Jekyll2022-11-22T06:33:29+00:00https://11011110.github.io/blog/feed.xml11011110Geometry, graphs, algorithms, and moreDavid EppsteinA straight line through every face2022-11-21T22:08:00+00:002022-11-21T22:08:00+00:00https://11011110.github.io/blog/2022/11/21/straight-line-through<p>While updating my online publications list for something else I noticed that I had neglected to discuss one of my papers from earlier this fall: “Geodesic paths passing through all faces on a polyhedron” (with Demaine, Demaine, Ito, Katayama, Maruyama, and Uno), in the <a href="https://www.rs.tus.ac.jp/jcdcggg/JCDCG3-2022Proceedings(r2).pdf">booklet of abstracts from JCDCG<sup>3</sup> 2022</a>, the Japanese Conference on Discrete and Computational Geometry, Graphs, and Games.</p>
<p>The paper is kind of telegraphic, but the question it considers is easily stated. On the surface of a polyhedron, the analogue of a straight line is a geodesic, the shortest curve between two points. Which polyhedra have geodesics that cross through all of their faces? Maybe the 2d version is easier to explain: any two points on a convex polygon split the polygon into two arcs, and a geodesic is the shorter of the two. Which polygons have at least one geodesic that includes a segment from each edge?</p>
<p style="text-align:center"><img src="/blog/assets/2022/2d-univ-geodesics.svg" alt="Geodesics through all edges of a kite and a trapezoid" style="width:100%;max-width:600px" /></p>
<p>The endpoints of such a geodesic \(A\) must be in different edges, because if they were in the same edge then the complementary arc \(\bar A\) would be a straight line segment, shorter than any other arc. Those two edges must be adjacent, because otherwise \(\bar A\) would include an edge missed by \(A\). And these two adjacent edges must be longer than the sum of all the other edges, so that \(A\) (a superset of the other edges) can be the shorter than \(\bar A\) (a subset of the two adjacent edges). That turns out to be an exact characterization: a convex polygon has two points whose geodesic passes through all edges, if and only if it has two adjacent edges that together have more than half the perimeter. For these polygons, the arc \(A\) can be chosen to have its endpoints near the outer vertices of the two long adjacent edges. So this is possible for all triangles, for any quadrilateral that is not a parallelogram, and for many other polygons of arbitrarily many sides. But it does not work for any centrally symmetric polygon, because each two adjacent sides are at least matched in length by the two opposite sides.</p>
<p>When we first discussed this problem (five years ago at a Barbados workshop), we started with the idea that no polyhedron with two parallel faces can have a geodesic through all faces. In particular, this would imply that the only regular polyhedron with a geodesic through all faces is a regular tetrahedron. But it’s not true! Instead, if \(P\) is any polygon with a geodesic through all edges, then long-enough right prisms over \(P\) have geodesics through all faces.</p>
<p>Geodesics on the surface of a convex polyhedron may be easier to understand by unfolding the polyhedron into a <a href="https://en.wikipedia.org/wiki/Net_(polyhedron)">net</a>, a flat system of polygons in the plane, drawing the line segment between the endpoints of the geodesics in the net, and then folding it back up. The complication is that the line segment needs to stay inside the net, and there may be many different nets with different line segments.</p>
<p>Suppose \(P\) is a polygon with a geodesic \(A\) through all edges, like the yellow kite above. The prism over \(P\) has two copies of \(P\), connected by rectangles. It can be unfolded by unrolling the rectangles into one long rectangular strip, and connecting the two copies of \(P\) to the top and bottom of the strip, as shown below. (The lightly shaded copy of \(P\) is an alternative placement on the top of the strip; you should only keep one of the two top copies.) To make a curve through all faces on a prism over \(P\),
attach each copy of \(P\) along one of its two adjacent long edges, and arrange the rectangular strip with these two attached copies at opposite ends. Then, connect a point on the top copy of \(P\), near the start of \(A\) on that copy, to another point on the bottom copy of \(P\), near the end of \(A\) on that copy. The resulting curve is shown below as the red segment on its net.</p>
<p style="text-align:center"><img src="/blog/assets/2022/prism-univ-geodesic.svg" alt="A geodesic through all faces of an unfolded prism over a kite (red) and a different curve that is not a geodesic (blue)" style="width:100%;max-width:600px" /></p>
<p>The red curve is definitely shorter than the curve that you would get by applying the same construction to \(\bar A\), which would be drawn in its unfolding as a segment with the same height but greater width. But what we have to worry more about is the blue curve in the figure, which cuts through one of the rectangular sides of the prism before cutting straight across one of the copies of \(P\). Could such a curve be shorter? It isn’t in the figure (I measured), but what about more generally?</p>
<p>When the height of the prism is small, the blue curve can be shorter. But in the limit as the height of the prism gets much larger than the size of \(P\), it cannot. The length of curves like the blue one, in the limit, approaches the height of the prism plus the height of the endpoint above the central rectangular region. Instead, in the limit, the length of curves like the red one approaches the height of the prism: the horizontal component of the curve contributes negligibly to its length. So for tall enough prisms the red curve is shorter than any curve like the blue one, and we have a universal geodesic. (You might think that attaching the top and bottom face in the middle of the rectangular strip would produce a shorter geodesic, and for the illustrated prism maybe it does, but as long as the endpoints are much closer to the edge of \(P\) than to the corner of \(P\), the same argument also applies to these alternative curves.)</p>
<p>Wataru Maruyama, a student of Hiro Ito and a coauthor of the paper, succeeded in proving that the other regular polyhedra indeed do not have universal geodesics. We could also prove that every tetrahedron or right prism over a triangle does have one. On the other hand, much more remains unknown. In particular, we do not have an answer to the following question: is there a centrally symmetric polyhedron with a universal geodesic?</p>
<p>(<a href="https://mathstodon.xyz/@11011110/109386054076782254">Discuss on Mastodon</a>)</p>David EppsteinWhile updating my online publications list for something else I noticed that I had neglected to discuss one of my papers from earlier this fall: “Geodesic paths passing through all faces on a polyhedron” (with Demaine, Demaine, Ito, Katayama, Maruyama, and Uno), in the booklet of abstracts from JCDCG3 2022, the Japanese Conference on Discrete and Computational Geometry, Graphs, and Games.Linkage2022-11-15T17:54:00+00:002022-11-15T17:54:00+00:00https://11011110.github.io/blog/2022/11/15/linkage<p>The massive influx of new users and new content to Mastodon has caused a greater number of these to be boosts of someone else’s post rather than posts of my own.</p>
<ul>
<li>
<p><a href="https://www.quantamagazine.org/in-math-and-life-svetlana-jitomirskaya-stares-down-complexity-20221101/"><em>Quanta</em> has a nice profile of my colleague at UC Irvine, Ukrainian-born mathematician Svetlana Jitomirskaya</a> <span style="white-space:nowrap">(<a href="https://mathstodon.xyz/@11011110/109269250786290297">\(\mathbb{M}\)</a>).</span></p>
</li>
<li>
<p><a href="https://westy31.home.xs4all.nl/Penrose/Carboard_Impossible_Penrose_triangle.html">A cardboard model of the “impossible” Penrose triangle</a> <span style="white-space:nowrap">(<a href="https://mathstodon.xyz/@GerardWestendorp/109259133843305538">\(\mathbb{M}\)</a>),</span> with construction instructions, a printable cutout, and a demo video link, by Gerard Westendorp.</p>
</li>
<li>
<p><a href="https://ventrella.com/organic-algorithm/">Organic algorithm series</a> <span style="white-space:nowrap">(<a href="https://mathstodon.xyz/@11011110/109282060578920502">\(\mathbb{M}\)</a>),</span> artworks by Jeffrey Ventrella.</p>
</li>
<li>
<p>Fatih Kızılkaya, a student of David Kempe at USC, gave an excellent talk in our theory seminar on <a href="https://arxiv.org/abs/2206.07098">“Plurality Veto” voting</a> <span style="white-space:nowrap">(<a href="https://mathstodon.xyz/@11011110/109288019438622582">\(\mathbb{M}\)</a>).</span> The idea is to count first-place votes per candidate, then let each voter cancel a single vote for their least-favorite remaining candidate (one voter at a time or simultaneously and fractionally) until only one candidate has votes left. With voters and candidates in a metric space and preferences by distance, this 3-approximates the min average distance to voters, best possible.</p>
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<p><a href="https://scholarlykitchen.sspnet.org/2022/11/01/guest-post-wikipedias-citations-are-influencing-scholars-and-publishers/">Getting an academic publication cited on Wikipedia tends to lead to more citations elsewhere, and open-access publications tend to be more frequently cited on Wikipedia</a> <span style="white-space:nowrap">(<a href="https://mathstodon.xyz/@11011110/109292417799090772">\(\mathbb{M}\)</a>,</span> <a href="https://retractionwatch.com/2022/11/05/weekend-reads-double-edition-sciences-nasty-photoshopping-problem-dr-ozs-publication-ban-image-manipulation-detection-software/">via</a>).</p>
</li>
<li>
<p><a href="https://gilkalai.wordpress.com/2022/10/19/james-davies-every-finite-colouring-of-the-plane-contains-a-monochromatic-pair-of-points-at-an-odd-distance-from-each-other/">Every finite colouring of the plane contains a monochromatic pair of points at an odd distance from each other</a> <span style="white-space:nowrap">(<a href="https://mathstodon.xyz/@11011110/109299861480951154">\(\mathbb{M}\)</a>),</span> based on <a href="https://arxiv.org/abs/2209.15598">a new arXiv preprint by James Davies</a>. The paper is pretty heavy going and the linked blog post doesn’t give much detail, so providing a more generally understandable version of this looks like a challenge.</p>
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<li>
<p><a href="http://fredrik-j.blogspot.com/2009/02/how-not-to-compute-harmonic-numbers.html">How (not) to compute harmonic numbers</a> <span style="white-space:nowrap">(<a href="https://mathstodon.xyz/@peterluschny/109280460006661224">\(\mathbb{M}\)</a>).</span> Like for factorials, exact computation using divide-and-conquer works much better than the obvious method of sequentially adding each unit fraction. But factorials have an even faster algorithm; it’s unclear whether the same idea can be made to work for the harmonic numbers.</p>
</li>
<li>
<p><a href="http://courses.csail.mit.edu/6.849/fall12/lectures/C01.html">Erik Demaine’s geometric folding algorithms course lectures</a>
<span style="white-space:nowrap">(<a href="https://mathstodon.xyz/@j2kun/109305200398869548">\(\mathbb{M}\)</a>).</span></p>
</li>
<li>
<p><a href="https://xkcd.com/2694/">xkcd on the bridges of Königsburg</a> <span style="white-space:nowrap">(<a href="https://mastodon.xyz/@xkcd/109288968362775008">\(\mathbb{M}\)</a>).</span> This one is definitely going into my lecture notes for the next time it comes around.</p>
<p style="text-align:center"><img src="https://imgs.xkcd.com/comics/konigsberg.png" alt="xkcd on the bridges of Königsburg" /></p>
</li>
<li>
<p>News I didn’t know about the university I work for, UC Irvine: apparently <a href="https://www.latimes.com/sports/story/2022-11-01/esports-uc-irvine-overwatch-league-of-legends-valorant">we’re an “esports powerhouse”</a> <span style="white-space:nowrap">(<a href="https://mathstodon.xyz/@11011110/109322880998262132">\(\mathbb{M}\)</a>,</span> <a href="https://web.archive.org/web/20221107065218/https://www.latimes.com/sports/story/2022-11-01/esports-uc-irvine-overwatch-league-of-legends-valorant">archive</a>). Definitely preferable to funneling all the alumni donations into football stadium construction! (Unlike many big US universities we don’t have a football team and I like it that way.)</p>
</li>
<li>
<p>Zhao Liang asks: <a href="https://mathstodon.xyz/@neozhaoliang/109324352766569049">which polyhedra have the property that if you make them out of mirrors and stand inside, you will see a tessellation of space?</a></p>
</li>
<li>
<p><a href="https://discrete-notes.github.io/mandatory-attendance">About mandatory attendance by presenters at CS research conferences</a> <span style="white-space:nowrap">(<a href="https://mathstodon.xyz/@11011110/109331612952132380">\(\mathbb{M}\)</a>).</span> I <a href="/blog/2022/11/13/report-from-latin.html">just attended LATIN</a>, a hybrid in person/online conference that ended up heavily tilted to in-person participation. In-person clearly leads to significantly greater interaction rather than mere passive reception of talks. I am very attracted to the idea of reducing the carbon footprint of my travel but in my experience we have not found a successful online replacement for that aspect of conferencing.</p>
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<p><a href="https://mathstodon.xyz/@divbyzero/109314296701448645">Dave Richeson folds a regular octagon from a strip of paper</a>.</p>
</li>
<li>
<p>Our graduate students and their union went on strike yesterday. The <a href="https://www.latimes.com/california/story/2022-11-14/photos-strike-by-48-000-university-of-california-academic-workers-causes-systemwide-disruptions"><em>LA Times</em> has a photo-essay</a> <span style="white-space:nowrap">(<a href="https://mathstodon.xyz/@11011110/109345285468088888">\(\mathbb{M}\)</a>,</span> <a href="https://web.archive.org/web/20221115011509/https://www.latimes.com/california/story/2022-11-14/photos-strike-by-48-000-university-of-california-academic-workers-causes-systemwide-disruptions">archive</a>). For the union’s demands and university’s counteroffers see <a href="https://www.statnews.com/2022/11/14/university-of-california-academic-workers-go-out-on-strike-to-demand-higher-wages/">this story</a>.
You can also find recent <a href="https://www.ucop.edu/academic-personnel-programs/_files/2022/oct-2021-scales/t22.pdf">grad student</a> and <a href="https://www.ucop.edu/academic-personnel-programs/_files/2022-23/oct-2022-salary-scales/t1.pdf">faculty</a> salary scales online, but note that notionally at least the student salaries are for halftime work.</p>
</li>
</ul>David EppsteinThe massive influx of new users and new content to Mastodon has caused a greater number of these to be boosts of someone else’s post rather than posts of my own.Report from LATIN2022-11-13T13:04:00+00:002022-11-13T13:04:00+00:00https://11011110.github.io/blog/2022/11/13/report-from-latin<p>I just returned from the 15th <a href="https://pakal.cs.cinvestav.mx/latin2022/">Latin American Symposium on Theoretical Informatics (LATIN 2022)</a>, held this year in Guanajuato, Mexico. LATIN is a regional conference that is only offered every two years, like WADS in Canada and SWAT in Scandinavia. I had a paper in the first one, in 1992, but this is only the second one I’ve attended, after <a href="/blog/2016/04/16/photos-from-latin.html">2016 in Ensenada</a>. This time, I was one of four invited speakers, speaking about reversible computing. <a href="https://www.ics.uci.edu/~eppstein/pubs/Epp-LATIN-22.pdf">My talk slides are online</a>, and I think that a talk video should be uploaded by the conference in a week or so.</p>
<p>In Guanajuato, the conference was held at <a href="https://en.wikipedia.org/wiki/Centro_de_Investigaci%C3%B3n_en_Matem%C3%A1ticas">CIMAT, the Centro de Investigación en Matemáticas</a>. This is a large center with a permanent research staff and frequent visitors, similar for instance to the Mathematical Sciences Research Institute in Berkeley, California. Wikipedia tells me that it also offers advanced mathematics courses and postdoctoral degrees. It’s set high on a hill over the town and has a crazy architecture, seemingly designed with a combination of the idea of what a space for mathematicians should look like, and the organic architecture of Guanajuato itself: staircases leading in every directions leading to narrow angular alleys or in some cases to nowhere, lots of repeated geometric shapes and patterns, and saturated with a coral red-orange color. I was inspired to use some of the conference break times to explore the center and take some photographs:</p>
<div><table style="margin-left:auto;margin-right:auto">
<tr style="text-align:center;vertical-align:middle">
<td style="padding:10px"><a href="https://www.ics.uci.edu/~eppstein/pix/cimat/1.html"><img src="https://www.ics.uci.edu/~eppstein/pix/cimat/1-s.jpg" style="border-style:solid;border-color:black;" width="288" height="162" alt="CIMAT" /></a></td>
<td style="padding:10px"><a href="https://www.ics.uci.edu/~eppstein/pix/cimat/2.html"><img src="https://www.ics.uci.edu/~eppstein/pix/cimat/2-s.jpg" style="border-style:solid;border-color:black;" width="193" height="256" alt="CIMAT" /></a></td>
<td style="padding:10px"><a href="https://www.ics.uci.edu/~eppstein/pix/cimat/3.html"><img src="https://www.ics.uci.edu/~eppstein/pix/cimat/3-s.jpg" style="border-style:solid;border-color:black;" width="193" height="256" alt="CIMAT" /></a></td>
</tr>
<tr style="text-align:center;vertical-align:middle">
<td style="padding:10px"><a href="https://www.ics.uci.edu/~eppstein/pix/cimat/4.html"><img src="https://www.ics.uci.edu/~eppstein/pix/cimat/4-s.jpg" style="border-style:solid;border-color:black;" width="250" height="199" alt="CIMAT" /></a></td>
<td style="padding:10px"><a href="https://www.ics.uci.edu/~eppstein/pix/cimat/5.html"><img src="https://www.ics.uci.edu/~eppstein/pix/cimat/5-s.jpg" style="border-style:solid;border-color:black;" width="193" height="256" alt="CIMAT" /></a></td>
<td style="padding:10px"><a href="https://www.ics.uci.edu/~eppstein/pix/cimat/6.html"><img src="https://www.ics.uci.edu/~eppstein/pix/cimat/6-s.jpg" style="border-style:solid;border-color:black;" width="193" height="256" alt="CIMAT" /></a></td>
</tr>
<tr style="text-align:center;vertical-align:middle">
<td style="padding:10px"><a href="https://www.ics.uci.edu/~eppstein/pix/cimat/7.html"><img src="https://www.ics.uci.edu/~eppstein/pix/cimat/7-s.jpg" style="border-style:solid;border-color:black;" width="193" height="256" alt="CIMAT" /></a></td>
<td style="padding:10px"><a href="https://www.ics.uci.edu/~eppstein/pix/cimat/8.html"><img src="https://www.ics.uci.edu/~eppstein/pix/cimat/8-s.jpg" style="border-style:solid;border-color:black;" width="287" height="162" alt="CIMAT" /></a></td>
<td style="padding:10px"><a href="https://www.ics.uci.edu/~eppstein/pix/cimat/9.html"><img src="https://www.ics.uci.edu/~eppstein/pix/cimat/9-s.jpg" style="border-style:solid;border-color:black;" width="193" height="256" alt="CIMAT" /></a></td>
</tr>
</table></div>
<p>The conference activities were all in a single session in a single conference room, and
the center is located some distance from downtown (maybe an hour by foot), with shuttles to our hotels provided only at the start and end of each day. I think this worked very well to bring all the participants together and helped make the conference a success. The topic of the conference is very broad and diverse (all of theoretical computer science) but I was pleasantly surprised how many times I went to a session whose topic I didn’t think I cared much about and found a talk that was interesting to me.</p>
<p>The <a href="https://doi.org/10.1007/978-3-031-20624-5">proceedings are already online</a>, published through Springer LNCS, and the conference web site promises free access until the end of November. Besides mine, the invited talks included Jeff Ullman on abstraction in parsing formal languages, Merav Parter on the connections between graph connectivity and the ability of distributed algorithms to resist passive and active attackers, and Mauricio Osorio on the use of non-monotonic and paraconsistent logic to model the ability of human reasoners to work with inconsistent and changing beliefs. There was also an initial day of tutorial sessions that I missed, and a roundtable memorial discussion for <a href="https://en.wikipedia.org/wiki/H%C3%A9ctor_Garc%C3%ADa-Molina">Héctor García-Molina</a>, focused more on his mentorship than his research.</p>
<p>The contributed papers were organized by topic, both in the conference sessions and in the proceedings, but the order of topics differed between the sessions and the proceedings. Among the contributed talks, some that stood out to me (listed in proceedings order) were:</p>
<ul>
<li>
<p>“<a href="https://doi.org/10.1007/978-3-031-20624-5_3">Median and hybrid median \(K\)-dimensional trees</a>” by Amalia Duch, Conrado Martínez, Mercè Pons, and Salvadour Roura. If you are given a geometric point set all at once, you can construct a \(K\)-d tree by median bisection with strict rotation through the dimensions, giving provably fast time bounds for certain types of range trees. What if the points come to you incrementally, but in random order? It turns out that some choices of dimension to cut work better than strict rotation.</p>
</li>
<li>
<p>“<a href="https://doi.org/10.1007/978-3-031-20624-5_6">Near-optimal search time in \(\delta\)-optimal space</a>”, by Tomasz Kociumaka, Gonzalo Navarro, and Francisco Olivares. This is on <a href="https://en.wikipedia.org/wiki/Grammar-based_code">grammar-based compression</a> of text in such a way that the compressed text can be used as an index for fast substring lookups. It turns out that this can be done with compressed size roughly proportional to the “substring complexity”</p>
\[\delta=\max_k\left\{\frac{d_k(S)}{k}\right\},\]
<p>where \(d_k(S)\) is the number of distinct length-\(k\) substrings of a given string.</p>
</li>
<li>
<p>“<a href="https://doi.org/10.1007/978-3-031-20624-5_8">Klee’s measure problem made oblivious</a>”, by Thore Thießen and Jan Vahrenhold. The main result is on the problem of computing the volume of the union of rectangles in the plane, or of hyperrectangles in higher dimensions. A known \(O(n\log n+n^{d/2})\) algorithm of Chan is made oblivious, meaning that its data access pattern depends only on \(n\) and not on the data. This allows the algorithm to be used in cloud computing in a privacy-preserving way. But beyond this specific computation, the paper presents a general approach that can be applied to many other geometric divide-and-conquer algorithms.</p>
</li>
<li>
<p>“<a href="https://doi.org/10.1007/978-3-031-20624-5_11">Pathlength of outerplanar graphs</a>”, by Thomas Dissaux and Nicolas Nisse. This is a variant of the more well-known <a href="https://en.wikipedia.org/wiki/Pathwidth">pathwidth</a> parameter in which one measures the maximum distance between vertices in a bag of a path-like tree-decomposition instead of the size of the bag. It can be computed exactly in outerplanar graphs but the runtime is impractically large, \(O(n^{11})\). The authors find a much faster algorithm that gets within additive error one.</p>
</li>
<li>
<p>“<a href="https://doi.org/10.1007/978-3-031-20624-5_14">Obtaining approximately optimal and diverse solutions via dispersion</a>”, by Jie Gao, Mayank Goswami, C. S. Karthik, Meng-Tsung Tsai, Shih-Yu Tsai, and Hao-Tsung Yang. An example of the type of problem solved here is to find many low-weight spanning trees such that the sum of their Hamming distances (as sets of edges) is as large as possible. Curiously, the authors were unable to extend their results from spanning trees to TSP tours (without repetition, on metric complete graphs). The tree-doubling heuristic turns trees into tours but their diversity could be lost if you shortcut repeated vertices incautiously.</p>
</li>
<li>
<p>“<a href="https://doi.org/10.1007/978-3-031-20624-5_17">Multidimensional Manhattan preferences</a>”, by Jiehua Chen, Martin Nöllenburg, Sofia Simola, Anaïs Villedieu, and Markus Wallinger. Given voter preferences for candidates, when is it possible to embed both voters and candidates into a Manhattan metric space so that the preferences are ordered by distance?</p>
</li>
<li>
<p>“<a href="https://doi.org/10.1007/978-3-031-20624-5_21">Binary completely reachable automata</a>”, by David Casas and Mikhail V. Volkov. This is on a generalization of <a href="https://en.wikipedia.org/wiki/Synchronizing_word">reset automata</a> which, when started in an unknown state, can be taken to any single state by an well-chosen input sequence. Here, one instead wants to find a collection of sequences that take the set of all states to each other set of states. For two-symbol automata, the existence of such a collection can be determined in polynomial time.</p>
</li>
<li>
<p>“<a href="https://doi.org/10.1007/978-3-031-20624-5_24">Embedding arbitrary Boolean circuits into fungal automata</a>”, by Augusto Modanese and Thomas Worsch. This concerns the <a href="https://en.wikipedia.org/wiki/Abelian_sandpile_model">abelian sandpile model</a>, a cellular automaton in which each cell has a small number of grains of sand and when it has enough, “topples”, sending one grain to each neighbor. Is there any way to determine what happens more quickly than step-by-step simulation? In one dimension, yes, and in three dimensions, no, but 2d remains open. This paper proves \(\mathsf{P}\)-completeness for a variation in which toppling is done to the vertical neighbors and to the horizontal neighbors in alternating steps, rather than in both directions in each step.</p>
</li>
<li>
<p>“<a href="https://doi.org/10.1007/978-3-031-20624-5_32">On the zombie number of various graph classes</a>”, by Jit Bose, Jean-Lou De Carufel, and Tom Shermer. This is on a variation of the pursuit-evasion game studied for <a href="https://en.wikipedia.org/wiki/Cop-win_graph">cop-win graphs</a>, where a set of cops pursue a robber along the edges of the graph, alternating moves in which the players can either move along one edge or stay put. A “zombie” is a cop who always shambles towards the robber, rather than more cleverly moving away when that might be needed to corner the robber.</p>
</li>
<li>
<p>“<a href="https://doi.org/10.1007/978-3-031-20624-5_33">Patterns in ordered (random) matchings</a>”, by Andrzej Dudek, Jarosław Grytczuk, and Andrzej Ruciński. A typical result: in a <a href="https://en.wikipedia.org/wiki/Chord_diagram_(mathematics)">chord diagram</a> with \(n\) chords, there must be a subset of \(\Omega(n^{1/3})\) of the chords that all nest, all cover disjoint arcs, or all cross. Can such a basic variation of the <a href="https://en.wikipedia.org/wiki/Erd%C5%91s%E2%80%93Szekeres_theorem">Erdős–Szekeres theorem</a> really be new?</p>
</li>
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<p>“<a href="https://doi.org/10.1007/978-3-031-20624-5_34">Tree \(3\)-spanners on generalized prisms of graphs</a>”, by Renzo Gómez, Flávio Miyazawa, and Yoshiko Wakabayashi. A tree \(k\)-spanner is a spanning tree of a graph such that every graph edge can be replaced by a path of at most \(k\) edges in the tree. \(2\)-spanners are easy and \(k\)-spanners are hard for \(k\ge 4\) but the complexity of finding \(3\)-spanners has been open since 1995.</p>
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<p>“<a href="https://doi.org/10.1007/978-3-031-20624-5_35">A general approach to Ammann bars for aperiodic tilings</a>”, by Carole Porrier and Thomas Fernique. Two of the many constructions for <a href="https://en.wikipedia.org/wiki/Penrose_tiling">Penrose tilings</a> are the cut and project method, in which the tiling is formed from the square faces in a 2d tube through a 5d hypercube tiling, and the method of Ammann bars, in which the prototiles are decorated with line segments that must link up to form the lines of a line arrangement. So far the cut and project method has appeared to be much more fertile, leading to many other types of aperiodic tiling. The authors find a general method for showing that some of these other tilings also can be constructed using Ammann bars. <a href="https://arxiv.org/abs/2205.13973">The arXiv version</a> has more including a link to <a href="https://github.com/cporrier/Cyrenaic">SAGE code for generating these tilings as svg images</a>.</p>
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<p>“<a href="https://doi.org/10.1007/978-3-031-20624-5_40">Improved parallel algorithms for generalized Baumslag groups</a>”, by Caroline Mattes and Armin Weiß. One of the pluses of having such a broad conference is that it brings together people with highly varied research specialties and produces synergies between areas that might otherwise not connect. This paper has the same flavor, bringing together parallel computational complexity theory and <a href="https://en.wikipedia.org/wiki/Combinatorial_group_theory">combinatorial group theory</a>.</p>
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<p>“<a href="https://doi.org/10.1007/978-3-031-20624-5_41">Complexity results on untangling red-blue matchings</a>”, by Arun Kumar Das, Sandip Das, Guilherme da Fonseca, Yan Gerard, and Bastien Rivier. Suppose you have a red-blue matching between \(n\) red and \(n\) blue points in the plane, and you want to remove all crossings by re-matching pairs of crossing edges. How many of these flips does it take? There are many unsolved questions. The problem may seem a little artificial (because you can get to an uncrossing matching directly rather than by flipping) but the real motivation is to understand how to turn an approximate TSP tour with crossings into a shorter uncrossed TSP tour, where the use of flips seems more essential.</p>
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<p>The best paper award (named in memory of Alejandro López-Ortiz) was given to “<a href="https://doi.org/10.1007/978-3-031-20624-5_10">Theoretical analysis of <code class="language-plaintext highlighter-rouge">git</code> bisect</a>” by Julien Courtiel, Paul Dorbec, and Romain Lecoq (who unfortunately could not present it). The conference also has a “test of time” award, named for <a href="https://en.wikipedia.org/wiki/Imre_Simon">Imre Simon</a>, which went to “<a href="https://doi.org/10.1007/978-3-642-12200-2_16">Optimal succinctness for range minimum queries</a>” by Johannes Fischer from 2010.</p>
<p>All in all, a very successful conference, and one I’m very happy to have traveled to.</p>
<p>(<a href="https://mathstodon.xyz/web/@11011110/109338576841926898">Discuss on Mastodon</a>)</p>David EppsteinI just returned from the 15th Latin American Symposium on Theoretical Informatics (LATIN 2022), held this year in Guanajuato, Mexico. LATIN is a regional conference that is only offered every two years, like WADS in Canada and SWAT in Scandinavia. I had a paper in the first one, in 1992, but this is only the second one I’ve attended, after 2016 in Ensenada. This time, I was one of four invited speakers, speaking about reversible computing. My talk slides are online, and I think that a talk video should be uploaded by the conference in a week or so.Halloween linkage2022-10-31T17:24:00+00:002022-10-31T17:24:00+00:00https://11011110.github.io/blog/2022/10/31/halloween-linkage<ul>
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<p>Perspective drawing often consists of mapping a curved world onto a flat screen. But <a href="http://www.szegedicsaba.com/curvedscreen.htm">what if you mapped a polyhedral world onto a curved screen</a> <span style="white-space:nowrap">(<a href="https://mastodon.social/@curved_ruler/109174245124286751">\(\mathbb{M}\)</a>)?</span> Artwork by Szegedi Csaba, 1986. <a href="https://www.szegedicsaba.hu/en">His more recent work</a> is different enough that it’s not obvious they’re the same person.</p>
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<p><a href="https://www.wired.com/story/wikipedia-state-sponsored-disinformation/">The hunt for Wikipedia’s disinformation moles</a> <span style="white-space:nowrap">(<a href="https://mathstodon.xyz/@11011110/109187943967850219">\(\mathbb{M}\)</a>).</span> <em>Wired</em> on coordinated long-term state-level disinformation campaigns on Wikipedia. I couldn’t find their link to the research they report on, but it appears to be “<a href="https://www.isdglobal.org/isd-publications/information-warfare-and-wikipedia/">Information Warfare and Wikipedia</a>“(by Carl Miller, Melanie Smith, Oliver Marsh, Kata Balint, Chris Inskip, and Francesca Visser of the Institute for Strategic Dialogue), which focuses on Russian disinformation in their war on Ukraine.</p>
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<p><a href="https://www.quantamagazine.org/mathematicians-surprised-by-hidden-fibonacci-numbers-20221017/">Fibonacci numbers and fractals hiding in geometric optimization problems in symplectic geometry</a> <span style="white-space:nowrap">(<a href="https://mathstodon.xyz/@11011110/109193514333145471">\(\mathbb{M}\)</a>).</span> I don’t understand symplectic geometry at all but this article kind of makes me want to try. Based on <a href="https://annals.math.princeton.edu/2012/175-3/p05">a 2012 paper by Dusa McDuff and Felix Schlenk</a>, and <a href="https://doi.org/10.1007/978-3-030-80979-9_2">a 2021 paper by
Maria Bertozzi, Tara S. Holm, Emily Maw, Dusa McDuff, Grace T. Mwakyoma, Ana Rita Pires, and Morgan Weiler</a> (<a href="https://arxiv.org/abs/2010.08567">arXiv:2010.08567</a>).</p>
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<p>My colleague and coauthor <a href="https://meetings.informs.org/wordpress/indianapolis2022/awards-hall/">Vijay Vazirani wins the INFORMS John von Neumann Theory Prize</a> <span style="white-space:nowrap">(<a href="https://mathstodon.xyz/@11011110/109199025441051589">\(\mathbb{M}\)</a>)</span> “for his fundamental and sustained contributions to the design of algorithms, including approximation algorithms, computational complexity theory, and algorithmic game theory, central to operations research and the management sciences”.</p>
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<p>Lost documents from ancient Greek science are still being discovered in palimpsests, parchments that have been scraped clean and rewritten with something else, via modern multispectral imaging techniques <span style="white-space:nowrap">(<a href="https://mathstodon.xyz/@11011110/109204919184604129">\(\mathbb{M}\)</a>).</span> The latest: <a href="https://arstechnica.com/science/2022/10/part-of-lost-star-catalog-of-hipparchus-found-lurking-under-medieval-codex/">a constellation from a lost star catalog by Hipparchus</a> on the <a href="https://en.wikipedia.org/wiki/Codex_Climaci_Rescriptus">Codex Climaci rescriptus</a> from Saint Catherine’s Monastery on the Sinai Peninsula.</p>
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<p><a href="https://xorshammer.com/2008/09/04/a-geometrically-natural-uncomputable-function/">A geometrically natural uncomputable function</a> <span style="white-space:nowrap">(<a href="https://mathstodon.xyz/@11011110/109210523463519897">\(\mathbb{M}\)</a>,</span> <a href="https://news.ycombinator.com/item?id=33268451">via</a>; from 2008), reporting on Alexander Nebutovsky’s “<a href="https://doi.org/10.1002/cpa.3160480402">Non-recursive functions, knots ‘with thick ropes’, and self-clenching ‘thick’ hyperspheres</a>”. If you embed an <span style="white-space:nowrap">\(n\)-dimensional</span> sphere into an <span style="white-space:nowrap">\((n+1)\)-dimensional</span> one, you can always continuously move the embedding to the equator, but you may have to move the embedding closer to itself first. How much closer? It’s uncomputable!</p>
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<p><a href="https://origami.kosmulski.org/blog/2022-10-23-fujimoto-books-public-domain">Five geometric origami books by Shuzo Fujimoto now public-domain</a> <span style="white-space:nowrap">(<a href="https://mathstodon.xyz/@11011110/109218557060750300">\(\mathbb{M}\)</a>,</span> <a href="https://news.ycombinator.com/item?id=33307845">via</a>). They were originally published in the 1970s and 1980s and focus on polyhedra, tessellations, modular origami, and the like. In Japanese, but with lots of diagrams.</p>
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<p>Chris Purcell asks: <a href="https://mathstodon.xyz/@ccppurcell/109223240056862107">what do you call those fractal cracking patterns in paint, mud, etc.</a>?</p>
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<p><a href="https://arstechnica.com/science/2022/10/the-feds-new-open-access-policy-whos-gonna-pay-for-it/">Who will pay the publication fees under new grant-agency rules that all publications must be open access</a> <span style="white-space:nowrap">(<a href="https://mathstodon.xyz/@11011110/109232886231027636">\(\mathbb{M}\)</a>)?</span> “We face a growing risk that the ability to pay APCs – rather than the merits of the research – will determine what and who gets published.” … “I fear that forcing pay-to-play for every paper will end up amplifying existing inequities.”</p>
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<p>Claims to have proven <a href="https://doi.org/10.1007/s11225-022-10017-2">the twin prime conjecture</a> and <a href="https://doi.org/10.1007/s11225-022-10015-4">the existence of infinitely many Mersenne primes</a> <span style="white-space:nowrap">(<a href="https://mathstodon.xyz/@11011110/109237034145768039">\(\mathbb{M}\)</a>)</span> were published by Janusz Czelakowski in a peer-reviewed Polish logic journal. Beyond the obvious mistake in theorem 1.1 of the 1st link (the word “prime” is missing at an important point), other editors in a discussion on Wikipedia (where I found this) were skeptical that model theory and forcing are the right way to prove results like this. After <a href="https://mathoverflow.net/questions/433278/czelakowskis-claimed-proof-of-the-twin-prime-conjecture">the MathOverflow discussion</a> turned up more serious errors, the editor-in-chief promised a retraction.</p>
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<p><a href="https://en.wikipedia.org/wiki/Kite_(geometry)">New Wikipedia Good Article on kites</a> <span style="white-space:nowrap">(<a href="https://mathstodon.xyz/@11011110/109243040526023165">\(\mathbb{M}\)</a>),</span> the quadrilaterals, not the toys on strings and not the birds. The article briefly mentions that every non-rhombus kite has sides that are bitangents to two unequal circles. I was unable to source and did not include (although a figure makes it obvious) that this can be reversed. Every two unequal circles have four bitangents forming sides of exactly three quadrilaterals: a convex kite, a concave kite, and an antiparallelogram.</p>
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<p>The warm welcome given to edgelords on Twitter by the site’s new owner has led to another mass migration to Mastodon. One of the familiar names making the move is Vi Hart, who posted among other things about <a href="https://mastodon.social/@vihart/109244906964274293">60 whisks tangled together in icosahedral symmetry</a>.</p>
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<p>The usual version of the <a href="https://en.wikipedia.org/wiki/Chessboard_paradox">chessboard paradox</a> features seeming dissections of two different-area rectangles into the same set of triangles and trapezoids. Bruce35dc finds a variant with <a href="https://mathstodon.xyz/@Bruce35dc/109246400468582681">three dissections of the same rectangle</a>, into seven pieces, six out of the seven, and five of the six!</p>
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<p>It is a truth universally acknowledged that unless you write your papers with a catchy start, the referees will get bored and stop reading before getting to the important parts. <a href="https://igorpak.wordpress.com/2022/10/26/how-to-start-a-paper/">Igor Pak has some advice</a> <span style="white-space:nowrap">(<a href="https://mathstodon.xyz/@11011110/109261439776959800">\(\mathbb{M}\)</a>).</span></p>
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<p>A few years ago Google search was good enough, and Wikipedia’s bad enough, that I would regularly search Google for “something site:en.wikipedia.org”. Now the tables have turned: <a href="https://www.theverge.com/23416056/wikipedia-app-vs-google-mobile-search">James Vincent suggests switching to the Wikipedia mobile app as the default for searches</a> <span style="white-space:nowrap">(<a href="https://mathstodon.xyz/@11011110/109265698873986719">\(\mathbb{M}\)</a>,</span> <a href="https://en.wikipedia.org/wiki/Wikipedia:Wikipedia_Signpost/2022-10-31/In_the_media">via</a>) since much of the time what you are trying to find is there or linked from there. Of course, many of my searches seek sources for Wikipedia content, so that wouldn’t help me…</p>
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</ul>David EppsteinPerspective drawing often consists of mapping a curved world onto a flat screen. But what if you mapped a polyhedral world onto a curved screen (\(\mathbb{M}\))? Artwork by Szegedi Csaba, 1986. His more recent work is different enough that it’s not obvious they’re the same person.Repeated vertices in TSP tours2022-10-22T17:22:00+00:002022-10-22T17:22:00+00:00https://11011110.github.io/blog/2022/10/22/repeated-vertices-tsp<p>This week my graph algorithms course covered the traveling salesperson problem, which I usually describe in two equivalent forms:</p>
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<p>Given a distance matrix representing a metric space, find a cycle that passes through each point of the space exactly once, of minimum total length</p>
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<p>Given a connected positively-weighted undirected graph, find a closed walk that passes through each vertex at least once, of minimum total length</p>
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<p>To go from a distance matrix to a graph, we just use the complete graph, and skip any repeated vertices in its closed walk. To go from a graph to a distance matrix, compute all pairs shortest distances, and then form a closed walk by concatenating the shortest paths between consecutive vertices of the non-repeating cycle. But this concatenation may create many unavoidable repeated vertices. For instance, if your graph is an <span style="white-space:nowrap">\(n\)-vertex</span> star, then any closed walk through all the vertices must return to the central vertex \(n-1\) times, like the blue curve past all of the vertices in the nine-vertex star below.</p>
<p style="text-align:center"><img src="/blog/assets/2022/star-tour.svg" alt="A closed walk through all vertices of the star $$K_{1,8}$$ visits the central vertex eight times." /></p>
<p>It occurred to me to wonder: how many repetitions might be necessary, in total? The multigraph of edges used by the closed walk (with one copy for each time the walk uses each edge) is Eulerian, meaning that it connects all the vertices and has even degree at all of them. Any Eulerian multigraph has a closed walk visiting all the vertices, its Euler tour. Among these graphs, the TSP multigraph must be minimal: if it had an Eulerian subgraph we could walk on that instead. And any minimal Eulerian multigraph can be turned into a simple graph and weighted in such a way that all edges are used with their given multiplicities in the optimal TSP walk. So another, more combinatorial, way of asking the same question is: how many edges can a minimal Eulerian multigraph have?</p>
<p>The answer: <span style="white-space:nowrap">\(2n-2\).</span> More precisely, a graph is said to be <a href="https://en.wikipedia.org/wiki/Dense_graph">\((a,b)\)-sparse</a> if every <span style="white-space:nowrap">\(k\)-vertex</span> subgraph has at most \(ak-b\) edges. In this sense, the minimal Eulerian graphs are <span style="white-space:nowrap">\((2,2)\)-sparse.</span></p>
<p>If you were given an Eulerian graph that is not <span style="white-space:nowrap">\((2,2)\)-sparse,</span> it could not be minimal Eulerian. To see this, choose a minimal subset of \(k\) vertices that has more than \(2k-2\) edges. By deleting edges, you can find a subgraph that is <span style="white-space:nowrap">\((2,2)\)-tight:</span> it has exactly \(2k-2\) edges, and every subgraph is <span style="white-space:nowrap">\((2,2)\)-sparse.</span> A result of Nash-Williams from the 1960s states that a subgraph like this can always be decomposed into two spanning trees. But if you combine one of the deleted edges with a path between its endpoints in one of the trees, you get a cycle that you can remove without changing the parity of the vertex degrees. Removing this cycle still leaves a subgraph that is connected through the other spanning tree. Because there is a cycle you can remove leaving an Eulerian subgraph, your starting graph is not minimal.</p>
<p>The bound of \(2(n-1)\) on the number of edges in a minimal Eulerian multigraph cannot be made any smaller. One way to construct a minimal Eulerian multigraph with exactly this many edges (maybe the only way) is just to double all of the edges in a tree.</p>
<p>Instead of counting edges, another way to define sparse graphs involves forbidden <a href="https://en.wikipedia.org/wiki/Shallow_minor">shallow minors</a>. However, this does not work for minimal Eulerian graphs: they have no forbidden shallow minors. For instance, if you subdivide the edges of any Eulerian graph, such as a complete graph on an odd number of vertices, you will get a minimal Eulerian graph that has the complete graph as a depth-1 minor.</p>
<p style="text-align:center"><img src="/blog/assets/2022/subdivided-K7.svg" alt="Subdividing the edges of the complete graph $$K_7$$ produces a minimal Eulerian graph with $$K_7$$ as a 1-shallow minor." /></p>
<p>(<a href="https://mathstodon.xyz/@11011110/109214811068499556">Discuss on Mastodon</a>)</p>David EppsteinThis week my graph algorithms course covered the traveling salesperson problem, which I usually describe in two equivalent forms:Linkage2022-10-15T17:24:00+00:002022-10-15T17:24:00+00:00https://11011110.github.io/blog/2022/10/15/linkage<ul>
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<p><a href="https://actu.epfl.ch/news/age-old-technique-enhanced-by-computer-modeling/">Ultralightweight pavilion made from woven bamboo strips, aided by modern computer modeling</a> <span style="white-space:nowrap">(<a href="https://mathstodon.xyz/@11011110/109097228519637153">\(\mathbb{M}\)</a>).</span> Not very good for keeping sun or rain off in the form shown, but I imagine one could stretch a membrane over it if that were the goal.</p>
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<p><a href="https://support.google.com/photos/thread/180787712/corrupted-photos">Corruption in old images stored in Google Photos</a> <span style="white-space:nowrap">(<a href="https://mathstodon.xyz/@11011110/109104810612333424">\(\mathbb{M}\)</a>,</span> <a href="https://news.ycombinator.com/item?id=32970623">via</a>). Let this be a reminder that if you care about the permanence and stability of your data, keep a safe copy on media you own and control, not on someone else’s machine on the cloud somewhere.</p>
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<p>“<a href="https://www.aaup.org/news/university-idaho-should-rescind-guidance-speech-about-abortion">The University of Idaho administration has abandoned its duty to uphold the mission of the institution and signaled to all the world that the university is no longer committed to academic freedom</a>” <span style="white-space:nowrap">(<a href="https://mathstodon.xyz/@11011110/109108400913458700">\(\mathbb{M}\)</a>),</span> according to The American Association of University Professors. The context is a memo sent out by the university’s lawyers requesting faculty to “remain neutral on the topic of abortion”; for more on that, see <em>Inside Higher Ed</em>’s <a href="https://www.insidehighered.com/news/2022/09/27/university-tells-professors-stay-neutral-abortion">story</a> and <a href="https://www.insidehighered.com/quicktakes/2022/09/30/aaup-u-idaho-should-rescind-guidance-abortion-speech">editorial</a>.</p>
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<p><a href="http://blog.computationalcomplexity.org/2022/10/is-it-okay-for-paper-or-book-to-say-for.html">Gasarch asks for advice on whether it’s acceptable for an academic publication to cite Wikipedia</a> <span style="white-space:nowrap">(<a href="https://mathstodon.xyz/@11011110/109113654779787003">\(\mathbb{M}\)</a>).</span> Comments concern stability of Wikipedia articles, but that’s easy: every article has “cite this page” in the toolbar with sample citations that permalink stable versions. The bigger issue is the purpose of the citation. As background reading: fine. As credit for a figure: necessary. For a technical result: you should probably follow the references to a more-primary source.</p>
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<p><a href="https://www.deepmind.com/blog/discovering-novel-algorithms-with-alphatensor">Computer search for faster matrix multiplication algorithms</a> <span style="white-space:nowrap">(<a href="https://mathstodon.xyz/@j2kun/109116859743991084">\(\mathbb{M}\)</a>,</span> <a href="https://cp4space.hatsya.com/2022/10/06/matrix-multiplication-update/">see also</a>).</p>
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<p>In place of the prime numbers, consider <a href="http://oeis.org/A050376">the numbers \(p^{2^k}\) for \(p\) prime and integer \(i\ge 0\)</a> <span style="white-space:nowrap">(<a href="https://mathstodon.xyz/@11011110/109119821643711856">\(\mathbb{M}\)</a>).</span> Each positive integer has a unique factorization into a product of these without repetition. For example,</p>
\[21600 = 2\cdot 3\cdot 9\cdot 16\cdot 25.\]
<p>They are called the <a href="https://en.wikipedia.org/wiki/Fermi%E2%80%93Dirac_prime">Fermi–Dirac primes</a> because of an analogy to fermions and bosons from physics: like bosons, primes can appear repeatedly in an energy level (prime factorization), but these numbers appear only once.</p>
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<p>I linked to <a href="https://arxiv.org/abs/2205.09102">Milman and Neeman’s preprint on the triple bubble conjecture</a> <a href="/blog/2022/06/30/linkage.html">last June</a> <span style="white-space:nowrap">(<a href="https://mathstodon.xyz/@11011110/109131239935449901">\(\mathbb{M}\)</a>),</span> but now <a href="https://www.quantamagazine.org/monumental-math-proof-solves-triple-bubble-problem-and-more-20221006/"><em>Quanta</em> has a popularized explainer of it</a>.</p>
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<p><a href="https://amathr.org/an-aperiodic-set-of-eleven-wang-tiles/">An aperiodic set of 11 Wang tiles</a> <span style="white-space:nowrap">(<a href="https://mathstodon.xyz/@11011110/109136741064182404">\(\mathbb{M}\)</a>).</span> Wang tiles are edge-colored squares that tile the plane as a grid, without rotation and with matching edges. B. Durand and A. Shen explain a result of E. Jeandel and M. Rao from a <a href="https://arxiv.org/abs/1506.06492">2015 preprint</a> and <a href="https://amathr.org/an-aperiodic-set-of-eleven-wang-tiles/">2021 journal publication</a> using a computer search to prove that sets of 11 Wang tiles with 4 colors (but no fewer of either) can force the tiling to be aperiodic.</p>
<p>The first link is on the site of an organization led by people who have publicly opposed mandatory statements of support for diversity or inclusiveness, and the organization itself conspicuously lacks any statement of support for those values. This led to some discussion on Mathstodon over whether we should link to even purely-mathematical content by these people. <a href="https://mathstodon.xyz/@11011110/109144568072282691">My position is that it would be a mistake to shun them</a>. I think their position that mathematics can and should be above such concerns is naive and overly idealistic, but we cannot find solutions to societal problems such as institutionalized discrimination if we shut down free and open discussions of alternatives by banning anyone at the slightest misstep from the political orthodoxy of the minute. Also, doing so would strengthen their position by playing into their storyline of good mathematics getting pushed aside for political reasons. As I wrote in the linked comment, “Let’s not be the monsters they think we are.”</p>
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<p><a href="https://planetofthepaul.com/wikipedia-download-usb-flash/">How to make yourself a copy of Wikipedia on a flash drive, usable offline</a> <span style="white-space:nowrap">(<a href="https://mathstodon.xyz/@11011110/109142543607772779">\(\mathbb{M}\)</a>,</span> <a href="https://news.ycombinator.com/item?id=33114107">via</a>).</p>
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<p><a href="https://gafferongames.com/post/shape_of_the_go_stone/">The shape of a Go stone</a> <span style="white-space:nowrap">(<a href="https://mathstodon.xyz/@11011110/109147037847419575">\(\mathbb{M}\)</a>):</span> A project from 2013 to create “a physically accurate computer simulation of a Go board and stones” starts by trying to understand the shape of the stones, settling on an intersection between two balls, modified by using a torus to bevel the sharp edge where they meet. Which sort of looks right, but <a href="https://forums.online-go.com/t/the-shape-of-the-stones/27557">a 2020 discussion suggests that a more accurate model needs to take into account how real Go stones are made</a>.</p>
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<p><a href="https://www.youtube.com/watch?v=a-767WnbaCQ">Percolation: a Mathematical Phase Transition</a> <span style="white-space:nowrap">(<a href="https://mathstodon.xyz/@11011110/109153531832896359">\(\mathbb{M}\)</a>).</span> One of many recent mathematics exposition videos from the “Summer of Math Exposition 2”; this one is mostly about the bond percolation on an infinite square grid, the study of the connectivity of random subgraphs of the grid. There’s <a href="https://www.metafilter.com/196698/Summer-of-Math-Exposition-2">a more complete listing of the other videos on Metafilter</a>.</p>
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<p><a href="https://uxdesign.cc/how-apple-makes-you-think-green-bubbles-gross-e03b52b12fed">Intentionally-bad and anti-accessible user interface design by Apple, in order to undercut the competition</a> <span style="white-space:nowrap">(<a href="https://mathstodon.xyz/@11011110/109156160184481951">\(\mathbb{M}\)</a>,</span> <a href="https://boingboing.net/2022/10/12/apple-intentionally-made-the-green-chat-bubbles-of-android-text-messages-look-gross.html">via</a>, <a href="https://news.ycombinator.com/item?id=33176668">via2</a>).</p>
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<p><a href="https://github.com/mathjax/MathJax-src/releases/tag/4.0.0-alpha.1">MathJax 4.0.0-alpha.1 now available</a> <span style="white-space:nowrap">(<a href="https://mathstodon.xyz/@11011110/109165272680486533">\(\mathbb{M}\)</a>,</span> <a href="https://groups.google.com/g/mathjax-users/c/WhMiWK9Ld7k?pli=1">via</a>, <a href="https://www.ams.org/news?news_id=7098">via2</a>). Because it’s still an alpha-test version, it’s probably not the right time to switch unless you really need the new features (more fonts, better line breaking, html within math expressions).</p>
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<p><a href="https://www.boristhebrave.com/2021/05/23/triangle-grids/">An appreciation of triangular grids</a> <span style="white-space:nowrap">(<a href="https://mathstodon.xyz/@jsiehler/109166889958867986">\(\mathbb{M}\)</a>).</span></p>
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<p><a href="https://en.wikipedia.org/wiki/Interesting_number_paradox">The interesting number paradox</a> <span style="white-space:nowrap">(<a href="https://mathstodon.xyz/@11011110/109173873212315062">\(\mathbb{M}\)</a>)</span> is the argument that there cannot be a smallest uninteresting natural number, because that property would make it interesting. I had thought that it dated from <a href="https://www.jstor.org/stable/24942039">a 1958 “Mathematical Games” column by Martin Gardner in <em>Scientific American</em></a>, but now another Wikipedia editor has found an earlier reference, <a href="https://doi.org/10.2307/2305682">a 1945 letter to the American Mathematical Monthly by Edwin F. Bechenbach</a> (see bottom of last page of link).</p>
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</ul>David EppsteinUltralightweight pavilion made from woven bamboo strips, aided by modern computer modeling (\(\mathbb{M}\)). Not very good for keeping sun or rain off in the form shown, but I imagine one could stretch a membrane over it if that were the goal.Linkage2022-09-30T18:28:00+00:002022-09-30T18:28:00+00:00https://11011110.github.io/blog/2022/09/30/linkage<ul>
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<p><a href="https://cohost.org/0xabad1dea/post/112175-the-exciting-new-wor">The exciting new world of AI prompt injection</a> <span style="white-space:nowrap">(<a href="https://mathstodon.xyz/@11011110/109010772219100161">\(\mathbb{M}\)</a>,</span> <a href="https://lobste.rs/s/v9skyo/exciting_new_world_ai_prompt_injection">via</a>). Promote your business by running a bot that uses other people’s social media post text to prompt a text-writing AI that generates customized responses to those posts. What could go wrong?</p>
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<p><a href="https://community.wolfram.com/groups/-/m/t/2617634">Ed Pegg constructs and visualizes Engel’s 38-sided space-filling polyhedron</a> <span style="white-space:nowrap">(<a href="https://mathstodon.xyz/@11011110/109018050225431095">\(\mathbb{M}\)</a>),</span> the most possible for a Voronoi cell of an isohedral Voronoi tessellation, surrounded by 38 copies of itself.</p>
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<p><a href="https://mastodon.social/@curved_ruler/109015404349191976">Cyclography</a>, an old form of data visualization in which 3d points are visualized as 2d circles, with the third coordinate used as their radius. Sort of an inverse to the lifting transformation in computational geometry, which turns 2d circles into 3d points in order to use point-based algorithms on them.</p>
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<p><a href="https://terrytao.wordpress.com/2022/09/19/a-counterexample-to-the-periodic-tiling-conjecture/">Terry Tao on his new preprint with Rachel Greenfeld</a>, “<a href="https://arxiv.org/abs/2209.08451">A counterexample to the periodic tiling conjecture</a>” <span style="white-space:nowrap">(<a href="https://mathstodon.xyz/@11011110/109029559029937263">\(\mathbb{M}\)</a>).</span> The <a href="https://en.wikipedia.org/wiki/Gyrobifastigium">Schmitt-Conway-Danzer biprism</a> and <a href="https://en.wikipedia.org/wiki/Socolar%E2%80%93Taylor_tile">Socolar–Taylor tile</a> tile \(\mathbb{R}^3\) and \(\mathbb{R}^2\times{}\)finite only aperiodically. The <a href="https://en.wikipedia.org/wiki/Einstein_problem">einstein problem</a> asks if \(\mathbb{R}^2\) has an aperiodic tile. This work looks at analogous questions for tiling by translation of \(\mathbb{Z}^2\).</p>
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<p><a href="https://doi.org/10.1090/noti2539">Descriptive combinatorics and distributed algorithms</a> <span style="white-space:nowrap">(<a href="https://mathstodon.xyz/@11011110/109037227806099068">\(\mathbb{M}\)</a>).</span> Nice survey article by Anton Bernshteyn in <em>Notices of the AMS</em> about implications and in some cases equivalences between topological statements about whether certain infinite sets are Borel or measurable, and whether certain corresponding finite computational problems have distributed algorithms with sub-logarithmic round complexity.</p>
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<p><a href="https://www.libraryassociation.ie/irish-librarians-condemn-publisher-wileys-removal-of-hundreds-of-titles-from-ebook-collections/">Irish librarians protest</a> <span style="white-space:nowrap">(<a href="https://mathstodon.xyz/@11011110/109037822255336737">\(\mathbb{M}\)</a>,</span> <a href="https://news.ycombinator.com/item?id=32926378">via</a>) as Wiley suddenly removes over 1300 ebooks from the existing subscription packages of academic libraries, in order to convert them to a fee-per-student individual-textbook subscription model. <a href="https://www.insidehighered.com/news/2022/09/28/publisher-blocks-access-ebooks-students-faculty-scramble">Now also affecting US universities</a>.</p>
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<p>Do you need another demonstration that the physics of liquids is strange and counterintuitive <span style="white-space:nowrap">(<a href="https://mathstodon.xyz/@11011110/109054291006791921">\(\mathbb{M}\)</a>)?</span> I learned from <a href="https://www.youtube.com/watch?v=Vrl23FOgUck]">this “What’s eating Dan?” video</a> that, if you have the kind of peanut butter that needs mixing, but is too liquid (swimming in extra peanut oil), you can make it thicker by mixing in a little bit of water. The water droplets in the oil make an emulsion that is thicker than either the water or oil would be by themselves. I had occasion to try it recently and it worked! Science strikes again.</p>
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<p><a href="https://mathstodon.xyz/@ngons/109037311387589665">Two swirly rhombus tilings of 4-subdivided 20-gons</a>.</p>
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<p><a href="https://www.quantamagazine.org/the-new-math-of-wrinkling-patterns-20220922/">The new math of wrinkling</a> <span style="white-space:nowrap">(<a href="https://mathstodon.xyz/@11011110/109063241525281205">\(\mathbb{M}\)</a>).</span> <em>Quanta</em> on the research of Ian Tobasco on the way that crumpling thin surfaces (like paper) can sometimes lead to disordered folds and sometimes lead to regular patterns, like <a href="https://en.wikipedia.org/wiki/Yoshimura_buckling">Yoshimura buckling</a>, depending in part on local curvature. Based on two papers by Tobasco, “<a href="https://doi.org/10.1007/s00205-020-01566-8">Curvature-driven wrinkling of thin elastic shells</a>” (2021, <a href="https://arxiv.org/abs/1906.02153">arXiv:1906.02153</a>) and “<a href="https://doi.org/10.1038/s41567-022-01672-2">Exact solutions for the wrinkle patterns of confined elastic shells</a>” (2022, <a href="https://arxiv.org/abs/2004.02839">arXiv:2004.02839</a>).</p>
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<p><a href="https://www.wonkette.com/girls-who-code-books-banned"><em>Girls Who Code</em> appears on this year’s list of books banned by US schools</a> <span style="white-space:nowrap">(<a href="https://mathstodon.xyz/@11011110/109067124900840971">\(\mathbb{M}\)</a>).</span> Apparently this happened not directly because the kind of people who ban books want women to be ignorant, but rather because these books appeared on a diversity resource list and the kind of people who ban books oppose diversity (meaning anything that would challenge the white cis male evangelical-Christian point of view) in all forms. Fortunately local protests got the ban rescinded. More <a href="https://boingboing.net/2022/09/26/girls-who-code-book-series-banned-by-a-pennsylvania-school-district.html">on BoingBoing</a> and <a href="https://www.theguardian.com/us-news/2022/sep/26/pennsylvania-book-ban-girls-who-code">in <em>The Guardian</em></a>. The statement in <em>The Guardian</em> from a book-banning spokesperson “This book series has not been banned, and they remain available in our libraries” appears to actually mean that the ban blocked students from reading the books but failed to remove them permanently from the libraries, and that they became available because the ban was rescinded.</p>
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<p><a href="https://en.wikipedia.org/wiki/Mutilated_chessboard_problem">Mutilated chessboard problem</a> <span style="white-space:nowrap">(<a href="https://mathstodon.xyz/@11011110/109074266481611822">\(\mathbb{M}\)</a>):</span> remove opposite corners from a chessboard and try to cover the rest with dominos. It’s just planar bipartite perfect matching, easy for algorithms. There’s a cute trick for human problem solving that I won’t spoil. And yet, a logical formulation has been a test case for automated reasoning for nearly 60 years, and is provably hard for some systems (especially resolution). How can it be so easy and so hard? New Wikipedia Good Article.</p>
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<p><a href="https://jamesheathers.medium.com/publication-laundering-95c4888afd21">Publication laundering</a> <span style="white-space:nowrap">(<a href="https://mathstodon.xyz/@11011110/109080279085285756">\(\mathbb{M}\)</a>,</span> <a href="https://retractionwatch.com/2022/09/28/can-you-explain-what-these-1500-papers-are-doing-in-this-journal/">via</a>): James Heathers on how “proceedings journals” that accept whole special-issues without any internal oversight over relevance or quality ease the collaboration among academics desperate for publications, middlemen who sell authorship slots on mass-produced junk, and big publishers hungry for that publication-fee and subscription-fee cash as long as they can point the blame elsewhere.</p>
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<p><a href="https://mathstodon.xyz/@jsiehler/109082598903294416">Foxagon: A hexagon with exactly one line of reflective symmetry and one reflex angle</a>. Or maybe more specifically it’s what you get when you glue equilateral triangles onto two adjacent sides of a square. The more specific version tiles the plane; the tiling below hides a <a href="https://en.wikipedia.org/wiki/Snub_square_tiling">snub square tiling</a> but other tilings are possible.</p>
<p style="text-align:center"><img src="/blog/assets/2022/foxagons.svg" alt="Tiling by foxagons" /></p>
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<p><a href="https://twitter.com/robinhouston/status/1556407331344244743">The strange Wiki-history of Sethahedra and Chestahedra</a> <span style="white-space:nowrap">(<a href="https://mathstodon.xyz/@11011110/109090046984135256">\(\mathbb{M}\)</a>,</span> <a href="https://aperiodical.com/2022/09/carnival-of-maths-208/">via</a>, <a href="https://alephjamesa.co.uk/posts.php?data=FoMSept22">via2</a>). The Chestahedron is a polyhedron whose faces are four equilateral triangles and three kites of the same area as the triangles. <a href="http://frankchester.com/project/chestahedron/">Frank Chester makes bronze sculptures of it</a>. The Sethahedron is a nonexistent variation with golden-ratio dimensions. If made of paper it will fold along kite diagonals to form ten faces. The promoter of the Sethahedron has been edit-warring to keep their erroneous version in Wikipedia.</p>
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</ul>David EppsteinThe exciting new world of AI prompt injection (\(\mathbb{M}\), via). Promote your business by running a bot that uses other people’s social media post text to prompt a text-writing AI that generates customized responses to those posts. What could go wrong?Counting paths in convex polygons2022-09-21T17:32:00+00:002022-09-21T17:32:00+00:00https://11011110.github.io/blog/2022/09/21/counting-paths-convex<p>Let’s count non-crossing paths through the all points of a convex polygon.
There is a very simple formula for this, \(n2^{n-3}\) undirected paths through an \(n\)-gon, but why? Here’s a simple coloring-based argument that immediately gives this formula.</p>
<p>Choose a coloring for the points of the polygon, red and blue, and choose a starting point for the path. Build a path, starting from this point, by the following rule: if you are at a red point, go to the next available point clockwise, and if you are at a blue point, go to the next available point counterclockwise.</p>
<p style="text-align:center"><img src="/blog/assets/2022/colored-ham.svg" alt="Generating a non-crossing path through all points of a convex polygon, by using a 2-coloring of the points to determing the direction of each step" /></p>
<p>There are \(n2^n\) choices of starting point and coloring, but each path is counted eight times, because the colors of the last two points on the path don’t make a difference to where it goes, and because each path is also traced in the opposite direction using the other end as its starting point. Dividing \(n2^n\) by eight gives the formula.</p>
<p>This same idea also works to count non-crossing paths that are allowed to skip some of the points of the polygon. Now, color each point red, blue, or yellow. Use the same rule for building a path, but ignore the yellow points: start on a red or blue point, and when searching for an available point only go to another red or blue point.</p>
<p style="text-align:center"><img src="/blog/assets/2022/colored-path.svg" alt="Generating a non-crossing path through some points of a convex polygon, by using a 3-coloring of the points to determing the direction of each step" /></p>
<p>There are \(3^n\) choices of coloring. They have different numbers of choices of starting point, but by cyclically permuting the colors you can group them into \(3^{n-1}\) triples of colorings that together have exactly \(2n\) available (non-yellow) starting points. Each path is counted eight times just like before, so this argument would seem to give the formula \(2n\cdot 3^{n-1} / 8\) for the number of paths. But it’s not quite right. For one thing, it’s not even an integer.</p>
<p>The problem is, what happens when you color all but one of the points yellow, and that one remaining point red or blue? You get a sequence of one point only: does that count as a path? If we count these as length-zero paths (as I would prefer), then they are undercounted, because they do not have two ends, and they only have one point whose coloring (red or blue) is irrelevant, rather than the usual two points. When we divide by eight we make their contribution too small. If we don’t count them (as <a href="http://oeis.org/A261064">OEIS tells me</a> was the definition used in a 2020 Bulgarian mathematics contest) then they are overcounted, because they contribute to the formula and shouldn’t.</p>
<p>Adjusting for these one-point paths gives two alternative formulas:</p>
\[\frac{n}{4}(3^{n-1}+3)\]
<p>if we are counting one-point zero-length paths, or</p>
\[\frac{n}{4}(3^{n-1}-1),\]
<p>the formula from OEIS, if we are not counting them.</p>
<p>(<a href="https://mathstodon.xyz/@11011110/109039377346914779">Discuss on Mastodon</a>)</p>David EppsteinLet’s count non-crossing paths through the all points of a convex polygon. There is a very simple formula for this, \(n2^{n-3}\) undirected paths through an \(n\)-gon, but why? Here’s a simple coloring-based argument that immediately gives this formula.Linkage2022-09-15T22:36:00+00:002022-09-15T22:36:00+00:00https://11011110.github.io/blog/2022/09/15/linkage<ul>
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<p>Lithophanes, a 19th-century art medium involving backlit translucent engravings, <a href="https://arstechnica.com/science/2022/08/19th-century-art-form-revived-to-make-tactile-science-graphics-for-the-blind/">revived via 3d printing as a single format for scientific images that blind people can read by feeling and sighted people can see</a> <span style="white-space:nowrap">(<a href="https://mathstodon.xyz/@11011110/108927524393418764">\(\mathbb{M}\)</a>,</span> <a href="https://doi.org/10.1126/sciadv.abq2640">original research paper</a>).</p>
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<p>A quick “not a Reuleaux triangle” link: <a href="https://www.instructables.com/Lego-Triangle/">Lego triangles</a>, incorrectly embellished as <a href="https://makezine.com/article/maker-news/lego-reuleaux-triangles/">Lego Reuleaux triangles</a> <span style="white-space:nowrap">(<a href="https://mathstodon.xyz/@11011110/108933430914328377">\(\mathbb{M}\)</a>).</span> The second link even goes on to say “I don’t think [they] are quite convex enough to be proper Reuleaux triangles”. And in this it’s correct, if “convex” is interpreted to mean “bulgy”. You can tell because the corners are right angles. Proper Reuleaux triangles would have \(120^\circ\) corners.</p>
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<p>As part of a search for real-world applications of combinatorial design theory, Jeremy Kun asks: “<a href="https://mathstodon.xyz/@j2kun/108936354352902797">What is the authoritative text on combinatorial designs, anyway?</a>” I suspect that the shakeup in the area caused by Peter Keevash’s use of the probabilistic method has caused what used to be the authoritative texts to become obsolete.</p>
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<p><a href="https://youtube.com/shorts/KYMYshbhKcw">Squaring the circle illusion</a> <span style="white-space:nowrap">(<a href="https://mathstodon.xyz/@henryseg/108940642070353471">\(\mathbb{M}\)</a>).</span> Henry Segerman 3d-prints a shape that, when rotated \(180^\circ\), changes appearance from a circle to a square. With the same area in the viewing frame, even.</p>
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<p>Finding a vertex-to-vertex and edge-to-edge mapping (“homomorphism”) from an input directed graph to a fixed oriented tree or cycle can be either polynomial-time or <span style="white-space:nowrap">\(\mathsf{NP}\)-complete,</span> depending on the target. But <a href="https://cstheory.stackexchange.com/questions/33836/complexity-of-digraph-homomorphism-to-an-oriented-cycle">the targets for which it is <span style="white-space:nowrap">\(\mathsf{NP}\)-complete</span> are surprisingly complicated!</a> <span style="white-space:nowrap">(<a href="https://mathstodon.xyz/@11011110/108950099488030475">\(\mathbb{M}\)</a>)</span> <a href="https://arxiv.org/abs/2205.07528">A recent search</a> found the smallest hard tree to have <span style="white-space:nowrap">\(20\) vertices,</span> and the smallest hard cycle to <span style="white-space:nowrap">have \(26\).</span></p>
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<p>Tamal Dey and Yusu Wang have a new graduate-level textbook on computational topology <span style="white-space:nowrap">(<a href="https://mathstodon.xyz/@11011110/108953135639829468">\(\mathbb{M}\)</a>):</span> <em><a href="https://www.cs.purdue.edu/homes/tamaldey/book/CTDAbook/CTDAbook.pdf">Computational Topology for Data Analysis</a></em> (<a href="https://doi.org/10.1017/9781009099950">Cambridge University Press, 2022</a>). <a href="https://www.maa.org/press/maa-reviews/computational-topology-for-data-analysis">Ellen Gasparovic reviews it for the MAA</a>.</p>
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<p><a href="https://web.archive.org/web/20220908210937/https://www.nytimes.com/interactive/2022/09/08/world/europe/succession-royal-family.html">In case anyone else is looking for a topical real-world example of a depth-first traversal of a tree</a> <span style="white-space:nowrap">(<a href="https://mathstodon.xyz/@11011110/108965503136254522">\(\mathbb{M}\)</a>).</span></p>
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<p>Chegg stops pretending not to have coursework-cheating as their main business; <a href="https://www.chronicle.com/article/some-students-use-chegg-to-cheat-the-site-has-stopped-helping-colleges-catch-them">will no longer cooperate with university cheating investigations</a> <span style="white-space:nowrap">(<a href="https://mathstodon.xyz/@11011110/108970379605304595">\(\mathbb{M}\)</a>).</span></p>
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<p><a href="http://www.math.brown.edu/reschwar/PosterPappus/pappus.png">Schwartz’s Pappus fractal</a> <span style="white-space:nowrap">(<a href="https://mathstodon.xyz/@11011110/108972366289751471">\(\mathbb{M}\)</a>,</span> <a href="http://www.math.brown.edu/reschwar/PosterPappus/pappus.pdf">explanation</a>). Start with two separate lines of three points \(abc\) <span style="white-space:nowrap">and \(ABC\).</span> By <a href="https://en.wikipedia.org/wiki/Pappus%27s_hexagon_theorem">Pappus’s theorem</a> the diagonal crossing points <span style="white-space:nowrap">\(\alpha=aB\cdot Ab\),</span> <span style="white-space:nowrap">\(\beta=aC\cdot Ac\),</span> and \(\gamma=bC\cdot Bc\) form another line of three <span style="white-space:nowrap">points \(\alpha\beta\gamma\).</span> Now recurse with \(abc\) <span style="white-space:nowrap">and \(\alpha\beta\gamma\),</span> and again with \(\alpha\beta\gamma\) <span style="white-space:nowrap">and \(ABC\).</span> You get a nice lightning-bolt fractal shape.</p>
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<p><a href="https://arxiv.org/html/2209.04402">This year’s <em>Graph Drawing</em> conference proceedings are online</a> <span style="white-space:nowrap">(<a href="https://mathstodon.xyz/@11011110/108982719390301261">\(\mathbb{M}\)</a>).</span> As in past years, they’re using a system where the proceedings are published both on arXiv and in Springer LNCS.</p>
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<p><a href="https://ris.utwente.nl/ws/portalfiles/portal/275927505/3e2a9e5b2fad237a3d35f36fa2c5f44552f2.pdf">An analysis of online exam-proctoring tool Proctorio finds it completely ineffective at catching cheaters</a> <span style="white-space:nowrap">(<a href="https://mathstodon.xyz/@11011110/108984339518916646">\(\mathbb{M}\)</a>,</span> <a href="https://news.ycombinator.com/item?id=32744976">via</a>). “The use of online proctoring is therefore best compared to taking a placebo: it has some positive influence, not because it works but because people believe that it works … policy makers would do well to balance the cost of deploying it (which can be
considerable) against the marginal benefits of this placebo effect.”</p>
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<p><a href="https://langorigami.com/publication/paper-pentasia-an-aperiodic-surface-in-modular-origami/">Lang’s paper pentasia</a> <span style="white-space:nowrap">(<a href="https://mathstodon.xyz/@11011110/108995556838713267">\(\mathbb{M}\)</a>).</span> The kite and dart Penrose tiling lifts to a surface in 3d with two equilateral triangles per tile, overhanging for the darts. Robert Lang made these nice physical models. Explanations by <a href="https://link.springer.com/article/10.1007/s00283-021-10088-4">Barry Cipra on the 1993 discovery of this surface with Conway</a> and <a href="https://langorigami.com/wp-content/uploads/2015/09/paper-pentasia.pdf">Lang and Barry Hayes, also on a related 3d surface for the rhombic Penrose tiling</a>.</p>
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<p>Three more new Wikipedia Good Articles <span style="white-space:nowrap">(<a href="https://mathstodon.xyz/@11011110/108998160700266138">\(\mathbb{M}\)</a>):</span></p>
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<p><a href="https://en.wikipedia.org/wiki/Euclidean_minimum_spanning_tree">Euclidean minimum spanning tree</a></p>
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<p><a href="https://en.wikipedia.org/wiki/Doyle_spiral">Doyle spiral</a>, spiraling circle packings in which each circle is surrounded by a ring of six tangent circles</p>
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<p><a href="https://en.wikipedia.org/wiki/Laves_graph">Laves graph</a>, a highly-symmetric 3-regular infinite graph in 3d</p>
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</ul>
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<p>An algorithmic version of <a href="en.wikipedia.org/wiki/Mantel's theorem">Mantel’s theorem</a> <span style="white-space:nowrap">(<a href="https://mathstodon.xyz/@11011110/109004244575186009">\(\mathbb{M}\)</a>):</span> If an <span style="white-space:nowrap">\(n\)-vertex</span> graph has more than <span style="white-space:nowrap">\(\bigl\lfloor\tfrac{n^2}{4}\bigr\rfloor\) edges</span> then we can find a triangle in it in linear time.</p>
<p><em>Proof:</em> Delete min-degree vertices, maintaining the edge excess, until the remaining \(k\) vertices all have <span style="white-space:nowrap">degree \(\ge\tfrac{k}{2}\).</span> If <span style="white-space:nowrap">\(k\) is odd,</span> all <span style="white-space:nowrap">have \(\ge \tfrac{k+1}{2}\);</span> if <span style="white-space:nowrap">\(k\) is even,</span> at least one <span style="white-space:nowrap">has \(\ge \tfrac{k}{2}+1\).</span> Then a max-degree vertex and any neighbor have together <span style="white-space:nowrap">\(\ge k+1\) incidences,</span> so by the pigeonhole principle they have a common neighbor forming a <span style="white-space:nowrap">triangle. \(\Box\)</span></p>
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</ul>David EppsteinLithophanes, a 19th-century art medium involving backlit translucent engravings, revived via 3d printing as a single format for scientific images that blind people can read by feeling and sighted people can see (\(\mathbb{M}\), original research paper).Comparing distances along lines2022-09-10T17:26:00+00:002022-09-10T17:26:00+00:00https://11011110.github.io/blog/2022/09/10/comparing-distances-lines<p>I’ve written here several times about <a href="https://en.wikipedia.org/wiki/Gilbert_tessellation">Gilbert tessellations</a>, most recently in <a href="/blog/2021/11/02/gilbert-tessellations-cellular.html">last year’s post about cellular automata that naturally generate them</a>. These are polygonal subdivisions of the plane, generated from the tracks of particles moving at the same speed, where the particles start as oppositely-moving pairs with random locations and directions, and continue moving until they crash into the track of another particle. Here’s Wikipedia’s illustration of these things, generated in 2012 by Claudio Rocchini:</p>
<p style="text-align:center"><a href="https://commons.wikimedia.org/wiki/File:Gilbert_tessellation.svg"><img src="/blog/assets/2018/Gilbert-tessellation.svg" alt="A Gilbert tessellation, by Claudio Rocchini" /></a></p>
<p>Suppose someone gives you a picture like this and tells you it’s a Gilbert tessellation, like I just did. How can you tell that it’s genuine? The starting positions of the particles are not marked on the picture, so what needs to be done is to infer their locations (or a system of points that could be their locations) and to check that particles, starting at those locations and moving along the lines, crash in the order that the picture shows. Wherever two segments intersect in the picture, one of them ends, and it must be the case that the starting position on the segment that ends is farther away than the starting position on the other segment. Otherwise, it would have been the other particle that crashed and the other segment that ended.</p>
<p>One can set up a big system of inequalities where each variable describes the starting position of one of the particles, for instance by giving its distance from one end of its segment. Each inequality comes from an intersection of two segments and asks for the particle on one segment to be farther than the particle on the other segment. The distance between two positions \(x\) and \(y\) on the same segment, parameterized linearly, is <span style="white-space:nowrap">just \(\vert x-y\vert\).</span> So when a particle on <span style="white-space:nowrap">line \(j\)</span> crashes into a track on <span style="white-space:nowrap">line \(i\)</span> we get an inequality like</p>
\[\vert x_i-e_{ij}\vert\le \vert x_j-e_{ji}\vert,\]
<p>where \(x_i\) is the starting positions of the particle on <span style="white-space:nowrap">line \(i\),</span> \(e_i\) is the known position on <span style="white-space:nowrap">line \(i\)</span> of its intersection with <span style="white-space:nowrap">line \(j\),</span> \(\vert x_i-e_{ij}\vert\) is the distance from the starting position to the intersection, and the smaller distance for the particle on <span style="white-space:nowrap">line \(i\)</span> means that it gets to the intersection first, causing the other particle to crash into its track. We also have some inequalities specifying that each starting position lies between the endpoints of its line segment. This almost looks like a linear program (or rather a linear programming feasability problem, with only linear constraints but no objective function), but the absolute values make it nonlinear. We can get rid of them by two tricks:</p>
<ul>
<li>
<p>Because the left absolute value occurs on the left hand side of a <span style="white-space:nowrap">\(\le\) constraint,</span> it is equivalent to replace it by two constraints, one for each choice of sign:</p>
\[x_i-e_{ij}\le \vert x_j-e_{ji}\vert,\]
<p>and</p>
\[e_{ij}-x_i\le \vert x_j-e_{ji}\vert.\]
</li>
<li>
<p>Because <span style="white-space:nowrap">line \(j\)</span> ends at its intersection with <span style="white-space:nowrap">line \(i\),</span> we know on which side of the intersection to find the starting <span style="white-space:nowrap">point \(x_j\),</span> and therefore, we know the sign <span style="white-space:nowrap">of \(x_j-e_{ji}\).</span> We can use this knowledge to replace \(\vert x_j-e_{ji}\vert\) by either \(x_j-e_{ji}\) or \(e_{ji}-x_j\) (whichever of these two expressions is guaranteed to be non-negative), eliminating the right absolute value.</p>
</li>
</ul>
<p>Once these replacements have been made, we have a linear program with two variables per inequality, which can be solved in (strongly) polynomial time.</p>
<p>But all this made me wonder: can we always turn a system of inequaties between linear distances like this into a linear program? Suppose we have an arbitrary system of variables \(x_i\) and an arbitrary system of <span style="white-space:nowrap">inequalities \(\vert x_i-e_{ij}\vert\le \vert x_j-e_{ji}\vert\),</span> not coming from Gilbert tessellation recognition. Can we determine whether such a system has a solution, efficiently? Can we maybe turn any system of inequalities like this into a linear program, and solve it that way?</p>
<p>No! The problem is <span style="white-space:nowrap">\(\mathsf{NP}\)-complete.</span> The illustration below shows the key gadget, a system of six inequalities for two variables (each shown with a small blue double wedge in the plane pointing towards the points that obey the inequality). These inequalities have only the three red points shown as their solutions. You might want only one inequality per pair of variables, but in that case you can get the same effect by replacing each inequality by a pair of inequalities involving an additional dummy variable. The <span style="white-space:nowrap">\(\mathsf{NP}\)-completeness</span> proof is a reduction from graph 3-coloring that translates each vertex into one of these gadgets, and represents the color of a vertex by the choice of which red point is used to solve these inequalities. I haven’t shown the edge gadgets, but it’s easy to find more inequalities to use for a pair of vertices, to force them to have different red points as their solutions. For instance, the <span style="white-space:nowrap">inequality \(\vert x_i\vert\ge\vert y_j-1\vert\)</span> fails only for two vertices that both use the same red point \((0,0)\).</p>
<p style="text-align:center"><img src="/blog/assets/2022/vertex-gadget.svg" alt="Vertex gadget for reduction from 3-coloring to linear distance comparison, consisting of six inequalities abs(x) ≤ abs(y-4), abs(x+2) ≥ abs(y-2), abs(x-2) ≥ abs(y-2), abs(x+1) ≥ abs(y+1), abs(x-1) ≥ abs(y+1), and abs(x) ≤ abs(y+2), that force the two variables (x,y) to have one of the three combinations of values (3,1), (-3,1), or (0,0)" /></p>
<p>So somehow, the geometry of Gilbert tessellations is essential for the ability to convert their recognition problem into a linear program, and solve it efficiently.</p>
<p>(<a href="https://mathstodon.xyz/@11011110/108977122069764388">Discuss on Mastodon</a>)</p>David EppsteinI’ve written here several times about Gilbert tessellations, most recently in last year’s post about cellular automata that naturally generate them. These are polygonal subdivisions of the plane, generated from the tracks of particles moving at the same speed, where the particles start as oppositely-moving pairs with random locations and directions, and continue moving until they crash into the track of another particle. Here’s Wikipedia’s illustration of these things, generated in 2012 by Claudio Rocchini: