when all Internet Archive employees (in California) are asleep, making manual review of these bad takedowns difficult.</p>
</li>
<li>
<p><a href="https://www.siam.org/Conferences/CM/Main/apocs20">SIAM-ACM Conference on Algorithmic Principles of Computer Systems, APOCS</a> (<a href="https://mathstodon.xyz/@11011110/101909431804808574"><script type="math/tex">\mathbb{M}</script></a>). This is a new conference to be held with SODA, next January in Salt Lake City, covering “all areas of algorithms and architectures that offer insight into the performance and design of computer systems”. Submission titles and abstracts are due August 9 (with full papers due a week later) so if this is an area you’re interested in there’s still plenty of time to come up with something to submit.</p>
</li>
<li>
<p><a href="https://blog.computationalcomplexity.org/2019/04/elwyn-berlekamp-died-april-9-2019.html">Sad news from Berkeley</a>: <a href="https://en.wikipedia.org/wiki/Elwyn_Berlekamp">Elwyn Berlekamp</a> has died (<a href="https://mathstodon.xyz/@11011110/101921251002340100"><script type="math/tex">\mathbb{M}</script></a>, <a href="https://aperiodical.com/2019/04/elwyn-berlekamp-has-left-us/">see also</a>). Berlekamp made significant contributions to combinatorial game theory (motivated, as I understand it, by the mathematical study of Go endgames), coding theory, and algorithms for polynomials.</p>
</li>
<li>
<p><a href="https://www.berlintransitmap.de/">An unofficially-proposed new Berlin transit map replaces stylized axis-parallel and diagonal line segments with smooth curves</a> (<a href="https://mathstodon.xyz/@11011110"><script type="math/tex">\mathbb{M}</script></a>, <a href="https://www.metafilter.com/180431/Berlin-Transit-Map-now-with-pleasing-Curves">via</a>). The old design was seen as “out of style”, “too robotized”, and too difficult to follow routes. There’s still a strong preference for axis-parallel and diagonal lines in the new map, but the connections between them have been smoothed out.</p>
</li>
</ul>David Eppstein“You know how the \hat command in LaTeΧ puts a caret above a letter? … Well I was thinking it would be funny if someone made a package that made the \hat command put a picture of an actual hat on the symbol instead?” And then Matthew Scroggs and Adam Townsend went ahead and did it ().Monochromatic grids in colored grids2019-04-14T17:02:00+00:002019-04-14T17:02:00+00:00https://11011110.github.io/blog/2019/04/14/monochromatic-grids-colored<p>Color the points of an <script type="math/tex">n\times n</script> grid with two colors. How big a monochromatic grid-like subset can you find? By “grid-like” I mean that it should be possible to place equally many horizontal and vertical lines, partitioning the plane into <script type="math/tex">k\times k</script> cells each of which contains a single point.</p>
<p>So for the coloring of the <script type="math/tex">8\times 8</script> grid below, there are several <script type="math/tex">4\times 4</script> monochromatic grid-like subsets. The image below shows one, and the completely red and blue southwest and northeast quadrants provide two others.
The blue quadrant prevents any red grid-like subset from being larger than <script type="math/tex">4\times 4</script>, and vice versa, so these are the largest grid-like subsets in this grid.</p>
<p style="text-align:center"><img src="/blog/assets/2019/subgrid.svg" alt="Monochromatic grid-like subset in a colored grid" /></p>
<p>It’s not hard to prove that there always exists a monochromatic grid-like subset of size at least <script type="math/tex">\lfloor \sqrt{n} \rfloor\times \lfloor \sqrt{n} \rfloor</script>.
Just use vertical and horizontal lines to partition the big grid into blocks of that size. If one of those blocks is monochromatic, then it’s the grid-like subset you’re looking for. And if not, you can choose a red point from each block to get a grid-like subset of the same size.</p>
<p>In the other direction, there exist colorings of an <script type="math/tex">n\times n</script> grid for which the largest monochromatic grid-like subset has size only a little bigger, <script type="math/tex">O(\sqrt{n\log n})\times O(\sqrt{n\log n})</script>. To find such a coloring, partition the big grid into square blocks of size <script type="math/tex">O(\sqrt{n/\log n})\times O(\sqrt{n/\log n})</script>, and make each block monochromatic with a randomly chosen color.</p>
<p style="text-align:center"><img src="/blog/assets/2019/blocked-coloring.svg" alt="Coloring a grid by dividing into square blocks and coloring each block randomly" /></p>
<p>Now, consider any partition by axis-parallel lines into (irregular) rectangles, each containing one of the points of a grid-like subset. Only one row or column of the rectangles can cross each line of the partition into square blocks, so the number of rectangles that include parts of two or more square blocks is <script type="math/tex">O(\sqrt{n\log n})\times O(\sqrt{n\log n})</script>. Any remaining rectangles of the partition must come from a grid-like subset of square blocks that are all colored the same as each other. But with a random coloring, the expected size of this largest monochromatic subset of square blocks is <script type="math/tex">O(\log n)\times O(\log n)</script>. Therefore, the number of rectangles that stay within a single square block is limited to the total number of points in this grid-like subset of square blocks, which is again <script type="math/tex">O(\sqrt{n\log n})\times O(\sqrt{n\log n})</script>.</p>
<p>I’m not sure how to eliminate the remaining <script type="math/tex">O(\sqrt{\log n})</script> gap between these two bounds, or which way it should go.</p>
<p>One application of these ideas involves the theory of <a href="https://en.wikipedia.org/wiki/Superpattern">superpatterns</a>, permutations that contain as a <a href="https://en.wikipedia.org/wiki/Permutation_pattern">pattern</a> every smaller permutation up to some size <script type="math/tex">n</script>.
If <script type="math/tex">\pi</script> is a superpattern for the permutations of size <script type="math/tex">n</script>, then we can obtain a point set by interpreting the position and value of each element of <script type="math/tex">\pi</script> as Cartesian coordinates. This point set includes a grid-like subset of size <script type="math/tex">\lfloor \sqrt{n} \rfloor\times \lfloor \sqrt{n} \rfloor</script>, coming from a permutation of size <script type="math/tex">n</script> that translates to a grid-like set of points.
If the elements of the superpattern are colored with two colors, there still exists a monochromatic grid-like subset of size <script type="math/tex">O(n^{1/4})\times O(n^{1/4})</script>.
And this monochromatic grid-like subset corresponds to a superpattern, for permutations of size <script type="math/tex">O(n^{1/4})</script>. So, whenever the elements of a superpattern are colored with two (or finitely many) colors, there remains a monochromatic subset of elements that is still a superpattern for permutations of some smaller but non-constant size.</p>
<p>(<a href="https://mathstodon.xyz/@11011110/101927319466372642">Discuss on Mastodon</a>)</p>David EppsteinColor the points of an grid with two colors. How big a monochromatic grid-like subset can you find? By “grid-like” I mean that it should be possible to place equally many horizontal and vertical lines, partitioning the plane into cells each of which contains a single point.Coloring kinggraphs2019-04-11T21:28:00+00:002019-04-11T21:28:00+00:00https://11011110.github.io/blog/2019/04/11/coloring-kinggraphs<p>Draw a collection of quadrilaterals in the plane, meeting edge to edge, so that they don’t surround any open space (the region they cover is a topological disk) and every vertex interior to the disk touches at least four quadrilaterals. Is it always possible to color the corners of the quadrilaterals with four colors so that all four colors appear in each quadrilateral?</p>
<p style="text-align:center"><img src="/blog/assets/2019/colored-kinggraph.svg" alt="4-colored kinggraph" /></p>
<p>The graph of corners and quadrilateral edges is a <a href="https://en.wikipedia.org/wiki/Squaregraph">squaregraph</a> but this question is really about coloring a different and related graph, called a kinggraph, that also includes as edges the diagonals of each quad. It’s called that because one example of this kind of graph is the <a href="https://en.wikipedia.org/wiki/King's_graph">king’s graph</a> describing the possible moves of a chess king on a chessboard.</p>
<p style="text-align:center"><img src="/blog/assets/2019/kings-graph.svg" alt="The king's graph" /></p>
<p>The king’s graph, and kinggraphs more generally, are examples of 1-planar graphs, graphs drawn in the plane in such a way that each edge is crossed at most once. The edges of the underlying squaregraph are not crossed at all, and the diagonals of each quad only cross each other. Squaregraphs are bipartite (like every planar graph in which all faces have an even number of edges), so they can be colored with only two colors. 1-planar graphs, in general, can require six colors (for instance you can draw the complete graph <script type="math/tex">K_6</script> as a 1-planar graph by adding diagonals to the squares of a triangular prism) and this is tight.
And you can easily 4-color the king’s graph by using two colors in alternation across the even rows of the chessboard, and a different two colors across the odd rows. So the number of colors for kinggraphs should be somewhere between these two extremes, but where?</p>
<p>One useful and general graph coloring method is based on the <a href="https://en.wikipedia.org/wiki/Degeneracy_(graph_theory)">degeneracy</a> of graphs. This is the largest number <script type="math/tex">d</script> such that every subgraph has a vertex with at most <script type="math/tex">d</script> neighbors; one can use a <a href="https://en.wikipedia.org/wiki/Greedy_coloring">greedy coloring algorithm</a> to color any graph with <script type="math/tex">d+1</script> colors. Kinggraphs themselves always have a vertex with at most <script type="math/tex">3</script> neighbors, but unfortunately they do not have degeneracy <script type="math/tex">3</script>. If you form a king’s graph on a <script type="math/tex">4\times 4</script> chessboard, and then remove its four corners, you get a subgraph in which all vertices have at least four neighbors.
This turns out to be as large as possible: every kinggraph has degeneracy at most four. This is because, if you consider the zones of the system of quads (strips of quads connected on opposite pairs of edges), there always exists an “outer zone”, a zone with nothing on one side of it (see the illustration, below). You can remove the vertices of the outer zone one at a time, in order from one end to the other, always removing a vertex of degree at most four, and then repeat on another outer zone until the whole graph is gone. So the degeneracy and greedy coloring method shows that you can 5-color every kinggraph, better than the 6-coloring that we get for arbitrary 1-planar graphs.</p>
<p style="text-align:center"><img src="/blog/assets/2019/outer-zone.svg" alt="An outer zone" /></p>
<p>This turns out to be optimal! For a while I thought that every kinggraph must be 4-colorable, because it was true of all the small examples that I tried. But it’s not true in general, and here’s why. If you look at the zones of the 4-colored kinggraph above, you might notice a pattern. The edges that connect pairs of quads from the same zone have colorings that alternate between two different pairs of colors. For instance, we might have a zone that has red–yellow edges alternating with blue–green edges, or another zone that has red–blue edges alternating with green–yellow edges. This is true whenever a kinggraph can be 4-colored. But there are only three ways of coloring a zone (that is, of partitioning the four colors into two disjoint pairs, which alternate along the zone). And when two zones cross each other, they must be colored differently. So every 4-coloring of a kinggraph turns into a 3-coloring of its zones. But the graph that describes the zones and its crossings is a triangle-free <a href="https://en.wikipedia.org/wiki/Circle_graph">circle graph</a>, and vice versa: every triangle-free circle graph describes a kinggraph. And triangle-free circle graphs may sometimes need as many as five colors, in which case so does the corresponding kinggraph.</p>
<p>I posted <a href="/blog/2008/03/23/ageevs-squaregraph.html">an example of a squaregraph whose circle graph needs five colors</a> on this blog in 2008. Here’s a slightly different drawing of the same graph from <a href="/blog/2009/05/29/congratulations-dr-wortman.html">a later post</a>.
Because its circle graph is not 3-colorable, the corresponding kinggraph is not 4-colorable.</p>
<p style="text-align:center"><img src="/blog/assets/2009/cd220c.svg" alt="A squaregraph whose kinggraph is not 4-colorable" /></p>
<p>There are simpler examples of squaregraphs whose circle graph needs four colors. As long as the number of colors of the circle graph is more than three, the number of colors of the kinggraph will be more than four.</p>
<p>On the other hand, if you can color the circle graph with three colors, then it’s also possible to translate this 3-coloring of zones back into a 4-coloring of the kinggraph. Just remove an outer zone, color the remaining graph recursively, add the removed zone back, and use the color of the zone you removed to decide which colors to assign to its vertices. Unfortunately, I don’t know the computational complexity of testing whether a circle graph is 3-colorable. There was a conference paper by Walter Unger in 1992 that claimed to have a polynomial time algorithm, but without enough details and it was never published in a journal. I think we have to consider the problem of finding a coloring as still being open.</p>
<p>The same method also leads to an easy calculation of the number of 4-colorings (in the same sense) of the usual kind of chessboard with <script type="math/tex">n\times n</script> squares, or of a king’s graph with <script type="math/tex">(n+1)^2</script> vertices. In this case, the zones are just the rows and columns of the chessboard. We can use one color for the rows and two for the columns, or vice versa, so the number of 3-colorings of the zones (accounting for the fact that the 2-colorings get counted twice) is <script type="math/tex">3(2^{n+1}-2)</script>. And once the coloring of the zones is chosen, the coloring of the chessboard itself is uniquely determined by the color of any of its squares, so the total number of chessboard colorings is <script type="math/tex">12(2^{n+1}-2)</script>.</p>
<p>(<a href="https://mathstodon.xyz/@11011110/101911453128265954">Discuss on Mastodon</a>)</p>David EppsteinDraw a collection of quadrilaterals in the plane, meeting edge to edge, so that they don’t surround any open space (the region they cover is a topological disk) and every vertex interior to the disk touches at least four quadrilaterals. Is it always possible to color the corners of the quadrilaterals with four colors so that all four colors appear in each quadrilateral?Photos from Barbados2019-04-07T18:52:00+00:002019-04-07T18:52:00+00:00https://11011110.github.io/blog/2019/04/07/photos-from-barbados<p>I spent this year’s spring break at Erik Demaine’s annual computational geometry workshop in Barbados again. A few photos of workshop participants:</p>
<div><table style="margin-left:auto;margin-right:auto">
<tr style="text-align:center;vertical-align:middle">
<td style="padding:10px"><img src="http://www.ics.uci.edu/~eppstein/pix/bellairs19/9-m.jpg" alt="Hannah Alpert and Tom Hull" width="360" style="border-style:solid;border-color:black;" /></td>
<td style="padding:10px"><img src="http://www.ics.uci.edu/~eppstein/pix/bellairs19/13-m.jpg" alt="Jonathan Shoemaker, Ryuhei Uehara, and Bob Hearn" width="266" style="border-style:solid;border-color:black;" /></td>
</tr><tr style="text-align:center;vertical-align:middle">
<td style="padding:10px"><img src="http://www.ics.uci.edu/~eppstein/pix/bellairs19/26-m.jpg" alt="Belén Palop and Vera Sacristán" width="240" style="border-style:solid;border-color:black;" /></td>
<td style="padding:10px"><img src="http://www.ics.uci.edu/~eppstein/pix/bellairs19/10-m.jpg" alt="Yushi Uno, Walker Anderson, Ryan Williams, Dylan Hendrickson, and Jayson Lynch" width="360" style="border-style:solid;border-color:black;" /></td>
</tr></table></div>
<p>This time around, as well as the usual suspects, Erik included two high school students, incoming to MIT. You can see them at the left of the second and fourth photos.</p>
<p><a href="http://www.ics.uci.edu/~eppstein/pix/bellairs19/index.html">Here are the rest of my photos</a>.</p>
<p>(<a href="https://mathstodon.xyz/@11011110/101888686019377462">Discuss on Mastodon</a>)</p>David EppsteinI spent this year’s spring break at Erik Demaine’s annual computational geometry workshop in Barbados again. A few photos of workshop participants:Linkage2019-03-31T16:48:00+00:002019-03-31T16:48:00+00:00https://11011110.github.io/blog/2019/03/31/linkage<ul>
<li>
<p><a href="https://arxiv.org/abs/1903.04304">A 3-regular matchstick graph of girth 5 consisting of 54 vertices</a>, Mike Winkler, Peter Dinkelacker, and Stefan Vogel (<a href="https://mathstodon.xyz/@11011110/101761530841835018"><script type="math/tex">\mathbb{M}</script></a>). The previous smallest-known graph with these properties had 180 vertices, but this one might still not be optimal, as the known lower bound is only 30. I found it difficult to understand the connectivity of the graph from <a href="https://commons.wikimedia.org/wiki/File:Winkler_3-reg_girth5_54.svg">its matchstick representation</a> so I made another drawing of the same graph in a different style:</p>
<p style="text-align:center"><img src="/blog/assets/2019/girth-5-matchstick.svg" alt="The 54-vertex 3-regular girth-5 matchstick graph of Winkler, Dinkelacker, and Vogel in Lombardi drawing style" /></p>
</li>
<li>
<p><a href="https://www.reddit.com/r/plexodus/comments/az285j/saving_of_public_google_content_at_the_internet/">Saving public Google+ content at the Internet Archive</a> (<a href="https://mathstodon.xyz/@11011110/101767934776473984"><script type="math/tex">\mathbb{M}</script></a>, <a href="https://news.ycombinator.com/item?id=19407865">via</a>). Archive.org and the <a href="https://www.archiveteam.org/">Archive Team</a> (a separate “loose collective of rogue archivists, programmers, writers and loudmouths dedicated to saving our digital heritage”) are undertaking an organized effort to capture it all before Google takes it down at the end of the month.</p>
</li>
<li>
<p><a href="https://mathstodon.xyz/@shonk/101807604452877379">Symmetric minimality</a>, Clayton Shonkwiler. An animation of a symmetric minimal-length grid realization of a trefoil knot, posted after I complained that <a href="https://mathstodon.xyz/@shonk/101768569662399661">his earlier animation</a> was too asymmetric, and suggested using the realization below (drawn using <a href="http://vzome.com/home/">vZome</a>). Clayton also posted a <a href="https://mathstodon.xyz/@shonk/101819922406308000">bcc figure-8</a> and <a href="https://mathstodon.xyz/@shonk/101825678378019686">bcc <script type="math/tex">7_7</script></a>. Not every symmetric knot has a symmetric lattice realization (e.g. the <script type="math/tex">(5,2)</script> lattice knot doesn’t) but it may be an interesting question whether every lattice-realizable symmetry of a knot can be realized by a minimal lattice knot.</p>
<p style="text-align:center"><img src="/blog/assets/2019/zomeknot.png" alt="Symmetric lattice trefoil knot drawn using vZome" /></p>
</li>
<li>
<p><a href="https://en.wikipedia.org/wiki/Karen_Uhlenbeck">Karen Uhlenbeck</a> becomes <a href="http://www.abelprize.no/c73996/seksjon/vis.html?tid=74011&strukt_tid=73996">the first woman to win the Abel Prize</a> (<a href="https://mathstodon.xyz/@JordiGH/101778053801507834"><script type="math/tex">\mathbb{M}</script></a>).</p>
</li>
<li>
<p><a href="http://corner.mimuw.edu.pl/?p=1076">The curse of the Euclidean metric</a> (<a href="https://mathstodon.xyz/@11011110/101785024820380513"><script type="math/tex">\mathbb{M}</script></a>). Krzysztof Fleszar posts about a big difficulty with algorithms for problems like Euclidean shortest paths where the answer is a sum of distances: we don’t know how to compare two solutions efficiently. Fortunately in the case of <a href="https://epubs.siam.org/doi/10.1137/1.9781611975482.67">Krzysztof’s SODA 2019 paper on approximate TSP of hyperplanes</a> (find a short tour that touches each given hyperplane) it’s possible to use approximate numerical comparisons.</p>
</li>
<li>
<p><a href="https://hal.archives-ouvertes.fr/hal-02070778/document">Integer multiplication in time <script type="math/tex">O(n\log n)</script></a> (<a href="https://mathstodon.xyz/@erou/101787744661719386"><script type="math/tex">\mathbb{M}</script></a>, <a href="https://rjlipton.wordpress.com/2019/03/29/integer-multiplication-in-nlogn-time/">see also</a>).</p>
</li>
<li>
<p><a href="https://www.latimes.com/entertainment/arts/la-ca-cm-inksap-linda-lack-20190314-story.html">An odd couple of Los Angeles street art</a> (<a href="https://mathstodon.xyz/@11011110/101794533654899105"><script type="math/tex">\mathbb{M}</script></a>): 20-something Inksap, a son of Vietnamese refugees, and 70-something dance teacher Linda Lack.</p>
</li>
<li>
<p><a href="https://archive.org/stream/BrainfillingCurves-AFractalBestiary/BrainFilling">Brain-filling curves: A fractal bestiary</a> (<a href="https://mathstodon.xyz/@11011110/101801960587162487"><script type="math/tex">\mathbb{M}</script></a>). A 200-page compendium by Jeffrey Ventrella of space-filling curves, generated by subdivision rules that expand line segments to lattice paths whose sum of squared edge lengths equals the square of the segment.</p>
</li>
<li>
<p><a href="http://www.graphics.rwth-aachen.de/software/zometool">Automatically translate any 3d model into a zometool approximation of it</a> (<a href="https://mathstodon.xyz/@11011110/101807397098535057"><script type="math/tex">\mathbb{M}</script></a>, <a href="http://www.zometool.com/news/amazing-app-zometool-shape-approximation/">via</a>). With obligatory Stanford bunny. See also their <a href="https://www.youtube.com/watch?v=piDPsHTLV1A">1-minute video introduction</a>.</p>
</li>
<li>
<p><a href="https://arstechnica.com/science/2019/03/physicists-are-decoding-math-y-secrets-of-knitting-to-make-bespoke-materials/">Physicists are decoding math-y secrets of knitting to make bespoke materials</a> (<a href="https://mathstodon.xyz/@11011110/101813393444829450"><script type="math/tex">\mathbb{M}</script></a>). Jennifer Ouelette writes on <em>Ars Technica</em> that by varying the stitching one can control both the stretchiness and shape of the resulting knit.</p>
</li>
<li>
<p><a href="https://mathml.igalia.com/news/2019/02/12/launch-of-the-project/">The MathML people are trying again</a> (<a href="https://mathstodon.xyz/@11011110/101820557668090856"><script type="math/tex">\mathbb{M}</script></a>, <a href="https://news.ycombinator.com/item?id=19344843">via</a>). Much as I think MathML has failed and that promotion by its proponents after its failure was already obvious directly caused the continued bad mathematics rendering on Wikipedia, I wish them well. A working web mathematics markup language (if it can be achieved) would be a step up from Javascripted LaTeX. Of course I’m not about to give up LaTeX in my source but that’s what markup processors are for.</p>
</li>
<li>
<p><a href="https://mathstodon.xyz/@ccppurcell/101822142519831628">Chris Purcell asks “who is the mathematician/scientist who has the most Wikipedia citations without having a Wikipedia page?”</a>. <a href="https://mathstodon.xyz/@11011110/101822690106826262">My guess</a> is Heidi Burgiel, co-author of frequently-cited book <em>The Symmetries of Things</em>.</p>
</li>
<li>
<p><a href="https://jmfork.github.io/2048/">Deterministic 2048</a> (<a href="https://mathstodon.xyz/@11011110/101830138376282978"><script type="math/tex">\mathbb{M}</script></a>). The next tile is always a 2 at the first available square. It should be possible to describe a strategy that applies to any <script type="math/tex">n\times n</script> board and that provably achieves a score of <script type="math/tex">2^{\omega(n)}</script> but apparently this is not known.</p>
</li>
<li>
<p><a href="https://adamsheffer.wordpress.com/2019/03/11/incidences-open-problem-part-1/">Adam Sheffer points out that, despite recent advances in incidence geometry, many problems remain open, and sets out on a project to catalog them</a> (<a href="https://mathstodon.xyz/@11011110/101837928344081808"><script type="math/tex">\mathbb{M}</script></a>). His starting points are the unit distances problem and incidences with nonlinear algebraic plane curves.</p>
</li>
<li>
<p><a href="https://gizmodo.com/rss-is-better-than-twitter-1833624929">Why you should be keeping up with news and long-form websites using RSS rather than twitter</a> (<a href="https://mathstodon.xyz/@11011110/101840580734668234"><script type="math/tex">\mathbb{M}</script></a>, <a href="https://news.ycombinator.com/item?id=19529442">via</a>). If you want to incorporate RSS feeds into your Mastodon stream, <a href="https://friend.camp/@darius/100897757870017125">Darius Kazemi’s RSS-to-Mastodon tool</a> looks useful, but I haven’t tried it, because so far I’m happy reading those feeds through NetNewsWire.</p>
</li>
<li>
<p><a href="https://www.insidehighered.com/news/2019/03/28/scholars-complain-visa-problems-ahead-international-conference-canada">International Studies Association participants unable to get visas to go to their conference in Toronto</a> (<a href="https://mathstodon.xyz/@11011110/101847752696450725"><script type="math/tex">\mathbb{M}</script></a>), a repeat of an issue I posted about for <a href="/blog/2019/01/31/linkage.html">NeurIPS in January</a>. According to the story, the proportion of participants with visa issues is similar to recent years when the conference has been in the US. Which is not a great target for Canada to aspire to…</p>
</li>
</ul>David EppsteinA 3-regular matchstick graph of girth 5 consisting of 54 vertices, Mike Winkler, Peter Dinkelacker, and Stefan Vogel (). The previous smallest-known graph with these properties had 180 vertices, but this one might still not be optimal, as the known lower bound is only 30. I found it difficult to understand the connectivity of the graph from its matchstick representation so I made another drawing of the same graph in a different style:Linkage2019-03-15T22:53:00+00:002019-03-15T22:53:00+00:00https://11011110.github.io/blog/2019/03/15/linkage<p>It’s the last day of classes for the winter quarter here at UCI, and a good time for some spring cleaning of old bookmarked links. Probably also a good time for a reminder that Google+ is shutting down in two weeks so if, like me, you still have links to it then you don’t have long to replace them with archived copies before it gets significantly more difficult.</p>
<ul>
<li>
<p><a href="https://www.universityofcalifornia.edu/press-room/uc-terminates-subscriptions-worlds-largest-scientific-publisher-push-open-access-publicly">The University of California cancels its subscriptions to Elsevier journals after failing to agree on open access</a> (<a href="https://mathstodon.xyz/@11011110/101672959195728560"><script type="math/tex">\mathbb{M}</script></a>). I support this decision, but it means that new papers in Elsevier journals will be harder for me to read. (I will still have access to pre-2019 papers.) More recently, <a href="https://www.insidehighered.com/quicktakes/2019/03/13/norwegian-universities-ditch-elsevier">Norway has also cancelled its subscriptions</a>.</p>
</li>
<li>
<p>My keyboard needed replacement but fortunately the <a href="https://www.apple.com/support/keyboard-service-program-for-macbook-and-macbook-pro/">Apple Keyboard Service Program</a> came through (<a href="https://mathstodon.xyz/@11011110/101683848227213394"><script type="math/tex">\mathbb{M}</script></a>).</p>
<p style="text-align:center"><img src="https://www.ics.uci.edu/~eppstein/pix/keyboard/keyboard-m.jpg" alt="Broken MacBook Pro Keyboard" style="border-style:solid;border-color:black;" /></p>
</li>
<li>
<p><a href="https://inference-review.com/article/a-crisis-of-identification">A crisis of identification</a> (<a href="https://mathstodon.xyz/@11011110/101689960199735892"><script type="math/tex">\mathbb{M}</script></a>, <a href="https://thehighergeometer.wordpress.com/2019/03/02/something-i-wrote/">via</a>). David Roberts summarizes the state of play in the claimed proof of the <a href="https://en.wikipedia.org/wiki/abc_conjecture"><script type="math/tex">abc</script> conjecture</a>. The linked site, <em>Inference Review</em>, looks like an interesting platform for <a href="https://inference-review.com/topic/mathematics/">essays on mathematics</a>.</p>
</li>
<li>
<p><a href="https://en.wikipedia.org/wiki/Plan_S">Plan S</a> is a push for funding agencies to require research to be open access. While it’s no surprise that <a href="https://www.insidehighered.com/news/2019/02/19/publishers-express-concern-about-unintended-consequences-plan-s">publishers would push back</a>, there are concerns from the side of open access “that the technical requirements are too high and will result in only large, well-funded publishers and repositories to become compliant”. See the opinions from the <a href="https://euro-math-soc.eu/news/19/02/8/feedback-ems-implementation-plan-s">European Mathematical Society</a>, <a href="https://blogs.cornell.edu/arxiv/2019/02/04/arxivs-feedback-on-the-guidance-on-the-implementation-of-plan-s/">arXiv</a>, and <a href="https://www.coar-repositories.org/news-media/coar-feedback-on-the-guidance-on-implementation-of-plan-s/">Confederation of Open Access Repositories</a> (<a href="https://mathstodon.xyz/@11011110/101699226916200843"><script type="math/tex">\mathbb{M}</script></a>).</p>
</li>
<li>
<p><a href="https://arstechnica.com/science/2019/03/nasa-visualizes-supersonic-shockwaves-in-a-new-awe-inspiring-way/">Visualization of the shockwaves created by supersonic aircraft</a> (<a href="https://mathstodon.xyz/@11011110/101704826218861968"><script type="math/tex">\mathbb{M}</script></a>), created by NASA using aerial <a href="https://en.wikipedia.org/wiki/Schlieren_photography">schlieren photography</a> and stunt piloting.</p>
</li>
<li>
<p>I’m currently preparing a couple of papers, in the LaTeX format for the <a href="https://www.dagstuhl.de/publikationen/lipics/">LIPIcs computer science conference proceedings series</a>. I’ve written here before about <a href="/blog/2018/05/25/lipics-autoref-lemma.html">the trickery needed to get the hyperref package and autoref macro to work in LIPIcs</a>. Now, finally, with the v2019 version of LIPIcs format, it’s much easier: just add [autoref] to the options in the documentclass. Why they don’t turn this on by default is beyond me (<a href="https://mathstodon.xyz/@11011110/101705996579583347"><script type="math/tex">\mathbb{M}</script></a>).</p>
</li>
<li>
<p><a href="http://digg.com/2019/straight-line-uk-to-new-zealand">Today we learned you can sail in a straight line from the UK to New Zealand</a> (<a href="https://mathstodon.xyz/@jhertzli/101558721134407781"><script type="math/tex">\mathbb{M}</script></a>).</p>
</li>
<li>
<p><a href="https://mathscinet.ams.org/mathscinet-getitem?mr=3791763">László Szabó reviews my book for MathSciNet</a> (<a href="https://mathstodon.xyz/@11011110/101718165708839697"><script type="math/tex">\mathbb{M}</script></a>, subscription required).</p>
</li>
<li>
<p><a href="https://doi.org/10.1080/00029890.2019.1535735">Cantarella, Needham, Shonkwiler, and Stewart write in the <em>Monthly</em> about an interesting smooth probability distribution on the shapes of triangles</a> (<a href="https://mathstodon.xyz/@shonk/101723146578421634"><script type="math/tex">\mathbb{M}</script></a>).</p>
</li>
<li>
<p><a href="https://en.wikipedia.org/wiki/Sums_of_three_cubes">Sums of three cubes</a> (<a href="https://mathstodon.xyz/@11011110/101725333452262574"><script type="math/tex">\mathbb{M}</script></a>, <a href="https://gilkalai.wordpress.com/2019/03/09/8866128975287528%C2%B3-8778405442862239%C2%B3-2736111468807040%C2%B3/">via</a>), a notoriously hard Diophantine equation for which <a href="https://people.maths.bris.ac.uk/~maarb/papers/cubesv1.pdf">Andrew Booker has found</a> the new solution</p>
<script type="math/tex; mode=display">33 = 8866128975287528^3+(-8778405442862239)^3+(-2736111468807040)^3.</script>
<p>But don’t listen to Stephen Wolfram when he tells you that <a href="https://www.wolframscience.com/nks/p789--implications-for-mathematics-and-its-foundations/">the simplest representation for <script type="math/tex">2</script></a> is</p>
<script type="math/tex; mode=display">2 = 1214928^3+3480205^3+(-3528875)^3.</script>
</li>
<li>
<p><a href="https://hungarytoday.hu/mta-head-minister-sign-letter-of-intent-on-future-of-academy-research-network/">László Lovász has agreed to allow the <script type="math/tex">\approx 5000</script> researchers of the Hungarian Academy of Sciences (MTA) to be separated from the academy itself</a> (<a href="https://mathstodon.xyz/@11011110/101735747787440743"><script type="math/tex">\mathbb{M}</script></a>). And <a href="https://www.chronicle.com/blogs/letters/open-letter-to-the-president-of-the-hungarian-academy-of-sciences-laszlo-lovasz/">an open letter</a> asks him to resist pressure from the Hungarian government to dismantle the MTA’s research centers and place researchers under “direct political control”. I don’t know enough to tell whether his agreement is resistance or a concession, but it warrants continued attention.</p>
</li>
<li>
<p><a href="https://www.youtube.com/watch?v=imfqczglelI">Vi Hart’s <script type="math/tex">\pi</script>-day video</a> (<a href="https://mathstodon.xyz/@11011110/101753216836690906"><script type="math/tex">\mathbb{M}</script></a>) conveys what I feel about the significance of random concatenations of digits. <a href="https://mathstodon.xyz/@andrewt">Andrew Taylor’s</a> Aperiodical piece on <a href="https://aperiodical.com/2019/03/buzz-in-when-you-think-you-know-the-answer/">average numbers of representations as sums of squares</a> is good too.</p>
</li>
<li>
<p><a href="https://www.quantamagazine.org/math-duo-maps-the-infinite-terrain-of-minimal-surfaces-20190312/"><em>Quanta</em> on the work of Fernando Codá Marques and André Neves on the existence of minimal surfaces within arbitrary geometric manifolds</a> (<a href="https://mathstodon.xyz/@11011110/101758577189877053"><script type="math/tex">\mathbb{M}</script></a>). The vague non-technical language is a little frustrating but I think that’s what they mean by “shape”.</p>
</li>
</ul>David EppsteinIt’s the last day of classes for the winter quarter here at UCI, and a good time for some spring cleaning of old bookmarked links. Probably also a good time for a reminder that Google+ is shutting down in two weeks so if, like me, you still have links to it then you don’t have long to replace them with archived copies before it gets significantly more difficult.