Jekyll2021-03-01T01:04:49+00:00https://11011110.github.io/blog/feed.xml11011110Geometry, graphs, algorithms, and moreDavid EppsteinLinkage2021-02-28T16:24:00+00:002021-02-28T16:24:00+00:00https://11011110.github.io/blog/2021/02/28/linkage<ul>
<li>
<p><a href="http://acm-stoc.org/stoc2021/accepted-papers.html">STOC 2021 accepted papers</a> (<a href="https://mathstodon.xyz/@11011110/105748480557219533">\(\mathbb{M}\)</a>).</p>
</li>
<li>
<p><a href="https://randomascii.wordpress.com/2021/02/16/arranging-invisible-icons-in-quadratic-time/">Arranging invisible icons in quadratic time</a> (<a href="https://mathstodon.xyz/@11011110/105756917532905626">\(\mathbb{M}\)</a>, <a href="https://news.ycombinator.com/item?id=26152335">via</a>). Yet another instance where using a too-slow algorithm causes a UI hang, with the twist that the better solution would not be to replace it with a faster algorithm, but instead to not do the useless thing that the bad algorithm does at all.</p>
</li>
<li>
<p><a href="https://joshdata.me/iceberger.html">Fun with shapes: draw an iceberg and see which way up and how deep it would float</a> (<a href="https://mathstodon.xyz/@11011110/105768276511155377">\(\mathbb{M}\)</a>, <a href="https://news.ycombinator.com/item?id=26201160">via</a>, <a href="https://www.metafilter.com/190533/Iceberger">via2</a>, <a href="https://boingboing.net/2021/02/20/make-your-own-iceberg-with-iceberger.html">via3</a>). Inspired by <a href="https://mobile.twitter.com/GlacialMeg/status/1362557149147058178">a twitter thread by Megan Thompson-Munson</a> pointing out that many supposed photos or illustrations of icebergs are fake and wrong.</p>
</li>
<li>
<p>Draw an infinite subgraph of the 3d integer lattice in which each vertex has four co-planar neighbors, in a perpendicular plane to each of its neighbors (<a href="https://mathstodon.xyz/@11011110/105771494222747316">\(\mathbb{M}\)</a>). This completely determines the subgraph, which is 4-regular and highly symmetric. It is the graph of adjacencies of the cubes in the <a href="https://en.wikipedia.org/wiki/Tetrastix">tetrastix structure</a>. Does this graph have a name and history?</p>
</li>
<li>
<p><a href="https://cacm.acm.org/magazines/2021/3/250708-gender-trends-in-computer-science-authorship">Gender trends in computer science authorship</a> (<a href="https://mathstodon.xyz/@11011110/105781841287243050">\(\mathbb{M}\)</a>). Takeaways for me (mostly from the barely-readable Fig. 4) are:</p>
<ul>
<li>
<p>Roughly one in four coauthors of CS research publications are currently female, up from a big dip of one in seven in the 1970s to 1990s.</p>
</li>
<li>
<p>Mathematics started lower and is currently more or less the same.</p>
</li>
<li>
<p>We are not on track to gender parity.</p>
</li>
</ul>
</li>
<li>
<p>I’m sad that the only way to find a viewable version of the 1991 short film <em><a href="https://en.wikipedia.org/wiki/Not_Knot">Not Knot</a></em> (on the hyperbolic geometry of knot complements) seems to be through pirate copies (<a href="https://mathstodon.xyz/@11011110/105785401264334824">\(\mathbb{M}\)</a>). Or you could pay $45 to Amazon for a copy on DVD. Do most people still have DVD players? At least they’re not still trying to sell it on VHS only.</p>
</li>
<li>
<p><a href="https://cscresearchblog.wordpress.com/2018/11/16/karp-sipser-heuristic-and-reductions/">On the slow spread of knowledge of nice theorems</a> (<a href="https://mathstodon.xyz/@11011110/105793165233864617">\(\mathbb{M}\)</a>), an amusing cartoon at the end of a longer blog post on fast graph matching heuristics.</p>
</li>
<li>
<p>Today’s LaTeX formatting tip (<a href="https://mathstodon.xyz/@11011110/105796107362586793">\(\mathbb{M}\)</a>): You know that bug where amsthm + hyperref, with one numbering for theorems and lemmas and corollaries and whatever, causes <code class="language-plaintext highlighter-rouge">\autoref</code> to call them theorems even when they’re really lemmas and corollaries and whatever? If you don’t, you’re lucky. Anyway, there’s a very simple workaround: after loading amsthm and hyperref, add one more package:</p>
<p><code class="language-plaintext highlighter-rouge">\usepackage[capitalize,nameinlink]{cleveref}</code></p>
<p>Then, just use <code class="language-plaintext highlighter-rouge">\cref</code> everywhere you were using <code class="language-plaintext highlighter-rouge">\autoref</code>. Problem solved!</p>
</li>
<li>
<p><a href="https://en.wikipedia.org/wiki/Lloyd%27s_algorithm">Lloyd’s algorithm</a> animated for 3d points (<a href="https://mathstodon.xyz/@tpfto/105553548210257285">\(\mathbb{M}\)</a>). See also <a href="https://mathstodon.xyz/@tpfto/105803635782297523">the spherical version</a>.</p>
</li>
<li>
<p><a href="https://rjlipton.wordpress.com/2021/02/27/new-old-ancient-results/">Applications of the no-3-in-line problem and cap-sets to complexity theory</a> (<a href="https://mathstodon.xyz/@11011110/105807834096788492">\(\mathbb{M}\)</a>). “What is most curious to us is that for matrix multiplication, the cap-set related technique frustrates a better complexity upper bound, whereas [for linear algebraic circuits] it frustrates a better lower bound.”</p>
</li>
<li>
<p><a href="https://www.bldgblog.com/2013/08/tensioned-suspension/">Tensioned suspension</a> (<a href="https://mathstodon.xyz/@11011110/105811049795181041">\(\mathbb{M}\)</a>, <a href="https://news.ycombinator.com/item?id=9093187">via</a>): sculptures by Dan Grayber in which the weight of mechanical linkages causes them to push out against the sides of their glass enclosures, seemingly causing them to hang suspended in air. More at <a href="http://www.dangrayber.com/">Grayber’s web site</a>.</p>
</li>
</ul>David EppsteinSTOC 2021 accepted papers (\(\mathbb{M}\)).Loops, degrees, and matchings2021-02-19T18:57:00+00:002021-02-19T18:57:00+00:00https://11011110.github.io/blog/2021/02/19/loops-degrees-matchings<p>A student in my graph algorithms class asked how <a href="Loop (graph theory)">self-loops</a> in undirected graphs affect the vertex degrees and matchings of a graph. The standard answer is that a self-loop adds two to the degree (because each edge has two endpoints) and that they are useless in matching because matchings should have at most one incidence to each vertex, not two. But that’s just a convention; one could reasonably declare that the contribution of a self-loop to the degree is one, and I’m pretty sure I’ve seen sources that do just that. With that alternative convention, it should be possible to include a self-loop in a matching, and use it to match only a single vertex.</p>
<p>However, this turns out not to make much difference to many matching problems, because the following simple transformation turns a problem with self-loops (allowed in matchings in this way) into a problem with no self-loops (so it doesn’t matter whether they are allowed or not). Simply form a <a href="https://en.wikipedia.org/wiki/Covering_graph">double cover</a>\(^*\) of the given graph (let’s call it the “loopless double cover”) by making two copies of the graph and replacing all corresponding pairs of loops by simple edges from one copy to the other. In weighted matching problems, give the replacement edges for the loops the sum of the weights of the two loops they replace; all other edges keep their original weights.</p>
<p style="text-align:center"><img src="/blog/assets/2021/loopless-double-cover.svg" alt="The loopless double cover of a graph and of one of its loopy matchings" /></p>
<p>Then (unlike the <a href="https://en.wikipedia.org/wiki/Bipartite_double_cover">bipartite double cover</a>, which also eliminates loops) the cardinality or optimal weight of a matching in the loopy graph can be read off from the corresponding solution in its loopless double cover. Any matching of the original loopy graph can be translated into a matching of the loopless cover by applying the same loopless cover translation to the matching instead of to the whole graph; this doubles the total weight of the matching and the total number of matched vertices. And among matchings on the loopless cover, when trying to optimize weight or matched vertices, it is never helpful to match the two copies differently, so there is an optimal solution that can be translated back to the original graph without changing its optimality.</p>
<p>This doesn’t quite work for the problem of finding a matching that maximizes the total number of matched edges, rather than the total number of matched vertices. These two problems are the same in simple graphs, but different in loopy graphs. However, in a loopy graph, if you are trying to maximize matched edges, you might as well include all loops in the matching, and then search for a maximum matching of the simple graph induced by the remaining unmatched vertices. Again, in this case, you don’t get a problem that requires any new algorithms to solve it.</p>
<p>In the case of my student, I only provided the conventional answer, because really all they wanted to know was whether these issues affected how they answered one of the homework questions, and the answer was that the question didn’t involve and didn’t need loops. However it seems that the more-complicated answer is that even if you allow loops to count only one unit towards degree, and to be included in matchings, they don’t change the matching problem much.</p>
<p>\(^*\) This is only actually a covering graph under the convention that the degree of a loop is one. For the usual degree-2 convention for loops, you would need to replace each loop by a pair of parallel edges, forming a multigraph, to preserve the degrees of the vertices.</p>
<p>(<a href="https://mathstodon.xyz/@11011110/105762400402127534">Discuss on Mastodon</a>)</p>David EppsteinA student in my graph algorithms class asked how self-loops in undirected graphs affect the vertex degrees and matchings of a graph. The standard answer is that a self-loop adds two to the degree (because each edge has two endpoints) and that they are useless in matching because matchings should have at most one incidence to each vertex, not two. But that’s just a convention; one could reasonably declare that the contribution of a self-loop to the degree is one, and I’m pretty sure I’ve seen sources that do just that. With that alternative convention, it should be possible to include a self-loop in a matching, and use it to match only a single vertex.Lattice Borromean rings2021-02-16T16:11:00+00:002021-02-16T16:11:00+00:00https://11011110.github.io/blog/2021/02/16/lattice-borromean-rings<p>A lot of topology is finding ways to prove things that are really obvious but where explaining why they’re obvious can be difficult. So I want to do this for a discrete analogue of <a href="https://en.wikipedia.org/wiki/Ropelength">ropelength</a>, the length of the shortest lattice representation, for the <a href="https://en.wikipedia.org/wiki/Borromean_rings">Borromean rings</a>. You can find several pretty lattice (and non-lattice) representations of the Borromean rings in a paper by Verhoeff & Verhoeff, “<a href="https://archive.bridgesmathart.org/2015/bridges2015-53.pdf">Three families of mitered Borromean ring sculptures</a>” [<em>Bridges</em>, 2015]; the one in the middle of their figure 2, thinned down to use only lattice edges and not thick solid components, is the one I have in mind. It is formed by three \(2\times 4\) rectangles, shown below next to <a href="https://en.wikipedia.org/wiki/Jessen%27s_icosahedron">Jessen’s icosahedron</a> which has the same vertex coordinates. (You can do the same thing with a regular icosahedron but then you get non-lattice golden rectangles.)</p>
<p style="text-align:center"><img src="/blog/assets/2021/Borromean-Jessen.svg" alt="Lattice Borromean rings and Jessen's icosahedron" /></p>
<p>Each of the three rectangles has perimeter \(12\), so the total length of the whole link is \(36\). Why should this be the minimum possible? One could plausibly run a brute force search over all small-enough realizations, but this would be tedious and some effort would be needed to prune the search enough to get it to run at all. Instead, I found an argument based on the lengths of the individual components of the link, allowing me to analyze them (mostly) separately.</p>
<p>Each component is unknotted, so it can be the boundary of a disk in space. Importantly, for the Borromean rings, every disk spanned by one of the components must be crossed at least twice by other components. If we could find a disk spanned by one component that was not crossed at all by other components, then we could shrink the first component topologically within its disk down to a size so small that it could easily be pulled apart from the other two components, something that is not possible with the Borromean rings. And if we could find a disk that was only crossed once by another component, then the <a href="https://en.wikipedia.org/wiki/Linking_number">linking number</a> of the two components would be one, something that doesn’t happen for the Borromean rings.</p>
<p>If you travel in some consistent direction around a cycle in a 3d lattice, every step in one direction along a coordinate axis must be cancelled by a step in the opposite direction elsewhere along the ring. So if a lattice cycle has length \(\ell\), there must be \(\ell/2\) pairs of opposite steps, partitioned somehow among the three dimensions. If the bounding box of the cycle has size \(a\times b\times c\), then we must have \(a+b+c\le\ell/2\), and we can classify the possible shapes of lattice cycles of length \(\ell\) by the possible shapes of their bounding boxes. This gives us the following cases:</p>
<ul>
<li>
<p>A lattice cycle of length \(\ell=4\) can only be a square, with bounding box dimensions \(1\times 1\times 0\) (the zero means that it lies in a single plane in 3d, not that it doesn’t exist at all). The square itself is a disk not crossed by any other lattice path, unusable as a component of the Borromean rings.</p>
</li>
<li>
<p>A lattice cycle of length \(\ell=6\) can be a rectangle with bounding box \(2\times 1\times 0\), or fully 3-dimensional with bounding box \(1\times 1\times 1\). There are two fully 3-dimensional cases, one that avoids two opposite vertices of the bounding box and one that avoids two adjacent vertices. The rectangle can be its own spanning disk, and in the 3-dimensional cases we can use a spanning disk connecting the center of the bounding cube by a line segment to each point along the ring. Neither of these types of disks is crossed by any other lattice path.</p>
<p style="text-align:center"><img src="/blog/assets/2021/grid-6-cycles.svg" alt="Spanning disks for three grid 6-cycles" /></p>
</li>
<li>
<p>A lattice cycle of length \(\ell=8\) can be a rectangle with bounding box \(3\times 1\times 0\), a square with bounding box \(2\times 2\times 0\) or fully 3-dimensional with bounding box \(2\times 1\times 1\). It can also double back on itself and cover all vertices of a cube, with bounding box \(1\times 1\times 1\). All cases except the \(2\times 2\times 0\) square can be handled as in the length \(6\) cases; for instance, for the \(2\times 1\times 1\) bounding box we form a disk at the center of the box, connected by a line segment to all points of the ring. These cases cannot be crossed by any other lattice path. The \(2\times 2\times 0\) square can be crossed by a lattice path, through its center point, but only by one path. We can see from this that the shortest lattice representation of the <a href="https://en.wikipedia.org/wiki/Hopf_link">Hopf link</a> (two linked circles) is the obvious one formed from two length-\(8\) squares. However, these squares are still too small to be used in the Borromean rings.</p>
</li>
<li>
<p>A lattice cycle of length \(\ell=10\) can be a rectangle with bounding box \(4\times 1\times 0\) or \(3\times 2\times 0\), or fully 3-dimensional with bounding box \(3\times 1\times 1\) or \(2\times 2\times 1\). The \(4\times 1\times 0\) rectangle and the center-point spanning disk of the \(3\times 1\times 1\) box cannot be crossed by any other lattice path, and the center-point spanning disk of the \(2\times 2\times 1\) can be crossed only by one, through its center edge. Using even-smaller bounding boxes doesn’t help.</p>
</li>
</ul>
<p>That leaves only one problematic case, the \(3\times 2\times 0\) rectangle, of perimeter \(10\), which is shorter than the rectangles of the optimal representation but can nevertheless be crossed by two other lattice paths. In fact, this rectangle can be used as a component in a representation of the Borromean rings. It is even possible to use two of them! (I’ll leave this as an exercise.) So we need some other argument to prove that, when we use one or two of these short rectangles, we have to make up for it elsewhere by making something else extra-long.</p>
<p>If a \(3\times 2\times 0\) rectangle is a component of the Borromean rings, it must be twice by one of the other components, because if its two crossings were from different components it would have nonzero linking number with both of them, different from what happens in the Borromean rings. And the crossings must happen at the two interior lattice points of the rectangle, through paths that (to avoid each other and the boundary of the rectangle) must pass straight across the rectangle, at least for one unit on each side. The component that crosses the rectangle in this way consists of two loops connecting the pairs of ends of these two straight paths; any other connection pattern would lead to linking number \(2\), not zero. We can think of these two loops as being separate cycles, shortcut by the lattice edges between the endpoints of the two straight paths. And any disk that spans either of these two loops must itself be crossed by another component of the Borromean rings, because if one of the loops had an uncrossed spanning disk then we could wrap a spanning disk for the rectangle around it (like a glove around a hand) and create an uncrossed spanning disk for the rectangle as well.</p>
<p style="text-align:center"><img src="/blog/assets/2021/glove.png" alt="Spanning disk wrapping around a loop like a glove around a hand, adapted from https://commons.wikimedia.org/wiki/File:Disposable_nitrile_glove_with_transparent_background.png" /></p>
<p>By the analysis above, in order to be crossed by something else, both of the two shortcut loops of the component that crosses the \(3\times 2\times 0\) rectangle must have length at least \(8\). Adding in the two straight paths (and removing the two shortcut edges) shows that the component itself must have length at least \(18\). And if we have one component of length \(18\) and two more of length \(10\), we get total length at least \(38\), more than the length of the minimal representation. Since all representations that use components of length less than \(12\) are too long, the representation in which all three component lengths are exactly \(12\) must be the optimal one, QED.</p>
<p>Researching all this led me to an interesting paper by Dai, Ernst, Por, and Ziegler, “<a href="https://doi.org/10.1142/S0218216519500858">The ropelengths of knots are almost linear in terms of their crossing numbers</a>” [<em>J. Knot Theory and its Ramifications</em>, 2019]. Ropelength is the minimum length of a 3d representation that can be thickened to a radius-1 tube without self-intersections. (Some sources use diameter in place of radius; this changes the numeric values by a factor of two but does not change the optimizing representations.) Doubling the dimensions of a lattice representation gives you such a representation, and on the other hand one can find short lattice representations by following the thickened tubes of a ropelength representation, so ropelength and lattice length are within constant factors of each other. Dai et al. use this to show that knots that can be drawn in the plane with few crossings also have small ropelength. It doesn’t work to use the plane embedding directly, adding a third coordinate to handle the crossings, because some planar graphs (like the <a href="https://en.wikipedia.org/wiki/Nested_triangles_graph">nested triangles graph</a>) have nonlinear total edge length in any planar lattice drawing. Instead, Dai et al show how to crumple up a planar drawing of any degree-four planar graph into a 3d integer lattice embedding of the graph, with near-linear total edge length, so that the faces of the drawing can also be embedded as disks that are not crossed by each other or the graph edges. One can then modify the lifted drawing to turn the degree-four vertices into crossings in the lifted topologically-planar surface formed by these faces, giving a grid representation of the original knot with near-linear total length.</p>
<p>The ropelength of the Borromean rings has also been the subject of some study. Doubling the grid rectangles and rounding off their corners produces three <a href="https://en.wikipedia.org/wiki/Stadium_(geometry)">stadia</a> with total perimeter \(12\pi+24\approx 61.7\). The same argument as above shows that each curve must be at least long enough for all its spanning disks to be crossable by two disjoint radius-1 tubes. Intuitively the smallest curve that can surround two tubes is a smaller stadium corresponding to the \(2\times 3\) rectangle, with length \(4\pi+4\). If so, this would give a lower bound of \(12\pi+12\approx 49.7\) for the total ropelength of the Borromean rings. The conjectured-optimal configuration, <a href="https://archive.bridgesmathart.org/2008/bridges2008-63.html">used for the logo of the International Mathematical Union</a>, uses three copies of a complicated two-lobed planar curve in roughly the same positions as the three rectangles or stadia; it is described carefully by Cantarella, Fu, Kusner, Sullivan, and Wrinkle, “<a href="http://dx.doi.org/10.2140/gt.2006.10.2055">Criticality for the Gehring link problem</a>” [<em>Geometry & Topology</em> 2006] (section 10), and has length \(\approx 58.006\). The intuition that the \(2\times 3\) stadium is the shortest curve that can surround two others also appears to be stated as proven in this paper, in section 7.1. But they state that the best lower bound for the Borromean ropelength is \(12\pi\) so maybe the \(12\pi+12\) argument above is new?</p>
<p><strong>Update, February 17:</strong> In email, John Sullivan pointed me to a paper by Uberti, Janse van Rensburg, Orlandinit, Tesi, and Whittington, “<a href="https://doi.org/10.1007/978-1-4612-1712-1_9">Minimal links in the cubic lattice</a>” [<em>Topology and Geometry in Polymer Science</em>, 1998; see table 2, p. 97], which does the tedious computer search and comes up with the same result, that the shortest length for a lattice representation of the Borromean rings is 36. (I had searched for papers on lattice representations of the Borromean rings but didn’t find this one, probably failing because it identifies the Borromean rings only by the <a href="https://en.wikipedia.org/wiki/Alexander%E2%80%93Briggs_notation">Alexander–Briggs notation</a> \(6_2^3\), which is hard to search for.) John also tells me that the proof of ropelength-minimality of the \(2\times 3\) stadium is only for links in which it is linked with two other components, different enough from the situation here in which every spanning disk is crossed twice that the same proof doesn’t apply. So the question of whether this stadium really is the ropelength minimizer for components satisfying this crossed-twice condition seems to fall into the category of obvious topological facts that are difficult to prove, rather than being already known.</p>
<p>(<a href="https://mathstodon.xyz/@11011110/105743724683948185">Discuss on Mastodon</a>)</p>David EppsteinA lot of topology is finding ways to prove things that are really obvious but where explaining why they’re obvious can be difficult. So I want to do this for a discrete analogue of ropelength, the length of the shortest lattice representation, for the Borromean rings. You can find several pretty lattice (and non-lattice) representations of the Borromean rings in a paper by Verhoeff & Verhoeff, “Three families of mitered Borromean ring sculptures” [Bridges, 2015]; the one in the middle of their figure 2, thinned down to use only lattice edges and not thick solid components, is the one I have in mind. It is formed by three \(2\times 4\) rectangles, shown below next to Jessen’s icosahedron which has the same vertex coordinates. (You can do the same thing with a regular icosahedron but then you get non-lattice golden rectangles.)Linkage2021-02-15T21:45:00+00:002021-02-15T21:45:00+00:00https://11011110.github.io/blog/2021/02/15/linkage<ul>
<li>
<p><a href="https://cacm.acm.org/opinion/articles/250078-lets-not-dumb-down-the-history-of-computer-science/fulltext">Let’s not dumb down the history of computer science</a> (<a href="https://mathstodon.xyz/@11011110/105663355639241434">\(\mathbb{M}\)</a>, <a href="https://www.metafilter.com/190214/Lets-Not-Dumb-Down-the-History-of-Computer-Science">via</a>). A 2014 plea from Knuth to historians of computer science to stop ignoring the technical parts of the history, reprinted this month in CACM.</p>
</li>
<li>
<p><a href="https://www.newscientist.com/article/2266041-tom-gaulds-runaway-lobster-telephone-problem/">Studies in ethical surrealism: the runaway lobster telephone problem</a> (<a href="https://mathstodon.xyz/@11011110/105664424099324982">\(\mathbb{M}\)</a>). I was pleased to learn that <a href="https://en.wikipedia.org/wiki/Lobster_Telephone">the lobster telephone depicted in this cartoon is a real objet d’art</a>.</p>
</li>
<li>
<p><a href="https://www.archim.org.uk/eureka/archive/">Archive of back issues of Eureka</a> (<a href="https://mathstodon.xyz/@11011110/105671433589477405">\(\mathbb{M}\)</a>, <a href="https://aperiodical.com/2021/02/aperiodical-news-roundup-january-2021/">via</a>), the recreational mathematics journal of the Cambridge Archimedeans, now online for open access. On Wikipedia, the popular articles from Eureka appear to be Dyson’s work on ranks of partitions, in #8, Haselgrove & Haselgrove on polyominoes, in #23, Penrose on pentaplexity, in #39, and Leinster on his eponymous groups, in #55.</p>
</li>
<li>
<p><a href="https://philpapers.org/rec/BOBFPT">In a new book chapter, Susanne Bobzien claims that famous philosopher of logic Gottlob Frege plagiarized extensively from the Stoic logicians</a> (<a href="https://mathstodon.xyz/@11011110/105680228594244342">\(\mathbb{M}\)</a>, <a href="https://dailynous.com/2021/02/03/frege-plagiarize-stoics/">via</a>, <a href="https://www.metafilter.com/190330/Frege-plagiarized-the-Stoics">via2</a>, <a href="https://handlingideas.blog/2021/02/05/the-stoic-foundations-of-analytic-philosophy-on-susanne-bobziens-groundbreaking-discovery-in-frege-and-prantl/">see also</a>).</p>
</li>
<li>
<p>How did I not know about the <a href="https://civs.cs.cornell.edu/">Condorcet Internet Voting Service</a> before (<a href="https://mathstodon.xyz/@11011110/105683306227631389">\(\mathbb{M}\)</a>)? Set up public or private polls and collate the results with your favorite Condorcet rank aggregation method (at least, if your favorite is one of the five they implement, which it probably is). Their public polls are kind of insipid, though, and in comments David Bremner brings up their past history of enabling online abusers.</p>
</li>
<li>
<p><a href="https://mathstodon.xyz/@RefurioAnachro/105684468712016832">Perkel’s graph and the 57-cell</a>, multi-post sequence on an abstract 4-polytope and associated distance-regular graph, by Refurio Anachro.</p>
</li>
<li>
<p><a href="https://blog.computationalcomplexity.org/2021/02/the-victoria-delfino-problems-example.html">The Victoria Delfino Problems</a> (<a href="https://mathstodon.xyz/@11011110/105694384937282737">\(\mathbb{M}\)</a>). Bill Gasarch blogs about mathematics problems named after non-mathematicians, in this case a Los Angeles based real estate agent.</p>
</li>
<li>
<p>The speech recognition system Zoom and/or my university are using to auto-caption my recorded lectures (whatever it is) really doesn’t like the word “bipartite”, heavily used in my lecture on matching (<a href="https://mathstodon.xyz/@11011110/105700305493373685">\(\mathbb{M}\)</a>). It came out “bipartisan”, “invite part tight”, “by party”, “by protect”, “by apartheid”, “by part aight”, and “by partnership”. Also “spanning forest” is now “Hispanic forest”, but mysteriously it got “spanning tree” right.</p>
</li>
<li>
<p><a href="https://retractionwatch.com/2021/02/09/20-ways-to-spot-the-work-of-paper-mills/">20 ways to spot the work of paper mills</a> (<a href="https://mathstodon.xyz/@11011110/105702699189961130">\(\mathbb{M}\)</a>). However one, using a non-institutional email address, is not “a bad global habit”, but deliberate. I have no thought of moving but do not want my entire professional life tied by email to my employer. My UCI address keeps student emails private but I tend to use gmail for off-campus concerns such as publishers. And not all scholars have institutions who can provide emails. If they refuse my email, I refuse to publish with them.</p>
</li>
<li>
<p><a href="https://arxiv.org/abs/2102.01543">Ben Green presents super-polynomial lower bounds for off-diagonal van der Waerden numbers \(W(3,k)\)</a> (<a href="https://mathstodon.xyz/@11011110/105713896982428000">\(\mathbb{M}\)</a>, <a href="https://gilkalai.wordpress.com/2021/02/08/to-cheer-you-up-in-difficult-times-20-ben-green-presents-super-polynomial-lower-bounds-for-off-diagonal-van-der-waerden-numbers-w3k/">via</a>). \(W(3,k)\) is the smallest \(N\) such that a 2-coloring of \([N]\) has a 3-term arithmetic progression of one color or a \(k\)-term progression of the other. It was previously known to be subexponential and thought to be only quadratic.</p>
</li>
<li>
<p><a href="https://mathoverflow.net/q/382940/440">The compound of an 11-simplex in an 11-hypercube (as a subset of its vertices) has the Mathieu group M11 as its symmetries</a> (<a href="https://mathstodon.xyz/@11011110/105717258433892969">\(\mathbb{M}\)</a>, <a href="https://cp4space.hatsya.com/2021/02/08/a-curious-construction-of-the-mathieu-group-m11/">via</a>). The via link goes on to describe how to find two dual 11-simplices in the same hypercube from the perfect ternary Golay code, much like the two simplices in a 3-cube that form the stella octangula.</p>
</li>
<li>
<p><a href="https://distill.pub/selforg/2021/textures/">Self-organizing textures</a> (<a href="https://mathstodon.xyz/@11011110/105719504474157451">\(\mathbb{M}\)</a>, <a href="https://news.ycombinator.com/item?id=26112959">via</a>). A small input image + “neural cellular automata” magic leads to organic-looking image textures.</p>
</li>
<li>
<p><a href="https://cse.buffalo.edu/socg21/accepted.html">Accepted papers for the Symposium on Computational Geometry</a> (SoCG; <a href="https://mathstodon.xyz/@11011110/105727235598338446">\(\mathbb{M}\)</a>). Decisions are out for the Symposium on Theory of Computing (STOC) but I haven’t seen a public list yet. Upcoming submission deadlines include the <a href="https://projects.cs.dal.ca/wads2021/">Algorithms and Data Structures Symposium</a> (WADS, Feb. 20), <a href="https://wg2021.mimuw.edu.pl/">Graph-Theoretic Concepts in Computer Science</a> (WG, Mar. 3), and the new <a href="https://www.siam.org/conferences/cm/conference/acda21">SIAM Conference on Applied and Computational Discrete Algorithms</a> (ACDA21, Mar. 1).</p>
</li>
<li>
<p><a href="http://jdh.hamkins.org/ode-to-hippasus/">A new contribution of Hypatia to mathematics</a> (<a href="https://mathstodon.xyz/@11011110/105732195870260348">\(\mathbb{M}\)</a>). Not the ancient Hypatia, but Hypatia Hamkins, and her parents, philosopher Barbara Gail Montero and logician Joel David Hamkins; the contribution is a verse proof of the irrationality of \(\sqrt{2}\).</p>
</li>
<li>
<p><a href="https://www.youtube.com/channel/UC8bRNi3tJX-tfR_RMtyWR7w">Computational Geometry YouTube channel</a> (<a href="https://mathstodon.xyz/@11011110/105739350560629546">\(\mathbb{M}\)</a>). This has been set up by Sariel Har-Peled and Sándor Fekete, and is recording talks from the New York Geometry Seminar. So far there are eleven, of roughly an hour length each.</p>
</li>
</ul>David EppsteinLet’s not dumb down the history of computer science (\(\mathbb{M}\), via). A 2014 plea from Knuth to historians of computer science to stop ignoring the technical parts of the history, reprinted this month in CACM.Linkage2021-01-31T15:57:00+00:002021-01-31T15:57:00+00:00https://11011110.github.io/blog/2021/01/31/linkage<ul>
<li>
<p><a href="https://www.quantamagazine.org/mathematicians-probe-unsolved-hilbert-polynomial-problem-20210114/">Hilbert’s 13th, unsolved</a> (<a href="https://mathstodon.xyz/@11011110/105569819827303922">\(\mathbb{M}\)</a>). You can solve polynomials of degree at most four using one-argument algebraic functions like \(\sqrt x\). If \(RD(n)\) denotes the number of arguments needed for degree-\(n\) polynomials, <span style="white-space:nowrap">then \(RD(4)=1\).</span> Hilbert asked whether \(RD(7)=2\). Vladimir Arnold showed in the 1950s that you can solve all polynomials with two-variable continuous (but not algebraic) functions, but mathematicians are only now catching on that the algebraic problem is still open. See also <a href="https://arxiv.org/abs/2001.06515">some recent bounds on \(RD\)</a>.</p>
</li>
<li>
<p><a href="https://twitter.com/joshmillard/status/1349979253937381379">Fogleworms</a> (<a href="https://mastodon.social/@joshmillard/105572806531271932">\(\mathbb{M}\)</a>), partitions of \(n\times n\) grids into \(n\)-vertex grid paths, their enumeration, and a crafty project to visualize them.</p>
</li>
<li>
<p><a href="https://doi.org/10.1007/s00283-020-10034-w">A figure with Heesch number 6: Pushing a two-decade-old boundary</a> (<a href="https://mathstodon.xyz/@11011110/105579308071467320">\(\mathbb{M}\)</a>), Bojan Bašić in the <em>Mathematical Intelligencer</em>. When a shape cannot tile the plane, its <a href="https://en.wikipedia.org/wiki/Heesch%27s_problem">Heesch number</a> measures how far you can tile before getting stuck: if you surround the shape by layers of the same shape, how many layers can you make? Casey Mann’s previous record of five looked like a row of five hexagons with extra crenellations. This one uses six hexagons, with simpler crenellations.</p>
</li>
<li>
<p><a href="https://blogs.scientificamerican.com/roots-of-unity/computation-in-service-of-poetry/">Exponentiation by squaring, in somewhat cryptic form, in a work by Pingala from India from over 2000 years ago</a> (<a href="https://mathstodon.xyz/@11011110/105586527250804920">\(\mathbb{M}\)</a>).</p>
</li>
<li>
<p>I’ve been trying to understand the structure of the Kurilpa Bridge in Brisbane, supposedly “the world’s largest tensegrity bridge” (<a href="https://mathstodon.xyz/@11011110/105592704371209563">\(\mathbb{M}\)</a>). The clearest description I’ve found is from <a href="http://tadashidesign.com/kurilpa-bridge">Tadashi Design</a>. <a href="https://en.wikipedia.org/wiki/Kurilpa_Bridge">Wikipedia</a> is more cagy, calling it a “hybrid tensegrity bridge”, as it also includes features of cable-stayed bridges where the deck hangs from cables attached to tower piers. But the Tadashi Design site shows long sections far from the piers, so it seems the tensegrity is not just for show.</p>
</li>
<li>
<p><a href="https://www.nationalgeographic.com/science/2021/01/we-need-better-face-masks-and-origami-might-help/">How origami folding patterns might help in the design of better face masks</a> (<a href="https://mathstodon.xyz/@11011110/105595412289181842">\(\mathbb{M}\)</a>).</p>
</li>
<li>
<p>More US universities using covid as an excuse to treat faculty badly (<a href="https://mathstodon.xyz/@11011110/105609583164670586">\(\mathbb{M}\)</a>): <a href="https://www.chronicle.com/article/kansas-regents-allow-sped-up-dismissals-of-tenured-faculty-members">the Kansas state university system guts tenure</a>, and <a href="https://www.insidehighered.com/news/2021/01/21/u-florida-asks-students-report-professors-who-arent-teaching-person">the University of Florida asks students to snitch on faculty who refuse to endanger themselves by teaching in person</a>.</p>
</li>
<li>
<p><a href="https://www.departures.com/lifestyle/architecture/hayri-atak-design-sarcostyle-building-manhattan-skyline">Proposed New York waterfront tower is a handlebody of high genus</a> (<a href="https://mathstodon.xyz/@11011110/105612316794720508">\(\mathbb{M}\)</a>, <a href="https://mastodon.social/@sarielhp/105612251833052747">via</a>, <a href="https://twitter.com/MathematicsUCL/status/1353328317781467137">via2</a>).</p>
</li>
<li>
<p>Although it also <a href="https://en.wikipedia.org/wiki/Book_(graph_theory)">has other names</a>, the graph \(K_{1,1,n}\) has been called the “thagomizer graph”, and its associated graphic matroid has been called the “thagomizer matroid” (<a href="https://mathstodon.xyz/@11011110/105620864011377814">\(\mathbb{M}\)</a>). The term appears to have been introduced by Katie Gedeon in <a href="https://arxiv.org/abs/1610.05349">arXiv:1610.05349</a> in honor of the famous Far Side cartoon, whose terminology has <a href="https://en.wikipedia.org/wiki/Thagomizer">also been adopted by some paleontologists</a>.</p>
</li>
<li>
<p>This new preprint looks interesting: <a href="https://arxiv.org/abs/2101.09592">Point-hyperplane incidence geometry and the log-rank conjecture, Noah Singer and Madhu Sudan, arXiv:2101.09592</a> (<a href="https://mathstodon.xyz/@11011110/105626816229116967">\(\mathbb{M}\)</a>). In the plane, \(n\) points and \(m\) lines can only touch \(\Theta\bigl((mn)^{2/3}+m+n\bigr)\) times. In 3d, points and planes can have mn incidences but only by sharing a common line. This paper connects similar problems in high dimensions to the <a href="https://en.wikipedia.org/wiki/Log-rank_conjecture">log-rank conjecture</a>, a famous unsolved problem in communication complexity.</p>
</li>
<li>
<p><a href="https://www.math.ucdavis.edu/research/seminars/?talk_id=6082">Unknot recognition in quasi-polynomial time</a> (<a href="https://mathstodon.xyz/@11011110/105630455655140054">\(\mathbb{M}\)</a>, <a href="https://www.scottaaronson.com/blog/?p=5270">via</a>). Title of talk announcement by Marc Lackenby. No details or preprint yet but judging solely from the title and non-fringe status of the author this sounds like big news.</p>
</li>
<li>
<p><a href="https://mathcs.clarku.edu/~fgreen/bookreviews/51-4.pdf">Frederic Green has published another review of my book “Forbidden Configurations in Discrete Geometry” in the latest <em>SIGACT News</em></a> (<a href="https://mathstodon.xyz/@11011110/105636589457644955">\(\mathbb{M}\)</a>, <a href="https://doi.org/10.1145/3444815.3444817">official but paywalled url</a>). Thanks to Joe O’Rourke for the heads-up: I last checked my mail at the office, where my physical copies of <em>SIGACT News</em> would go if they went anywhere, months ago, and even then it looked like magazines weren’t getting through.</p>
</li>
<li>
<p>Mathematics on the cutting block at Leicester again: <a href="https://gowers.wordpress.com/2021/01/30/leicester-mathematics-under-threat-again/">Gowers</a>, <a href="https://golem.ph.utexas.edu/category/2021/01/problems_at_the_university_of.html">nCat</a>, <a href="https://www.ipetitions.com/petition/mathematics-is-not-redundant">petition</a> (<a href="https://mathstodon.xyz/@11011110/105646674167288349">\(\mathbb{M}\)</a>). The plan is to eliminate research in pure mathematics at the University of Leicester, fire eight professors, and hire three back in purely teaching positions. I’m not sure who the eight are – the <a href="https://le.ac.uk/mathematics/people/academic-and-research">staff list</a> includes some other disciplines – but Leicester mathematicians in Wikipedia include <a href="https://en.wikipedia.org/wiki/Katrin_Leschke">Katrin Leschke</a> and <a href="https://en.wikipedia.org/wiki/Sergei_Petrovskii">Sergei Petrovskii</a>.</p>
</li>
<li>
<p>Two newly-listed Good Articles on Wikipedia: <a href="https://en.wikipedia.org/wiki/Curve_of_constant_width">Curve of constant width</a> and <a href="https://en.wikipedia.org/wiki/Ronald_Graham">Ronald Graham</a> (<a href="https://mathstodon.xyz/@11011110/105652461392545754">\(\mathbb{M}\)</a>).</p>
</li>
</ul>David EppsteinHilbert’s 13th, unsolved (\(\mathbb{M}\)). You can solve polynomials of degree at most four using one-argument algebraic functions like \(\sqrt x\). If \(RD(n)\) denotes the number of arguments needed for degree-\(n\) polynomials, then \(RD(4)=1\). Hilbert asked whether \(RD(7)=2\). Vladimir Arnold showed in the 1950s that you can solve all polynomials with two-variable continuous (but not algebraic) functions, but mathematicians are only now catching on that the algebraic problem is still open. See also some recent bounds on \(RD\).Which induced-subgraph problems are easy, and which are hard?2021-01-27T20:20:00+00:002021-01-27T20:20:00+00:00https://11011110.github.io/blog/2021/01/27/which-induced-subgraph<p>My latest preprint, <a href="https://arxiv.org/abs/2101.09918">arXiv:2101.09918</a> with Sid Gupta and Elham Havvaei, has a mouthful of a title: “Parameterized complexity of finding subgraphs with hereditary properties on hereditary graph classes”. It’s about finding large induced subgraphs with a given property within a larger graph, such as in an earlier paper I wrote on <a href="/blog/2014/08/26/planarization-by-vertex.html">finding large planar induced subgraphs</a>.</p>
<p>Since we’re looking for induced subgraphs, it makes sense to restrict attention to properties that behave nicely under induced subgraphs; these are called hereditary properties. And if there’s no restriction on what kind of graph the larger graph can be, then the parameterized complexity of this induced subgraph problem (parameterized by the number of vertices in the subgraph) is completely settled by a paper by Khot and Raman, “Parameterized complexity of finding subgraphs with hereditary properties” (<em>Theor. Comput. Sci.</em> 2002). The result depends only on whether the subgraphs you’re looking for include arbitrary large cliques or arbitrarily large independent sets. If both, then the answer to “is there a \(k\)-vertex induced subgraph with the property” is (for all sufficiently large inputs) yes, by <a href="https://en.wikipedia.org/wiki/Ramsey's_theorem">Ramsey’s theorem</a>, so the algorithmic problem is easy. If neither, then again Ramsey’s theorem says that (for all sufficiently large \(k\)) the answer is no, so again the problem is easy. And in all remaining cases, when the property includes large cliques but not large independent sets or vice versa, Khot and Raman show that the problem is hard for parameterized computation.</p>
<p>But what if the input is not allowed to be an arbitrary graph, but is restricted to another hereditary class of graphs? An example we consider is the problem of finding large planar induced subgraphs in <a href="https://en.wikipedia.org/wiki/Unit_disk_graph">unit disk graphs</a>. Planar graphs fall into the hard side of Khot and Raman’s dichotomy, but their hardness proof uses graphs that might not be unit disk graphs. Our results involve more case analysis based on cliques and independent sets, but for the input graph class as well as for the target subgraph class, so there are many more cases to consider. Again, Ramsey theory clears away many of them, either by saying that all sufficiently large inputs contain big subgraphs in the target class, or by limiting the number of subgraphs we need to consider to a finite set.</p>
<p>The case analysis from our paper focuses attention on one remaining case, where Ramsey-like arguments do not prevail. This is the case where the target subgraph class is one of the types that is hard for Khot and Raman’s dichotomy (such as finding induced planar subgraphs), and where the input subgraph class can contain both large cliques and large independent sets (as is true for the unit disk graphs). It would be nice to say that in this case everything is hard, providing a nice clean dichotomy, but some problems like this are not hard. For instance, when the input is a <a href="https://en.wikipedia.org/wiki/Cluster_graph">cluster graph</a>, a disjoint union of cliques, the largest induced planar subgraph is obtained by taking at most four vertices from each clique, and many other induced subgraph problems on cluster graphs are equally easy. We were at least able to find a couple of general hardness reductions that apply in many cases, including in the planar-in-unit-disk case. But the question of whether all problems in this case are either easy (fixed-parameter tractable) or hard, and if so whether there is an easy way of determining which side of the dichotomy any particular subgraph-searching problem falls into, remains open.</p>
<p>(<a href="https://mathstodon.xyz/@11011110/105631932001078385">Discuss on Mastodon</a>)</p>David EppsteinMy latest preprint, arXiv:2101.09918 with Sid Gupta and Elham Havvaei, has a mouthful of a title: “Parameterized complexity of finding subgraphs with hereditary properties on hereditary graph classes”. It’s about finding large induced subgraphs with a given property within a larger graph, such as in an earlier paper I wrote on finding large planar induced subgraphs.Bracing squaregraphs (and other rhombus tilings)2021-01-22T23:32:00+00:002021-01-22T23:32:00+00:00https://11011110.github.io/blog/2021/01/22/bracing-squaregraphs<p>A bookshelf or other structure made only of vertical and horizontal beams will easily fall over unless its joints are very strong, because axis-parallel structures are not inherently <a href="https://en.wikipedia.org/wiki/Structural_rigidity">rigid</a>. The vertical beams can tilt away from being perpendicular to the horizontals, without changing the relative spacing of the joints, and that can lead to books all over the floor. Here’s an example <a href="https://www.si.edu/newsdesk/photos/earthquake-damage-botany-library">from a 2011 earthquake at the Smithsonian</a> that didn’t get quite that far, but did damage some shelves beyond repair:</p>
<p style="text-align:center"><img src="/blog/assets/2021/nhb2011-01160-botany_library.jpg" alt="Damaged bookshelves in the Botany and Horticulture Library at the Smithsonian's National Museum of Natural History after the August 2011 Virginia earthquake, photographed by James DiLoreto, from https://www.si.edu/newsdesk/photos/earthquake-damage-botany-library" style="border-style:solid;border-color:black" width="65%" /></p>
<p>Triangular bookshelves would be annoying for different reasons, but they wouldn’t collapse like this, because triangular structures are more rigid: there is only one triangle that three sides of fixed lengths can form, unlike squares and rectangles which can flex into rhombi and parallelograms without changing their side lengths. So the standard solution to the bookshelf rigidity problem is to add <a href="https://en.wikipedia.org/wiki/Cross_bracing">cross bracing</a>, diagonal beams built into shelves or other rectangular structures that make them similarly rigid. But which parts to brace? Even if every vertical and horizontal segment of a square bookshelf is allowed to flex independently of the other ones on its same line, bracing all the squares of the bookshelf would be too many. The vertical segments in each row of squares must stay parallel (in any flex small enough to avoid self-crossings), and similarly the horizontal segments in each column of squares must stay parallel. So a square grid with \(r\) rows and \(c\) columns has only \(r+c-1\) degrees of freedom in its shape (the relative angles among the verticals of each of the \(r\) rows and the horizontals of each of the \(c\) columns), far fewer than the \(rc\) braces that would be obtained by bracing all the squares.</p>
<p style="text-align:center"><img src="/blog/assets/2021/grid-flex.svg" alt="Degrees of freedom of a flexible square grid" /></p>
<p>A beautiful theory showing how to minimally brace any square (or rectangular) grid, connecting this <a href="https://en.wikipedia.org/wiki/Grid_bracing">grid bracing problem</a> to the connectivity of bipartite graphs, was developed in the late 1970s and early 1980s by Ethan Bolker, <a href="https://en.wikipedia.org/wiki/Henry_Crapo_(mathematician)">Henry Crapo</a>, <a href="https://en.wikipedia.org/wiki/Jenny_Baglivo">Jenny Baglivo</a>, and Jack Graver.<sup id="fnref:bc" role="doc-noteref"><a href="#fn:bc" class="footnote">1</a></sup> <sup id="fnref:bg" role="doc-noteref"><a href="#fn:bg" class="footnote">2</a></sup> Their idea was to represent the grid and its bracing more abstractly, as a graph, whose vertices represent the rows and columns of grid squares. A braced square can be represented as an edge in this graph:</p>
<p style="text-align:center"><img src="/blog/assets/2021/grid-bracing.svg" alt="Representing a braced square grid as a graph" /></p>
<p>An edge in this graph, or more generally any chain of edges, fixes the relative angles between the vertical edges in the rows that it connects, and the horizontal edges of the columns. So if the whole graph is connected, as it is in the figure, all of these angles are fixed, and the grid is made rigid. In particular, any spanning tree of the complete bipartite graph on the rows and columns (such as the spanning tree shown in the figure) provides a bracing pattern that will suffice to make the grid rigid. It is also necessary for rigidity that the bracing pattern contain a spanning tree: any set of braces including a cycle is redundant, with one of the braces removable without changing the possible motions of the braced grid, and any forest that is not a tree has too few braces to eliminate all of the degrees of freedom. So a grid is braced rigidly if and only if its graph is connected, and the minimal rigid bracings are exactly the spanning trees.</p>
<p>The same theory can be (and has been) generalized in multiple ways. A grid is “double braced” if every brace is redundant: you can remove any one brace and the whole structure will stay rigid. This is true if and only if the corresponding graph has the property that you can remove any one edge and the whole graph will remain connected, which is true exactly for the connected <a href="https://en.wikipedia.org/wiki/Bridge_(graph_theory)">bridgeless graphs</a>. A version with directed graphs applies to bracing by strings or wires, strong under tension but useless under compression. These constrain the sides of the squares from turning only in one direction: if one side turns clockwise, the other side is forced to turn clockwise, or vice versa, but not both ways. So we can represent a tension brace in a square by a directed edge. The edge is directed from the square’s row to its column when a clockwise turn of the vertical sides in the row would force a clockwise turn of the horizontal sides in the column, and in the other direction otherwise. Then, the whole structure is rigid under this sort of tension bracing if and only if the resulting directed graph is strongly connected.</p>
<p>For this theory to work, it is necessary that the braced structure consist of quadrilaterals with parallel sides, but they don’t have to be squares: the same thing works for grids of rectangles, rhombi, or parallelograms. It would work for the deformed square grid in the right of the second figure, for instance. It’s also not necessary for the quadrilaterals to be connected in the pattern of a square grid. In 2006, Ture Wester observed that the same method should work for the rhombic version of the Penrose tiling, for instance, but was very vague about boundary conditions (not distinguishing carefully between infinite tilings and finite patches of them).<sup id="fnref:w" role="doc-noteref"><a href="#fn:w" class="footnote">3</a></sup> The correct boundary conditions are that this works for any tiling of a disk in the plane by finitely many parallel-sided quadrilaterals. For such tilings, one can group the quadrilaterals into zones, sequences of quadrilaterals connected edge-to-edge on opposite sides, so that each quadrilateral belongs to two zones (giving it a zone by itself if two opposite sides are both on the boundary of the disk). Without bracing, the shared parallel sides in each zone can turn independently of any other motion of the tiling; this is the part that requires that the tiled area be a disk rather than an annulus or more complicated shape, so that the two parts of the graph separated by each zone are disconnected from each other and can move independently. And the overall shape of the tiling is completely determined by the angles of the shared sides of all of its zones. Therefore, the number of degrees of freedom (factoring out motions in which the tiling translates without rotation, or rotates rigidly) is exactly one less than the number of zones. Any set of braced quadrilaterals can be represented as a subgraph of the intersection graph of the zones, and after the calculation of degrees of freedom, the same argument as for grids shows that the tiling is rigid if this subgraph connects all the zones, that it is double braced if this subgraph is connected and bridgeless, or that it is tension braced if the directed version of this subgraph is strongly connected.</p>
<p>So as Wester stated, this does work for rhombic Penrose tilings, as long as one considers disk-shaped patches of the tilings. Similar, it works for (simply-connected) polyominoes, or disk-shaped patches of the <a href="https://en.wikipedia.org/wiki/Rhombille_tiling">rhombille tiling</a>. It also works for <a href="https://en.wikipedia.org/wiki/Squaregraph">squaregraphs</a>, planar graphs in which all interior vertices have degree at least four and all interior faces are quadrilaterals, as long as those quadrilaterals are drawn with parallel sides. One of my old papers shows that these drawings always exist,<sup id="fnref:e" role="doc-noteref"><a href="#fn:e" class="footnote">4</a></sup> and another shows how to optimize them to avoid sharp angles.<sup id="fnref:ew" role="doc-noteref"><a href="#fn:ew" class="footnote">5</a></sup>. In all of these cases, the tiling can be made rigid by bracing tiles forming a spanning tree of its zones, doubly rigid by bracing tiles forming a bridgeless spanning subgraph, or rigid for tension bracing by adding braces that form a strongly connected graph.</p>
<p>However, some algorithmic aspects of this theory may depend more strongly on the fact that for square grids, the intersection graphs of their zones have a very simple structure. Square grids have complete bipartite zone intersection graphs, but more general tilings of disks by parallel-sided quadrilaterals have arbitrary <a href="https://en.wikipedia.org/wiki/Circle_graph">circle graphs</a> and squaregraphs have triangle-free circle graphs. Extending a partial bracing to a minimal rigid bracing is just a matter of extending a forest to a spanning tree, easy in all graphs, but the corresponding problem for tension bracing is more complicated. Gabow and Jordán solve it for square grids (equivalently, adding as few edges as possible to a bipartite directed graph to make it strongly connected, while respecting its given bipartition) in linear time.<sup id="fnref:gj" role="doc-noteref"><a href="#fn:gj" class="footnote">6</a></sup> But it is not at all obvious that it’s as easy to extend a directed subgraph of a circle graph to a minimal strongly connected subgraph of the same circle graph.</p>
<div class="footnotes" role="doc-endnotes">
<ol>
<li id="fn:bc" role="doc-endnote">
<p>Bolker, Ethan D., and Crapo, Henry (1977), “How to brace a one-story building”, <em>Environment and Planning B: Planning and Design</em> 4 (2): 125–152, <a href="https://doi.org/10.1068/b040125">doi:10.1068/b040125</a>. <a href="#fnref:bc" class="reversefootnote" role="doc-backlink">↩</a></p>
</li>
<li id="fn:bg" role="doc-endnote">
<p>Baglivo, Jenny A., and Graver, Jack E. (1983), “3.10 Bracing structures”, <em>Incidence and Symmetry in Design and Architecture</em>, Cambridge University Press, pp. 76–87. <a href="#fnref:bg" class="reversefootnote" role="doc-backlink">↩</a></p>
</li>
<li id="fn:w" role="doc-endnote">
<p>Wester, Ture (2006), “<a href="http://new.math.uiuc.edu/oldnew/quasicrystals/papers/TureWester.pdf">The structural morphology of Penrose and quasicrystal patterns, part I</a>”, <em>Adaptables2006, TU/e, International Conference On Adaptable Building Structures, Eindhoven, The Netherlands, 3–5 July 2006</em>, 10-290. <a href="#fnref:w" class="reversefootnote" role="doc-backlink">↩</a></p>
</li>
<li id="fn:e" role="doc-endnote">
<p>Eppstein, David (2004), “Algorithms for drawing media”, <em>Proc. 12th Int. Symp. Graph Drawing</em>, Springer, LNCS 3383, pp. 173–183, <a href="https://arxiv.org/abs/cs.DS/0406020">arXiv:cs.DS/0406020</a>. <a href="#fnref:e" class="reversefootnote" role="doc-backlink">↩</a></p>
</li>
<li id="fn:ew" role="doc-endnote">
<p>Eppstein, David, and Wortman, Kevin (2011), “Optimal angular resolution for face-symmetric drawings”, <em>J. Graph Algorithms and Applications</em> 15 (4): 551–564, <a href="https://doi.org/10.7155/jgaa.00238">doi:10.7155/jgaa.00238</a>. <a href="#fnref:ew" class="reversefootnote" role="doc-backlink">↩</a></p>
</li>
<li id="fn:gj" role="doc-endnote">
<p>Gabow, Harold N., and Jordán, Tibor (2000), “How to make a square grid framework with cables rigid”, <em>SIAM Journal on Computing</em> 30 (2): 649–680, <a href="https://doi.org/10.1137/S0097539798347189">doi:10.1137/S0097539798347189</a>. <a href="#fnref:gj" class="reversefootnote" role="doc-backlink">↩</a></p>
</li>
</ol>
</div>
<p>(<a href="https://mathstodon.xyz/@11011110/105603974591107625">Discuss on Mastodon</a>)</p>David EppsteinA bookshelf or other structure made only of vertical and horizontal beams will easily fall over unless its joints are very strong, because axis-parallel structures are not inherently rigid. The vertical beams can tilt away from being perpendicular to the horizontals, without changing the relative spacing of the joints, and that can lead to books all over the floor. Here’s an example from a 2011 earthquake at the Smithsonian that didn’t get quite that far, but did damage some shelves beyond repair:Linkage2021-01-15T18:18:00+00:002021-01-15T18:18:00+00:00https://11011110.github.io/blog/2021/01/15/linkage<ul>
<li>
<p><a href="https://cp4space.hatsya.com/2020/12/31/rigid-heptagon-linkage/">Rigid heptagon linkage</a> (<a href="https://mathstodon.xyz/@11011110/105484618560730198">\(\mathbb{M}\)</a>). Improvements in the size of unit distance graphs whose unit distance representation is forced to contain a regular heptagon, from 59 to 35 edges. Based on <a href="https://math.stackexchange.com/questions/3954719/is-this-braced-heptagon-a-rigid-graph">a math stackexchange question</a>. See also <a href="https://erich-friedman.github.io/mathmagic/0100.html">an old page on the same problem</a> from <a href="https://erich-friedman.github.io/">Erich Friedman’s site</a>, moved from its old location at Stetson University.</p>
</li>
<li>
<p><a href="https://shitpost.plover.com/g/graphviz-usa.html">How Graphviz thinks the USA is laid out</a> (<a href="https://mathstodon.xyz/@11011110/105487958342573170">\(\mathbb{M}\)</a>, <a href="https://news.ycombinator.com/item?id=25611053">via</a>).</p>
</li>
<li>
<p><a href="https://carlschwan.eu/2020/12/29/adding-comments-to-your-static-blog-with-mastodon/">Adding comments to your static blog with Mastodon</a> (<a href="https://mathstodon.xyz/@11011110/105496435649985707">\(\mathbb{M}\)</a>, <a href="https://news.ycombinator.com/item?id=25570268">via</a>). It’s a cute method for using JavaScript to pull the comments onto the blog page itself, but the method I’ve been using (just add a link from the page to the Mastodon thread and let people follow it from there to Mastodon) seems simpler.</p>
</li>
<li>
<p><a href="https://xkcd.com/2407/">xkcd on depth-first and breadth-first search</a> (<a href="https://mathstodon.xyz/@11011110/105499187435512986">\(\mathbb{M}\)</a>), just in time for my lecture reviewing these algorithms.</p>
</li>
<li>
<p><a href="https://www.alfonsobeato.net/math/the-many-ways-of-splitting-a-rectangle-in-many/">The many ways of splitting a rectangle — or, how to use mathematics to make using Zoom much more complicated for no particular benefit</a> (<a href="https://mathstodon.xyz/@11011110/105507831712183048">\(\mathbb{M}\)</a>, <a href="https://news.ycombinator.com/item?id=25633447">via</a>). I am very fond of rectangle partition problems but I don’t want to have to think about them just to talk to multiple people online.</p>
</li>
<li>
<p><a href="https://en.wikipedia.org/wiki/Doubly_triangular_number">Doubly triangular number</a> (<a href="https://mathstodon.xyz/@11011110/105511346991936371">\(\mathbb{M}\)</a>), new Wikipedia article on triangular numbers with a triangular-number index. These numbers come up in counting pairs of pairs of things, for instance in the illustration, which describes colorings of the corners of a square (up to symmetry) as pairs of colorings of pairs of opposite corners.</p>
<p style="text-align:center"><img src="/blog/assets/2021/Square_3-colorings.svg" alt="Coloring the corners of a square by combining colorings of pairs of opposite corners" /></p>
</li>
<li>
<p><a href="https://www.quantamagazine.org/some-math-problems-seem-impossible-that-can-be-a-good-thing-20201118/">How assigning impossible-to-solve problems can get mathematics students (or researchers) to take a step back and look at the bigger picture</a> (<a href="https://mathstodon.xyz/@11011110/105518753589309024">\(\mathbb{M}\)</a>). Patrick Honner in <em>Quanta</em>. I think it needs interactivity, though. It would be mean to do this on a problem set or exam.</p>
</li>
<li>
<p><a href="http://www.ams.org/publicoutreach/math-imagery/2020-Exhibition">Mathematical Art Exhibition from the January 2020 Joint Mathematical Meetings</a> (<a href="https://mathstodon.xyz/@11011110/105521691843483164">\(\mathbb{M}\)</a>, <a href="https://news.ycombinator.com/item?id=25687129">via</a>).</p>
</li>
<li>
<p>My latest puzzle (a belated Christmas present to myself) is the <a href="https://hanayama-toys.com/product/cast-twist/">Hanayama Twist</a> (<a href="https://mathstodon.xyz/@11011110/105527935339032381">\(\mathbb{M}\)</a>). I’m very pleased with it: an elegant symmetric two-piece design, solid feel in the hand, and a solution that is surprisingly complicated but not tediously long. My only complaint is that the solution is very linear, with almost no ways to go wrong if you keep moving on from things you’ve already done. Anyway <a href="/blog/assets/2021/twist-map.svg">here’s a map I drew to help me</a>, also used as an example of an implicit graph in my graph algorithms lectures.</p>
</li>
<li>
<p>In pre-covid lectures I hand-wrote notes as I talked, both to keep content fresh and to avoid racing through it too quickly. When we locked down I tried pre-recorded lectures with prepared slides, but it took too long to record and I missed the spontaneity. This term I’m back to live zoom lectures with prepared slides, and it seems to work: zoom chat keeps it spontaneous and I haven’t found myself going faster than when we were in person. Lesson learned, just in time to (I hope) unlock in fall. (<a href="https://mathstodon.xyz/@11011110/105540249087507369">\(\mathbb{M}\)</a>; see discussion for Pat Morin’s home whiteboard cat interruption.)</p>
</li>
<li>
<p><a href="https://bookzoompa.wordpress.com/2020/12/17/three-foldings-artful-modern-and-retro/">Two useful folds and a decorative one</a> (<a href="https://mathstodon.xyz/@11011110/105541359073675174">\(\mathbb{M}\)</a>). Paula Beardall Krieg. The useful ones seal a foil snack bag and wrap a sandwich securely in paper, without needing any fasteners.</p>
</li>
<li>
<p><a href="https://xenaproject.wordpress.com/2020/09/19/thoughts-on-the-pythagorean-theorem/">Thoughts on the Pythagorean theorem</a> (<a href="https://mathstodon.xyz/@11011110/105547435011690191">\(\mathbb{M}\)</a>). Or, what did Euclid actually mean by saying that two squares are equal to a third square? And how does this view relate to type-theoretic foundations? From the xena automatic theorem-proving project. Both the name and the horrifying illustrations for the blog posts come from the author’s daughter. For some heavier going, see <a href="https://xenaproject.wordpress.com/2020/06/05/the-sphere-eversion-project/">the post on perfectoid spaces and sphere eversion</a>.</p>
</li>
<li>
<p><a href="http://danbliss.blogspot.com/2011/11/">Regular star patterns for expanded US flags</a> (<a href="https://mathstodon.xyz/@11011110/105552860512312493">\(\mathbb{M}\)</a>, <a href="https://news.ycombinator.com/item?id=25772259">via</a>). I suspect that the likelihood of quickly adding Puerto Rico or the District of Columbia as states is not high, but just in case, there are some nice regular point arrangements available.</p>
</li>
<li>
<p>The figure below shows <a href="https://en.wikipedia.org/wiki/Desargues_configuration">two pentagons inscribed in each other</a>: each vertex of one pentagon lies on a line through a side of the other pentagon (<a href="https://mathstodon.xyz/@11011110/105555337730874472">\(\mathbb{M}\)</a>). You can’t do this with quadrilaterals in the Euclidean plane, but <a href="https://en.wikipedia.org/wiki/M%C3%B6bius%E2%80%93Kantor_configuration">you can in the complex projective plane</a>. From <a href="https://mathstodon.xyz/@christianp/105553391372028649">a thread of fun math facts</a> for Christian Lawson-Perfect’s birthday.</p>
<p style="text-align:center"><img src="/blog/assets/2021/Mutually-inscribed-pentagons.svg" alt="Two mutually-inscribed pentagons" /></p>
</li>
<li>
<p><a href="https://arxiv.org/abs/2101.04698">New preprint claims a proof of the Erdős-Faber-Lovász conjecture for all sufficiently large \(k\)</a> (<a href="https://mathstodon.xyz/@11011110/105561900948206748">\(\mathbb{M}\)</a>, <a href="https://gilkalai.wordpress.com/2021/01/14/to-cheer-you-up-in-difficult-times-17-amazing-the-erdos-faber-lovasz-conjecture-for-large-n-was-proved-by-dong-yeap-kang-tom-kelly-daniela-kuhn-abhishek-methuku-and-deryk-osthus/">via</a>), by Dong Yeap Kang, Tom Kelly, Daniela Kühn, Abhishek Methuku, and Deryk Osthus. The conjecture states that the union of \(k\) edge-disjoint \(k\)-cliques is \(k\)-colorable. See <a href="https://en.wikipedia.org/wiki/Erd%C5%91s%E2%80%93Faber%E2%80%93Lov%C3%A1sz_conjecture">its Wikipedia article</a> for a real-world-ish formulation with faculty committees as cliques and chairs in a common meeting room as colors.</p>
</li>
</ul>David EppsteinRigid heptagon linkage (\(\mathbb{M}\)). Improvements in the size of unit distance graphs whose unit distance representation is forced to contain a regular heptagon, from 59 to 35 edges. Based on a math stackexchange question. See also an old page on the same problem from Erich Friedman’s site, moved from its old location at Stetson University.Year-end linkage2020-12-31T21:32:00+00:002020-12-31T21:32:00+00:00https://11011110.github.io/blog/2020/12/31/year-end-linkage<ul>
<li>
<p><a href="https://www.quantamagazine.org/a-mathematicians-adventure-through-the-physical-world-20201216/">A mathematician’s unanticipated journey through the physical world</a> (<a href="https://mathstodon.xyz/@11011110/105394664282719115">\(\mathbb{M}\)</a>). <em>Quanta</em> profiles <a href="https://en.wikipedia.org/wiki/Lauren_Williams">Lauren Williams</a> and discusses her work on enumerating cells of Grassmannians and its unexpected connections with intersection patterns of solitons.</p>
</li>
<li>
<p><a href="https://www.insidehighered.com/news/2020/12/15/draconian-contract-proposals-connecticut">Connecticut state university system faculty contract negotiations go sour</a> (<a href="https://mathstodon.xyz/@11011110/105402517620055997">\(\mathbb{M}\)</a>). Proposals include increasing units taught per term, adding a term, halving units counted per hour of teaching, doubling required office hours, killing a cap on part-timers, adding required weekend teaching, eliminating faculty ownership of course content, eliminating funds for faculty travel and research, eliminating committee review of personnel actions, and monitoring emails for union activity.</p>
</li>
<li>
<p><a href="https://github.com/patmorin/lhp">Pat Morin implements the product structure for planar graphs in Python</a> (<a href="https://mathstodon.xyz/@patmorin/105396573803578266">\(\mathbb{M}\)</a>). This is part of a recent line of research in which planar graphs can be decomposed as subgraphs of strong products of paths with bounded-treewidth graphs. Pat writes: “Not exactly industrial-strength, and leans towards simplicity over performance. Still, it can decompose 100k-vertex triangulations in a few seconds.”</p>
</li>
<li>
<p><a href="http://make-origami.com/RonaGurkewitz/home.php">Modular origami polyhedra systems</a> (<a href="https://mathstodon.xyz/@11011110/105408938256987767">\(\mathbb{M}\)</a>). An old link by <a href="https://en.wikipedia.org/wiki/Rona_Gurkewitz">Rona Gurkewitz</a> from my Geometry Junkyard, moved to a new address.</p>
</li>
<li>
<p><a href="https://diagonalargument.com/2020/12/08/socrates-bad-guy/">Socrates as anti-democratic enabler of tyrannical coups</a> (<a href="https://mathstodon.xyz/@11011110/105419425427517260">\(\mathbb{M}\)</a>). None of this analysis is particularly new, but it’s not the version of Socrates you’ll see when you look at <a href="https://en.wikipedia.org/wiki/Socrates">the Wikipedia article</a>.</p>
</li>
<li>
<p><a href="https://en.wikipedia.org/wiki/Blichfeldt%27s_theorem">Blichfeldt’s theorem</a> (<a href="https://mathstodon.xyz/@11011110/105422739565106313">\(\mathbb{M}\)</a>): Any set in the plane of area greater than one can be translated to contain two integer points. New article on Wikipedia, connected to an expansion of the <a href="https://en.wikipedia.org/wiki/Hans_Frederick_Blichfeldt">biography of Hans Blichfeldt</a>, who came to the US from Denmark as a teenager in 1888 and worked for several years as a lumberman, railway worker, and surveyor before his mathematical talent was recognized and he became one of the first students at Stanford.</p>
</li>
<li>
<p>I’ve been experimenting with <a href="https://git-scm.com/docs/git-svn">git-svn</a> (<a href="https://mathstodon.xyz/@11011110/105426263611221446">\(\mathbb{M}\)</a>), for a new project whose coauthors chose svn. It somehow manages to be clunkier than either svn or git by themselves. I understand why svn’s linear history forces compromise, but mostly I tend to keep a linear history anyway. I just want git pull and push to work, but instead I have to learn new un-mnemonic commands (for which <a href="https://git.wiki.kernel.org/images-git/7/78/Git-svn-cheatsheet.pdf">the cheatsheet of equivalences from svn to git-svn</a> was helpful). The only advantage over svn that I found was you have a local copy of the history, although in the comments David Bremner suggests more.</p>
</li>
<li>
<p><a href="https://www.latimes.com/business/story/2020-12-22/agree-to-disagree">“Why I never ‘agree to disagree’ — I just tell you when you’re wrong”</a> (<a href="https://mathstodon.xyz/@11011110/105431453312440767">\(\mathbb{M}\)</a>). <em>Los Angeles Times</em> columnist Michael Hiltzik on truth versus neutrality, and why it is incorrect and intellectually lazy for public media to treat certain firmly-established facts — such as the existence of the COVID pandemic, the outcome of the recent US election, or human-driven climate change — as topics on which debate is still reasonable.</p>
</li>
<li>
<p><a href="https://www.atlasobscura.com/articles/best-christmas-cookie-cutter">The perfect Christmas cookie cutter: one that tessellates your cookie dough sheet with Christmas trees</a> (<a href="https://mathstodon.xyz/@11011110/105436497912189975">\(\mathbb{M}\)</a>).</p>
</li>
<li>
<p><a href="https://hardmath123.github.io/chaos-game-fractal-foliage.html">Using gradient descent to find Christmas-tree-shaped fractals</a> (<a href="https://mathstodon.xyz/@11011110/105439870724073086">\(\mathbb{M}\)</a>).</p>
</li>
<li>
<p><a href="https://en.wikipedia.org/wiki/Lars_Arge">Lars Arge</a> died of cancer on December 23 (<a href="https://mathstodon.xyz/@11011110/105447919719671152">\(\mathbb{M}\)</a>). You can read an <a href="https://cs.au.dk/news-events/news/show-news/artikel/in-memoriam-professor-lars-arge/">obituary by his department chair</a> and <a href="http://blog.geomblog.org/2020/12/lars-arge.html">another by Suresh Venkatasubramanian</a>. Lars was a leading researcher in algorithms for massive data, and an anchor for algorithms and computational geometry in Denmark. As Suresh writes, he was a larger-than-life figure; we’ll miss him.</p>
</li>
<li>
<p><a href="https://en.wikipedia.org/wiki/Reinhardt_polygon">Reinhardt polygons</a> (<a href="https://mathstodon.xyz/@11011110/105454229077474608">\(\mathbb{M}\)</a>). These polygons have equal side lengths and are inscribed in Reuleaux polygons. Among all convex polygons with the same number of sides (any number that is not a power of two), they have the largest possible perimeter for their diameter, the largest possible width for their diameter, and the largest possible width for their perimeter. New Wikipedia article.</p>
<p style="text-align:center"><img src="/blog/assets/2020/Reinhardt_15-gons.svg" alt="Reinhardt 15-gons" width="50%" /></p>
</li>
<li>
<p><a href="https://cacm.acm.org/magazines/2021/1/249441-reboot-the-computing-research-publication-systems/fulltext">Moshe Vardi suggests taking advantage of this year’s disruption to the dysfunctional computer science conference publication system by setting up a replacement that is more scalable and doesn’t involve large amounts of carbon-expensive travel</a> (<a href="https://mathstodon.xyz/@11011110/105460878037520050">\(\mathbb{M}\)</a>). He doesn’t really say what this new system (or old journal system?) should be, though, only that we should design it.</p>
<p>Relatedly, <a href="https://statmodeling.stat.columbia.edu/2020/12/23/update-on-ieees-refusal-to-issue-corrections/">IEEE relents on corrections to conference papers</a>.</p>
</li>
<li>
<p><a href="https://mathstodon.xyz/@christianp/105465163493715434">What winter looks like in different parts of the world</a>. Long photo-thread responding to a request by Christian Lawson-Perfect’s 3-year-old. Here’s <a href="https://mathstodon.xyz/@11011110/105466532506211765">my contribution</a>, with prickly pears and the distant snow-topped San Gabriel mountains in Southern California:</p>
<p style="text-align:center"><img src="https://www.ics.uci.edu/~eppstein/pix/pricklypearmountains/PricklyPearMountains-m.jpg" alt="Coastal prickly pear (opuntia littoralis) with visible fruit in the University of California, Irvine Ecological Preserve, looking north towards the San Gabriel Mountains, covered in snow from a recent storm" style="border-style:solid;border-color:black;" width="80%" /></p>
</li>
<li>
<p><a href="https://boingboing.net/2020/12/29/game-of-life-running-on-penrose-tiles.html">Game of Life running on Penrose tiles</a> (<a href="https://mathstodon.xyz/@11011110/105473798576179397">\(\mathbb{M}\)</a>). With links to a <em>New York Times</em> feature of “<a href="https://www.nytimes.com/2020/12/28/science/math-conway-game-of-life.html">short reflections from big thinkers on why Conway’s famous cellular-automata gewgaw remains so fascinating</a>”. From which I found Kjetil Golid’s generative-art <a href="https://generated.space/sketch/crosshatch-automata/">crosshatch automata</a>.</p>
</li>
<li>
<p>Two analyses of citation vs other impact in mathematics (<a href="https://mathstodon.xyz/@11011110/105477934629467245">\(\mathbb{M}\)</a>):</p>
<ul>
<li>
<p>In “<a href="https://www.ams.org/journals/notices/202101/rnoti-p114.pdf">Don’t count on it</a>” in the <em>Notices</em>, Edward Dunne compares highly cited mathematicians to winners of multiple prizes; high citations clustered in few topics, while prize winners were widely distributed across research areas.</p>
</li>
<li>
<p>In “<a href="https://arxiv.org/abs/2005.05389">Citations versus expert opinions</a>” (arXiv:2005.05389, <a href="https://retractionwatch.com/2020/12/19/weekend-reads-prof-sues-journal-school-after-demotion-following-retraction-researcher-fired-after-questioning-why-school-rejected-grant-the-authors-who-like-publish-or-perish/">via retractionwatch</a>), Smolinsky et al compare highly cited papers to MathSciNet featured reviews, again finding little overlap.</p>
</li>
</ul>
</li>
</ul>David EppsteinA mathematician’s unanticipated journey through the physical world (\(\mathbb{M}\)). Quanta profiles Lauren Williams and discusses her work on enumerating cells of Grassmannians and its unexpected connections with intersection patterns of solitons.Linkage2020-12-15T21:28:00+00:002020-12-15T21:28:00+00:00https://11011110.github.io/blog/2020/12/15/linkage<ul>
<li>
<p><a href="https://www.ams.org/journals/notices/202011/rnoti-p1692.pdf">3d-printed models of the chaotic attractors from dynamical systems</a> (<a href="https://mathstodon.xyz/@11011110/105309562849621245">\(\mathbb{M}\)</a>). Stephen K. Lucas, Evelyn Sander, and Laura Taalman in the cover article of the latest <em>Notices</em>.</p>
</li>
<li>
<p><a href="https://threadreaderapp.com/thread/1333670741590503425.html">Complete classification of tetrahedra whose angles are all rational multiples of \(\pi\)</a> (<a href="https://mathstodon.xyz/@11011110/105311921075649463">\(\mathbb{M}\)</a>, <a href="https://aperiodical.com/2020/12/aperiodical-news-roundup-november-2020/">via</a>). The original paper is “<a href="https://arxiv.org/abs/2011.14232">Space vectors forming rational angles</a>”, by Kiran S. Kedlaya, Alexander Kolpakov, Bjorn Poonen, and Michael Rubinstein.</p>
</li>
<li>
<p><a href="https://www.maa.org/programs/faculty-and-departments/classroom-capsules-and-notes/geometry-strikes-again">Geometry strikes again</a> (<a href="https://mathstodon.xyz/@11011110/105320692550081128">\(\mathbb{M}\)</a>, <a href="https://www.metafilter.com/189571/slaps-roof-this-bad-boy-can-fit-so-many-fucking-polyhedra-in-it">via</a>), Branko Grünbaum, <em>Math. Mag.</em> 1985. Somehow I don’t think I’d encountered this short paper before but it’s filled with many examples of horribly-drawn mathematics, one in the logo of the MAA. Worth reading as a warning for what not to do. Also for clear instructions on how to draw regular icosahedra correctly.</p>
</li>
<li>
<p><a href="https://www.technologyreview.com/2020/12/04/1013294/google-ai-ethics-research-paper-forced-out-timnit-gebru">Ethical issues in large-corpus natural language processing, or what’s behind the research that got Timnit Gebru kicked out of Google</a> (<a href="https://mathstodon.xyz/@11011110/105326128589157740">\(\mathbb{M}\)</a>, <a href="https://news.ycombinator.com/item?id=25311402">via</a>).</p>
</li>
<li>
<p><a href="https://jack.wrenn.fyi/blog/brown-location-surveillance/">How one university (Brown) tracks the physical locations of its students to ensure compliance with its pandemic safety policies</a> (<a href="https://mathstodon.xyz/@11011110/105330487336321470">\(\mathbb{M}\)</a>, <a href="https://news.ycombinator.com/item?id=25319392">via</a>). Most of it is pretty obvious: if you use a campus keycard or connect to a campus wireless network, they know you’re on campus.</p>
</li>
<li>
<p><a href="https://www.forbes.com/sites/madhukarpai/2020/11/30/how-prestige-journals-remain-elite-exclusive-and-exclusionary/?sh=1c14baef4d48">How prestige journals remain elite, exclusive and exclusionary</a> (<a href="https://mathstodon.xyz/@11011110/105345990074471874">\(\mathbb{M}\)</a>, <a href="https://retractionwatch.com/2020/12/05/weekend-reads-google-ai-researcher-fired-after-being-asked-to-retract-paper-journal-accused-of-stonewalling-on-paper-used-to-justify-human-rights-violations-reflecting-on-a-covid-19-retraction/">via</a>). Nature is charging up to €9,500 per paper in open-access fees, as much as some scientists in third-world countries earn in a year, making open-access publication inaccessible to people from low- and middle-income countries.</p>
</li>
<li>
<p><a href="https://en.wikipedia.org/wiki/Euclidean_distance">Euclidean distance</a> (<a href="https://mathstodon.xyz/@11011110/105348855359182047">\(\mathbb{M}\)</a>), now a Good Article on Wikipedia.</p>
</li>
<li>
<p><a href="https://www.quantamagazine.org/mathematician-solves-centuries-old-grazing-goat-problem-exactly-20201209/">Ingo Ullisch and the goats</a> (<a href="https://mathstodon.xyz/@btcprox/105354303333088090">\(\mathbb{M}\)</a>). A new solution to the problem of how to bisect the area of a circle by another circular arc centered on the first circle. But, given that it involves integrals and trig, is it really fair to call it “more exact” than the previous solution? I don’t think we even know whether the solution radius is transcendental (or transcendental over \(\pi\)).</p>
</li>
<li>
<p><a href="http://www.anilaagha.com/sculpturelooksee">Anila Quayyum Agha’s openwork sculptures cast intricate tessellated shadows on the surrounding surfaces</a> (<a href="https://mathstodon.xyz/@11011110/105360602658987905">\(\mathbb{M}\)</a>). See also <a href="https://en.wikipedia.org/wiki/Anila_Quayyum_Agha">her Wikipedia article</a> and two stories on her work, “<a href="http://canjournal.org/2019/11/between-light-and-shadow-at-the-toledo-museum-of-art/">Between light and shadow at the Toledo Museum of Art</a>” and “<a href="https://news.artnet.com/art-world/anila-quayyum-agha-interview-741371">Anila Quayyum Agha on drawing inspiration from darkness</a>”.</p>
</li>
<li>
<p>Pat Morin notes that it’s “good to see that the pandemic hasn’t affected every aspect of our lives”: <a href="https://mathstodon.xyz/@patmorin/105362925062596140">the registration fees for the online SODA conference are still way too high</a>.</p>
</li>
<li>
<p><a href="https://statmodeling.stat.columbia.edu/2020/12/10/ieees-refusal-to-issue-corrections/">IEEE has no mechanism to publish corrections or errata to conference proceedings papers</a> (<a href="https://mathstodon.xyz/@11011110/105369247233466112">\(\mathbb{M}\)</a>, <a href="https://retractionwatch.com/2020/12/12/weekend-reads-p-hacking-the-us-election-an-apparently-fake-author-sinks-a-stock-sued-for-using-a-research-tool/">via</a>), violating IEEE’s own code of ethics requiring authors “to acknowledge and correct errors”: Probably many other conference proceedings have similar issues.</p>
</li>
<li>
<p><a href="https://cp4space.hatsya.com/2020/12/13/shallow-trees-with-heavy-leaves/">Shallow trees with heavy leaves</a> (<a href="https://mathstodon.xyz/@11011110/105375238890363766">\(\mathbb{M}\)</a>). On “the general strategy of searching much fewer positions and expending more effort on each position”, and its application in using SAT solvers to find new spaceships in cellular automata.</p>
</li>
<li>
<p><a href="https://www.flyingcoloursmaths.co.uk/dictionary-of-mathematical-eponymy-the-xuong-tree">Dictionary of mathematical eponymy: The Xuong tree</a> (<a href="https://mathstodon.xyz/@11011110/105382771623486905">\(\mathbb{M}\)</a>, <a href="https://en.wikipedia.org/wiki/Xuong_tree">see also</a>), a special kind of spanning tree in graphs, used to embed them into surfaces with as high a genus as possible.</p>
</li>
<li>
<p><a href="https://journals.carleton.ca/jocg/index.php/jocg/article/view/461">An explicit PL-embedding of the square flat torus into \(\mathbb{E}^3\)</a> (<a href="https://mathstodon.xyz/@11011110/105385674493868301">\(\mathbb{M}\)</a>). The square torus is like the old Asteroids arcade game: a Euclidean square with boundary conditions that wrap around so if you move off one edge you re-enter at the corresponding point of the opposite edge. In 4d, it has a nice representation as the set \(\{(a,b,c,d)\mid a^2+b^2=c^2+d^2=1\}\), the Cartesian product of two circles. The <a href="https://en.wikipedia.org/wiki/Nash_embedding_theorem">Nash embedding theorem</a> gives it fractal embeddings in 3d, but Tanessi Quintanar finds it as a bona fide polyhedron.</p>
</li>
</ul>David Eppstein3d-printed models of the chaotic attractors from dynamical systems (\(\mathbb{M}\)). Stephen K. Lucas, Evelyn Sander, and Laura Taalman in the cover article of the latest Notices.