Jekyll2020-07-06T00:35:22+00:00https://11011110.github.io/blog/feed.xml11011110Geometry, graphs, algorithms, and moreDavid EppsteinThe shape of the Wankel rotor2020-07-05T14:59:00+00:002020-07-05T14:59:00+00:00https://11011110.github.io/blog/2020/07/05/shape-wankel-rotor<p>I’ve written a number of posts about curvilinear triangles that are not the <a href="https://en.wikipedia.org/wiki/Reuleaux_triangle">Reuleaux triangle</a>, including <a href="/blog/2016/04/30/shape-of-kresge.html">MIT’s Kresge Auditorium</a>, <a href="https://web.archive.org/web/20190217225035/https://plus.google.com/100003628603413742554/posts/DpF5krEaU9u">triforce string art</a>, <a href="/blog/2018/04/17/mythical-reuleaux-manhole.html">valve covers</a>, <a href="/blog/2018/06/24/la-maddalena-non-reuleaux.html">a patio table</a>, and <a href="/blog/2020/06/30/linkage.html">the logo of Whale Cove, Nunavut</a>. I’ve long intended to write about another obvious topic in this theme, the curved-triangle rotor of the <a href="https://en.wikipedia.org/wiki/Wankel_engine">Wankel engine</a>, but was finally pushed into doing so by seeing that two recent popular mathematics books, <em><a href="https://en.wikipedia.org/wiki/How_Round_Is_Your_Circle">How Round Is Your Circle?</a></em> (2008) and <em><a href="https://en.wikipedia.org/wiki/Icons_of_Mathematics">Icons of Mathematics</a></em> (2011) repeat the falsehood that Wankel rotors are Reuleaux triangles. They are not.</p>
<p>Wikipedia has <a href="https://commons.wikimedia.org/wiki/File:Wankel_Cycle_anim_en.gif">a good visualization of how Wankel engines work</a>, which I’ve copied below. They go through the same four steps as a conventional <a href="https://en.wikipedia.org/wiki/Four-stroke_engine">four-stroke combustion engine</a>, in which a piston pulls away from the combustion chamber, sucking in a mixture of fuel and air, pushes back towards the chamber, compressing the mixture, ignites the mixture, pushing the piston back out and applying force to the drive shaft, and then pushes back towards the chamber, pushing the exhaust out. The difference is that in a Wankel engine, these four steps happen at four different locations within the combustion chamber, as the gases within it are pushed around by a curved triangular piston, the rotor of the engine.</p>
<p style="text-align:center"><img src="/blog/assets/2020/animated-wankel.gif" alt="Animation of a Wankel engine by Y tambe from https://commons.wikimedia.org/wiki/File:Wankel_Cycle_anim_en.gif" /></p>
<p>The driveshaft in the engine is the fixed smaller gear in the center of the animation; in the actual engine, this gearwheel would itself be spinning, but this is not shown. The triangular rotor connects to the driveshaft by an eccentric planetary gear, and spins around the driveshaft like a hula hoop around a spinning dancer. The gears have teeth and radii in the ratio 3:2, causing the driveshaft to spin three times faster than the rotor. As it does so, the three corners of the rotor (the “apex seals”) stay in contact with the outer wall of the engine, called its stator, so that the gases in the engine do not leak between different phases.</p>
<p>The shape of the stator is not determined by the curve of the rotor itself, but only by the trajectory of the moving apex seals. This trajectory is a curve called an <a href="https://en.wikipedia.org/wiki/Epitrochoid">epitrochoid</a>. If you’ve ever played with a spirograph, you know what an epitroichoid is: it’s what you get by fixing one circular disk, letting another circular disk rotate around it, placing a point somewhere within the rotating disk, and tracing the curve that it follows. Here’s <a href="https://commons.wikimedia.org/wiki/File:EpitrochoidIn3.gif">another Wikipedia animation</a>:</p>
<p style="text-align:center"><img src="/blog/assets/2020/animated-epitrochoid.gif" alt="Animation of an epitrochoid by Sam Derbyshire from https://commons.wikimedia.org/wiki/File:EpitrochoidIn3.gif" /></p>
<p>Different ratios of radii between the inner and outer disk give you different numbers of lobes in the curve, and different placements of the moving point in the outer disk (closer to or farther from the disk center) give you curves that are closer to a circle or more curvy. Placing the moving point on the outer circle itself gives you pointy rather than curvy epitrochoids, and placing it even farther out turns the inner bulges of these curves into self-crossing loops.</p>
<p>Spirograph trajectories differ from rotating apex seal trajectories in at least three ways: in the Wankel engine, the central circle (the driveshaft) rotates rather than being held stationary, the outer circle (the planetary gear) surrounds the central circle rather than being outside it, and the point whose motion is being traced (the apex seal) is outside the outer circle rather than inside it. Nevertheless, the shape is still a two-lobed epitrochoid; see the “double generation theorem” of the Bernoullis, as described by Nash,<sup id="fnref:nash"><a href="#fn:nash" class="footnote">1</a></sup> for why the same curve can be generated in multiple ways. Modulo the scale of the whole system, there is one free parameter controlling the precise shape of this epitrochoid: the ratio of the distances from the center of the rotor to the apex seals and to the planetary gear. If the apex seals are too close in, the planetary gear will bash into into the stator; if they are too far out, the stator will be close to circular and there will be little change in pressure from one part of the combustion cycle to another, losing engine efficiency. The choice made in actual engines is not the one that places the apex seals as close as possible, but seems to involve more careful optimization that considers the shape and size of the regions formed by the rotor and stator at different stages of the combustion cycle.</p>
<p>Once the stator shape has been determined, one can then proceed to answer the question we started with: what is the shape of the rotor? The main design constraint is that it should touch or at least stay close to the inner bulge of the stator (on its “side seals”), to prevent exhaust gas from flowing back around to the intake. The shape that achieves this can be understood by a thought experiment in which we imagine the rotor as somehow being fixed in space while the vehicle containing it rotates around it, rather than vice versa. As the vehicle rotates, its stator passes through parts of the space that cannot be occupied by the rotor. The parts of space that remain untouched by the rotating stator are available to be used by the rotor, and should be used by it if we want a rotor that stays in contact with the stator on its side seals. Mathematically, this is described as an “envelope” of the positions of the rotating stator with respect to the fixed rotor. This envelope is a curved triangle, but not a Reuleaux triangle. Its curves are flatter than a Reuleaux triangle’s arcs, but also they are not circular arcs. As an envelope of algebraic curves, they are presumably algebraic themselves, but of higher order; trigonometric formulas are given by Shung and Pennock.<sup id="fnref:sp"><a href="#fn:sp" class="footnote">2</a></sup></p>
<p>In practice, the rotor shape varies from its ideal envelope-of-epitrochoid form, in a couple of different ways. First, as Drogosz explains,<sup id="fnref:drogosz"><a href="#fn:drogosz" class="footnote">3</a></sup> for ease of manufacturing it is often approximated by circular arcs rather than exactly following the envelope shape. As long as the approximation stays within the envelope, the rotor will avoid colliding with the stator, and the side seal contact is not so important near the corners of the triangle, so that’s where the approximation is most noticeable. Second, real Wankel rotors often have scoops taken out from the middles of their sides, to form mini-combustion chambers that guide and shape the combustion gases within the engine.</p>
<p>For more details of all this, see:</p>
<div class="footnotes">
<ol>
<li id="fn:nash">
<p>Nash, David H. (1977), “Rotary engine geometry”, <em>Mathematics Magazine</em> 2: 87–89, <a href="https://doi.org/doi:10.1080/0025570X.1977.11976621">doi:10.1080/0025570X.1977.11976621</a>, <a href="https://www.jstor.org/stable/2689731">JSTOR:2689731</a> <a href="#fnref:nash" class="reversefootnote">↩</a></p>
</li>
<li id="fn:sp">
<p>Shung, J. B. & Pennock, G. R. (1994), “Geometry for trochoidal-type machines with conjugate envelopes”, <em>Mechanism and Machine Theory</em> 29 (1): 25–42, <a href="https://doi.org/10.1016/0094-114X(94)90017-5">doi:10.1016/0094-114X(94)90017-5</a> <a href="#fnref:sp" class="reversefootnote">↩</a></p>
</li>
<li id="fn:drogosz">
<p>Drogosz, P. (2010), “Geometry of the Wankel rotary engine”, <em>Journal of KONES</em> 17 (3): 69–74, <a href="http://yadda.icm.edu.pl/yadda/element/bwmeta1.element.baztech-article-BUJ5-0031-0018">http://yadda.icm.edu.pl/yadda/element/bwmeta1.element.baztech-article-BUJ5-0031-0018</a> <a href="#fnref:drogosz" class="reversefootnote">↩</a></p>
</li>
</ol>
</div>
<p>(<a href="https://mathstodon.xyz/@11011110/104464015428969365">Discuss on Mastodon</a>)</p>David EppsteinI’ve written a number of posts about curvilinear triangles that are not the Reuleaux triangle, including MIT’s Kresge Auditorium, triforce string art, valve covers, a patio table, and the logo of Whale Cove, Nunavut. I’ve long intended to write about another obvious topic in this theme, the curved-triangle rotor of the Wankel engine, but was finally pushed into doing so by seeing that two recent popular mathematics books, How Round Is Your Circle? (2008) and Icons of Mathematics (2011) repeat the falsehood that Wankel rotors are Reuleaux triangles. They are not.Linkage2020-06-30T16:06:00+00:002020-06-30T16:06:00+00:00https://11011110.github.io/blog/2020/06/30/linkage<ul>
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<p><a href="https://scilogs.spektrum.de/hlf/the-five-bridges-puzzle/">The five bridges puzzle</a> (<a href="https://mathstodon.xyz/@11011110/104357604444143899"><script type="math/tex">\mathbb{M}</script></a>). Sort of like the bridges of Königsberg, but stochastic. A cute puzzle with a connection to percolation theory and connection games.</p>
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<p>Dubious journal publisher MDPI provides special-issue editors with some number of no-publication-charge slots, but requires that priority for these slots be given to first-world scholars, because they are the ones with “more abundant scientific research resources” (read: funds to pay publication charges). This didn’t sit well with three environmental health failure researchers, who <a href="https://retractionwatch.com/2020/06/16/failure-fails-as-publisher-privileges-the-privileged/">resigned their guest editorship over it</a> (<a href="https://mathstodon.xyz/@11011110/104363585936972806"><script type="math/tex">\mathbb{M}</script></a>, <a href="https://wash.leeds.ac.uk/what-the-f-how-we-failed-to-publish-a-journal-special-issue-on-failures/">see also</a>).</p>
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<p>On today’s edition of <a href="/blog/2018/06/24/la-maddalena-non-reuleaux.html">not the Reuleaux triangle</a>, we have <a href="https://twitter.com/Nukaq/status/1273803547574972416">the logo of Whale Cove, Nunavut</a> (<a href="https://mathstodon.xyz/@11011110/104368587916436396"><script type="math/tex">\mathbb{M}</script></a>, <a href="https://www.metafilter.com/187552/Nunavut-Aesthetics">via</a>). The sides of the triangle are straighter than a Reuleaux triangle would be, and its corners are slightly narrower than equilateral. Cool logo, though.</p>
<p style="text-align:center"><img src="/blog/assets/2020/beluga-reuleaux.png" alt="Logo of Whale Cove, Nunavut" /></p>
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<p><a href="https://computerhistory.org/blog/discovering-dennis-ritchies-lost-dissertation/">Discovering Dennis Ritchie’s lost dissertation</a> (<a href="https://mathstodon.xyz/@11011110/104375200069651123"><script type="math/tex">\mathbb{M}</script></a>, <a href="https://news.ycombinator.com/item?id=23582070">via</a>). Ritchie’s doctoral committee signed off in 1968, but the Harvard Library wanted a bound copy and he was unwilling to pay the binding costs so he never officially received his Ph.D. According to the post, the thesis defines a simple model of computation characterizing primitive recursion, within which one can prove that the complexity and growth rates of primitive recursive functions are equal.</p>
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<p><a href="https://mathstodon.xyz/@tpfto/104376254903441351">J.M. redraws the xkcd golden spiral in Mathematica</a>. The thing that annoys me about it is the non-monotonic curvature, but that appears to be unavoidable.</p>
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<p><a href="https://www.youtube.com/playlist?list=PLn0nrSd4xjjadfcMd5xvmJ_GNSLDi1ATn">Playlist of talk videos from this year’s Symposium on Theory of Computing</a> (<a href="https://mathstodon.xyz/@11011110/104389387469883738"><script type="math/tex">\mathbb{M}</script></a>).</p>
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<p><a href="https://www.youtube.com/watch?v=wujEE3PRVUo">Video on the projective geometry of sidewalk trompe-l’oeil chalk art</a> (<a href="https://mathstodon.xyz/@11011110/104397742913356589"><script type="math/tex">\mathbb{M}</script></a>), and <a href="https://www.youtube.com/watch?v=L95cNBEfi5I">another one with less mathematics, more flying pigs and cute space aliens</a>.</p>
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<p><a href="https://www.makeuseof.com/tag/reduce-video-file-size-without-sacrificing-quality/">Some advice I needed on how to reduce video file size</a> (<a href="https://mathstodon.xyz/@11011110/104402114011362297"><script type="math/tex">\mathbb{M}</script></a>). Their suggestion of Handbrake worked well on its default settings and easily reached the file size I needed to reach, without sacrificing quality. (This was for a video of voice over still slides, so it should have been easy to compress, but iMovie couldn’t do it.)</p>
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<p><a href="https://blogs.ams.org/beyondreviews/2020/06/29/the-mathematics-genealogy-project-moves-to-the-cloud/">The Mathematics Genealogy Project rises into the clouds</a> (<a href="https://mathstodon.xyz/@11011110/104407629370705734"><script type="math/tex">\mathbb{M}</script></a>) like a phoenix from the flames of its dead former server. Or maybe not quite as poetically as that, but I use this resource daily in Wikipedia biography editing, so it’s a relief to learn that it’s back after its recent outage.</p>
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<p><a href="https://dl.acm.org/doi/10.1145/3357713.3384232">QCSP monsters and the demise of the Chen conjecture</a> (<a href="https://mathstodon.xyz/@11011110/104414859746330563"><script type="math/tex">\mathbb{M}</script></a>, <a href="https://www.youtube.com/watch?v=c2HjFlcTjQ0">talk video</a>). In STOC’20, Zhuk and Martin show that <a href="https://en.wikipedia.org/wiki/Schaefer%27s_dichotomy_theorem">dichotomy for constraint satisfaction</a> gets messier for quantified CSP. Chen conjectured that QCSP problems are either in NP or PSPACE-complete, but this new paper shows that coNP-completeness can happen for 3 elements and more elements lead to even more classes. Relatedly, <a href="http://eatcs.org/index.php/component/content/article/1-news/2849-the-eatcs-bestows-the-presburger-award-2020">Zhuk just won the Presburger Award</a>.</p>
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<p>I learned while writing <a href="https://en.wikipedia.org/wiki/Doyle_spiral">a Wikipedia article on Doyle spirals</a> (<a href="https://mathstodon.xyz/@11011110/104418935123493700"><script type="math/tex">\mathbb{M}</script></a>) that although the pure-mathematics work in this area dates to Coxeter in 1968 and the work of Thurston and his followers in the 1980s and 1990s, the use of spiral patterns of tangent circles to model plant growth can be traced back much earlier, to Gerrit van Iterson in 1907. The image below, which I used as the lead for the article, is an illustration of phylogeny from <a href="https://archive.org/details/popularsciencemo79newy/page/450/mode/2up">a 1911 <em>Popular Science</em> story about mathematical patterns in nature</a>.</p>
<p style="text-align:center"><img src="/blog/assets/2020/Doyle.png" alt="Doyle spiral from _Popular Science_, 1911" /></p>
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<p><a href="https://www.lms.ac.uk/news-entry/26062020-1657/lms-prize-winners-2020">This year’s London Math Soc. prizewinners</a> (<a href="https://mathstodon.xyz/@11011110/104431647996227722"><script type="math/tex">\mathbb{M}</script></a>, <a href="https://twitter.com/hollykrieger/status/1276590144628416512">via</a>). For some reason they keep the <a href="https://www.lms.ac.uk/prizes/louisbachelierprize">Louis Bachelier Prize in a separate listing</a>. These results have already led me to add to Wikipedia brief articles on <a href="https://en.wikipedia.org/wiki/Maria_Bruna">Maria Bruna</a> (a Whitehead Prize winner) and <a href="https://en.wikipedia.org/wiki/Pauline_Barrieu">Pauline Barrieu</a> (Bachelier 2018).</p>
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<p><a href="https://www.chronicle.com/article/georgia-s-top-down/249095">A power struggle between the Georgia state university system and state government blocks Georgia Tech and other campuses from enacting any coronavirus safety rules</a> (<a href="https://mathstodon.xyz/@11011110/104435200227063842"><script type="math/tex">\mathbb{M}</script></a>).</p>
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</ul>David EppsteinThe five bridges puzzle (). Sort of like the bridges of Königsberg, but stochastic. A cute puzzle with a connection to percolation theory and connection games.Sorting with integer offsets2020-06-28T17:54:00+00:002020-06-28T17:54:00+00:00https://11011110.github.io/blog/2020/06/28/sorting-integer-offsets<p>Here’s a cute exercise for the next time you’re teaching radix sorting in an algorithms class:</p>
<p>Suppose you’re given as input a set of real numbers <script type="math/tex">\{x_i\mid 0\le i\lt n\}</script>, and an integer parameter <script type="math/tex">k</script>. Describe an algorithm for sorting the <script type="math/tex">kn</script> numbers <script type="math/tex">\{x_i+j \mid 0\le i\lt n, 0\le j\lt k\}</script> in time <script type="math/tex">O(kn+n\log n)</script>. You can assume that standard arithmetic operations on real numbers (including comparisons and rounding down to an integer) take constant time per operation.</p>
<p>Models of computation that mix constant-time real arithmetic and rounding operations can be problematic, as by building up and then rounding numbers with unlimited precision you can access a level of computational power beyond what actual computers can do, but I don’t think that’s a concern here. If someone wants to use bit-packing tricks to implement a crazy but fast sorting algorithm in this model, they’re beyond the level of this exercise.</p>
<p>The same method (which I’m not going to describe, to preserve its value as an exercise) more generally allows you to take as input pairs <script type="math/tex">(x_i,m_i)</script> and sort the numbers <script type="math/tex">\{x_i+j \mid 0\le i\lt n, 0\le j\lt m_i\}</script> in time <script type="math/tex">O(M+n\log n)</script> where <script type="math/tex">M=\sum m_i</script>. But it relies heavily on the fact that you’re adding integers to the <script type="math/tex">x</script>’s. For a problem that can’t be handled in this way, consider instead sorting the numbers <script type="math/tex">\{jx_i \mid 0\le i\lt n, 0\le j\lt m_i\}</script> where we multiply instead of adding. Or, if you prefer to view this as a type of <a href="https://en.wikipedia.org/wiki/X_%2B_Y_sorting"><script type="math/tex">X+Y</script> sorting</a> problem, take logs in your favorite base and sort the numbers <script type="math/tex">\{x_i+\log j \mid 0\le i\lt n, 0\le j\lt m_i\}</script>. It’s not at all obvious to me whether this can be done in the same <script type="math/tex">O(M+n\log n)</script> time bound.</p>
<p>The motivation for looking at all this is <a href="https://cstheory.stackexchange.com/q/47120/95">a question about how to implement the greedy set cover quickly</a>. You can find the unweighted <a href="https://en.wikipedia.org/wiki/Set_cover_problem#Greedy_algorithm">greedy set cover</a> in linear time (linear in the sum of the sizes of the input sets; this is an exercise in CLRS), and you can approximate the weighted greedy set cover very accurately in linear time using similar ideas. If you could sort <script type="math/tex">jx_i</script> quickly you could use the sorted order to compute the weighted greedy set cover exactly in the same time as the sorting algorithm. Which is totally useless because the greedy cover is already an approximation, so a fast and accurate approximation to the greedy cover is good enough. But I think the question of sorting <script type="math/tex">jx_i</script> is interesting despite its uselessness in this application.</p>
<p>(<a href="https://mathstodon.xyz/@11011110/104424777130789257">Discuss on Mastodon</a>)</p>David EppsteinHere’s a cute exercise for the next time you’re teaching radix sorting in an algorithms class:Subpract2020-06-21T16:29:00+00:002020-06-21T16:29:00+00:00https://11011110.github.io/blog/2020/06/21/subpract<p>I’ve <a href="/blog/2018/04/18/subtraction-games.html">written here before</a> about <a href="https://en.wikipedia.org/wiki/Subtraction_game">subtraction games</a>, two-player games in which the players remove tokens from a pile of tokens, the number of removed tokens is required to belong to a designated <em>subtraction set</em>, and the goal is to make the last move. For instance, <a href="https://en.wikipedia.org/wiki/Subtract_a_square">subtract a square</a>, a game <a href="https://doi.org/10.4230/LIPIcs.FUN.2018.20">I studied at FUN 2018</a>, is of this type, with the subtraction set being the square numbers.</p>
<p>At some point in studying these games I also briefly looked at the subtraction game whose subtraction set is the set of <a href="https://en.wikipedia.org/wiki/Practical_number">practical numbers</a>, the numbers whose sums of divisors include all values up to the given numbers. The sequence of these numbers begins</p>
<script type="math/tex; mode=display">1, 2, 4, 6, 8, 12, 16, 18, 20, 24, 28, 30, 32, 36, \dots</script>
<p>and it turns out to be important here that, after the first one, they’re all even. Let’s call the subtraction game with this subtraction set <em>subpract</em>.</p>
<p>For a subtraction game, or more generally any <a href="https://en.wikipedia.org/wiki/Impartial_game">impartial game</a>, the game states can be partitioned into <script type="math/tex">\mathcal{P}</script>-positions (where the player who played previously is winning with optimal play) and <script type="math/tex">\mathcal{N}</script>-positions (where the next player to move can force a win); the <script type="math/tex">\mathcal{P}</script>-positions tend to be rarer than the <script type="math/tex">\mathcal{N}</script>-positions, and it’s important to know where they are because the optimal strategy in the game is to move to a <script type="math/tex">\mathcal{P}</script>-position whenever possible.</p>
<p><a href="http://oeis.org/A275432">OEIS A275432</a> lists the <script type="math/tex">\mathcal{P}</script>-positions for subpract. They are:</p>
<script type="math/tex; mode=display">0, 3, 10, 13, 44, 47, 102, 105, 146, 149, 232, 235, \dots</script>
<p>For instance, it’s a winning move in subpract to move to a pile of ten tokens (if you can), because whatever move your opponent makes from there lets you win. If your opponent takes an even number of tokens, you will be able to take all the remaining tokens and win immediately. And if your opponent takes one token, leaving a pile of nine tokens, you can win by taking six more and leaving a pile of three tokens. Then, regardless of how your opponent responds, you will be able to take all the tokens on your next move.</p>
<p>An obvious pattern jumps out from this list of <script type="math/tex">\mathcal{P}</script>-positions: they come in pairs, spaced three apart. More precisely, an even number <script type="math/tex">2x</script> is a <script type="math/tex">\mathcal{P}</script>-position if and only if the odd number <script type="math/tex">2x+3</script> is a <script type="math/tex">\mathcal{P}</script>-position. It’s not just a coincidence, true at the start of the sequence and then false later on: it carries on throughout the entire sequence of <script type="math/tex">\mathcal{P}</script>-positions. More strongly, this same three-apart pairing of <script type="math/tex">\mathcal{P}</script>-positions holds for any subtraction game whose subtraction set contains <span style="white-space:nowrap"><script type="math/tex">1</script>, <script type="math/tex">2</script>, and <script type="math/tex">4</script>,</span> and does not contain any other odd numbers.</p>
<h1 id="proof-of-the-pairing-property">Proof of the pairing property</h1>
<p>To prove this, I need to show that <script type="math/tex">2x</script> is a <script type="math/tex">\mathcal{P}</script>-position if and only if <script type="math/tex">2x+3</script> is a winning position. We can prove this by induction, where we assume that all the <script type="math/tex">\mathcal{P}</script>-positions below <script type="math/tex">2x</script> and <script type="math/tex">2x+3</script> are paired up in the same way, and use it to prove that the same pairing holds for <script type="math/tex">2x</script> and <script type="math/tex">2x+3</script>. The basic idea of the proof is to assume that one of the two players has a winning strategy for the <span style="white-space:nowrap">position <script type="math/tex">2x</script>,</span> and to copy that strategy for <script type="math/tex">2x+3</script>, most of the time playing the same moves and responses that you would play for the smaller position. Whenever a sequence of moves is applicable to both <script type="math/tex">2x</script> and <script type="math/tex">2x+3</script> and preserves the parity of the starting position, the induction hypothesis shows that it has the same outcome for both starting positions. However, there are a few cases where you may be forced to deviate from this strategy:</p>
<ul>
<li>
<p>If <script type="math/tex">2x</script> is an <script type="math/tex">\mathcal{N}</script>-position, but its winning move is to subtract one token leading to an odd <script type="math/tex">\mathcal{P}</script>-position <script type="math/tex">2y+3</script>, then copying that move from the starting position <script type="math/tex">2x+3</script> would lead to the position <script type="math/tex">2y+6</script> which may not be a <script type="math/tex">\mathcal{P}</script>-position. Instead you should subtract four tokens to get to the position <script type="math/tex">2y+3</script> directly.</p>
</li>
<li>
<p>If <script type="math/tex">2x</script> is a <script type="math/tex">\mathcal{P}</script>-position, and you’re trying to copy its winning strategy in the position <script type="math/tex">2x+3</script>, your opponent may be able to subtract <script type="math/tex">2x+2</script>, a move that is not possible in <script type="math/tex">2x</script>, so you have no response to copy. But in this case the result is a pile of just one token, from which you can immediately win.</p>
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<p>Again, if you’re trying to copy the winning strategy for <script type="math/tex">\mathcal{P}</script>-position <script type="math/tex">2x</script> in the position <script type="math/tex">2x+3</script>, your opponent may subtract only one token. In this case, the winning response when starting from <script type="math/tex">2x</script> might be to subtract an even number, leading to an odd <script type="math/tex">\mathcal{P}</script>-position <script type="math/tex">2y+3</script>. If you copy this response, you will end up at <script type="math/tex">2y+6</script> which may not be a <script type="math/tex">\mathcal{P}</script>-position. But instead of copying the <script type="math/tex">2x</script>-strategy, you can simply subtract two tokens, leading to the position <script type="math/tex">2x</script> itself.</p>
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<p>Similarly, it may be the case that the winning response to an opponent’s even move from <script type="math/tex">2x</script> is to take a single token, leading to odd <script type="math/tex">\mathcal{P}</script>-position <script type="math/tex">2y+3</script>. Copying this strategy from the starting position <script type="math/tex">2x+3</script> would again lead to <script type="math/tex">2y+6</script>. But in this case you can subtract four tokens leading to <script type="math/tex">2y+3</script> again.</p>
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</ul>
<p>It’s tempting to guess that, more strongly than pairing <script type="math/tex">\mathcal{P}</script>-positions and <script type="math/tex">\mathcal{N}</script>-positions in this way, subpract and similar subtraction games have a pairing of their <a href="https://en.wikipedia.org/wiki/Sprague%E2%80%93Grundy_theorem">nim-values</a>, where the nim-value of an odd position always equals the nim-value of the even position three units smaller. But it’s not true. For instance, in subpract, a pile of four tokens has nim-value 1 while a pile of seven tokens has nim-value 4.</p>
<h1 id="other-subtraction-sets">Other subtraction sets</h1>
<p>Probably the most obvious choice of another subtraction set that begins <script type="math/tex">\{1, 2, 4,\dots\}</script> and has no larger odd numbers would be the powers of two, but they don’t give rise to an interesting subtraction game: the <script type="math/tex">\mathcal{P}</script>-positions are just the multiples of three. The same thing happens whenever there are no multiples of three in the subtraction set, as happens for instance with the <a href="Telephone number (mathematics)">telephone numbers</a> and <a href="http://oeis.org/A003422">left factorials</a>.</p>
<p>Another natural subtraction set to which this theory applies is the sequence of <a href="http://oeis.org/A025487">Hardy–Ramanujan integers (A025487)</a>, the numbers whose prime factorization <script type="math/tex">2^a\,3^b\,5^c\,7^d\cdots</script> has a non-increasing sequence of exponents <script type="math/tex">a\ge b\ge c\ge d\ge \cdots</script>. They are:</p>
<script type="math/tex; mode=display">1, 2, 4, 6, 8, 12, 16, 24, 30, 32, 36, 48, 60, 64, 72, 96, 120, \dots</script>
<p>These are a subset of the practical numbers so one would expect their subtraction game to have more-dense <script type="math/tex">\mathcal{P}</script>-positions. My implementation found that these <script type="math/tex">\mathcal{P}</script>-positions are:</p>
<script type="math/tex; mode=display">0, 3, 10, 13, 20, 23, 38, 41, 66, 69, 76, 79, 94, 97, 104, 107, \dots</script>
<p>again obeying the offset-by-three pairing as it should, and otherwise having somewhat irregular intervals between its <script type="math/tex">\mathcal{P}</script>-positions.</p>
<p>The <a href="http://oeis.org/A000084">enumeration function of the series-parallel graphs and the cographs</a> is even after its first term because of series-parallel duality; it begins</p>
<script type="math/tex; mode=display">1, 2, 4, 10, 24, 66, 180, 522, 1532, 4624, 14136, 43930, \dots</script>
<p>These are not all practical; for instance, 10, 1532, and 43930 are not practical. The sequence of <script type="math/tex">\mathcal{P}</script>-positions for their subtraction game begins</p>
<script type="math/tex; mode=display">0, 3, 6, 9, 12, 15, 18, 21, 26, 29, 32, 35, 38, 41, 44, 47, \dots</script>
<p>mostly differing by three between consecutive values but with occasional glitches where the larger multiple-of-three subtraction set values kick in.</p>
<p>And finally, if we subtract numbers that are one less than a prime, we get the subtraction set</p>
<script type="math/tex; mode=display">1, 2, 4, 6, 10, 12, 16, 18, 22, 28, 30, 36, 40, 42, \dots</script>
<p>and the sequence of <script type="math/tex">\mathcal{P}</script>-positions</p>
<script type="math/tex; mode=display">0, 3, 8, 11, 32, 35, 56, 59, 64, 67, 118, 121, 208, 211, 216, 219, \dots</script>
<p>Its small values have many five-unit gaps but that pattern appears to die out after the quadruple of <script type="math/tex">\mathcal{P}</script>-positions <script type="math/tex">712, 715, 720, 723</script>.</p>
<p>(<a href="https://mathstodon.xyz/@11011110/104384624632242432">Discuss on Mastodon</a>)</p>David EppsteinI’ve written here before about subtraction games, two-player games in which the players remove tokens from a pile of tokens, the number of removed tokens is required to belong to a designated subtraction set, and the goal is to make the last move. For instance, subtract a square, a game I studied at FUN 2018, is of this type, with the subtraction set being the square numbers.Linkage2020-06-15T17:54:00+00:002020-06-15T17:54:00+00:00https://11011110.github.io/blog/2020/06/15/linkage<ul>
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<p><a href="https://www.ics.uci.edu/~eppstein/261/">Graduata data structures online</a> (<a href="https://mathstodon.xyz/@11011110/104266993520726615"><script type="math/tex">\mathbb{M}</script></a>), finally done and graded. Warning: dry voice-over-slides videos, and some mistakes, because I didn’t have time to put together anything more sophisticated or edit more carefully.</p>
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<p><a href="https://boingboing.net/2020/06/02/list-100-times-law-enforceme.html">Freedom of the press under attack: 100+ times law enforcement violently assaulted journalists in US at George Floyd protests</a> (<a href="https://mathstodon.xyz/@11011110/104276714124643888"><script type="math/tex">\mathbb{M}</script></a>). Of course this is only a small piece of an enormous pattern of awfulness by the current administration, law enforcement, and the prison-industrial complex, but it’s a piece that I think is important to document.</p>
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<p><a href="https://blog.archive.org/2020/06/01/four-commercial-publishers-filed-a-complaint-about-the-internet-archives-lending-of-digitized-books/">Four book publishing corporations claim that what libraries always have done (lending out copies of books they have purchased as physical objects) is illegal, because computer, and are suing the Internet Archive over it</a> (<a href="https://mathstodon.xyz/@11011110/104283895750093044"><script type="math/tex">\mathbb{M}</script></a>, <a href="https://www.theverge.com/2020/6/1/21277036/internet-archive-publishers-lawsuit-open-library-ebook-lending">via</a>, <a href="https://torrentfreak.com/publishers-sue-the-internet-archive-over-its-open-library-declare-it-a-pirate-site-200601/">via2</a>). One of them, Wiley, is also a major publisher of academic works. Perhaps that should give some of us pause in which journals we send our papers to and referee for.</p>
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<p><a href="https://observablehq.com/@otaviocv/moire-patterns-from-random-dots">Moiré patterns from random dots</a> (<a href="https://mathstodon.xyz/@11011110/104290097720006041"><script type="math/tex">\mathbb{M}</script></a>). Overlaying the same random dot pattern on a translated and rotated copy of itself shows concentric dots around the center of rotation, illustrating <a href="https://en.wikipedia.org/wiki/Chasles%27_theorem_(kinematics)">Chasles’ theorem</a> that every rigid transformation of the plane is a translation or rotation. The effect seems to have first been observed by Leon Glass in “<a href="https://doi.org/10.1038/223578a0">Moiré patterns from random dots</a>” (<em>Nature</em>, 1969).</p>
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<p><a href="https://doi.org/10.1007/978-3-030-48966-3_3">“The Micro-world of Cographs”, Alecu, Lozin, and de Werra, <em>IWOCA</em> 2020</a> (<a href="https://mathstodon.xyz/@11011110/104295850760016598"><script type="math/tex">\mathbb{M}</script></a>). <a href="https://en.wikipedia.org/wiki/Cograph">Cographs</a> have a simple structure, but there’s still an interesting hierarchy of subclasses of graphs within them restricting different parameters of graph complexity to be bounded. A typical result: Every cograph with large h-index must contain a large complete graph, balanced bipartite graph, or forest of many high-degree stars.</p>
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<p><a href="https://www.theguardian.com/technology/2020/jun/01/cutting-edge-japanese-paper-art-inspires-a-non-slip-shoe">Japanese scientists use kirigami to design a shoe sole with pop-up non-slip spikes</a>.</p>
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<p><a href="https://mamot.fr/@starifi/104246098809527372">Ombre et lumière</a>. Artwork in which random-looking blocks on a wall create a recognizable shadow in side-light.</p>
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<p><a href="https://mathoverflow.net/a/362569/440">Brian Hopkins answers his own 9-year-old question on the history of Fibonacci numbers and compositions</a> (<a href="https://mathstodon.xyz/@11011110/104312192948087479"><script type="math/tex">\mathbb{M}</script></a>). The ancient Indians knew that compositions (ordered partitions of integers) into <script type="math/tex">1</script>’s and <script type="math/tex">2</script>’s are counted by Fibonacci numbers. For instance, there five ways of forming <script type="math/tex">4</script> as an ordered sum of <script type="math/tex">1</script>’s and <script type="math/tex">2</script>’s: <script type="math/tex">2+2=</script> <script type="math/tex">2+1+1=</script> <script type="math/tex">1+2+1=</script> <script type="math/tex">1+1+2=</script> <script type="math/tex">1+1+1+1</script>. Cayley knew that the compositions with all parts bigger than <script type="math/tex">1</script> have Fibonacci counts. But who first knew that compositions with all parts odd are also counted by Fibonacci? Hopkins suggests: de Morgan, 1846.</p>
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<p>Despite new US covid cases being more or less the same level (or worse) as the start of the lockdown in March, <a href="https://www.chronicle.com/article/Faculty-Want-a-Say-in-Whether/248951">some universities are telling their students that it’s safe to return to normal and at the same time telling their faculty that unless they’re close to retirement age and have additional medical conditions, they must teach face to face</a> (<a href="https://mathstodon.xyz/@11011110/104318055800990449"><script type="math/tex">\mathbb{M}</script></a>).</p>
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<p><a href="https://drericsilverman.wordpress.com/games/">A nice page of recent writings about abstract strategy games, mostly connection games</a> (<a href="https://mathstodon.xyz/@jsiehler/104241019767261031"><script type="math/tex">\mathbb{M}</script></a>).</p>
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<p><a href="https://news.mit.edu/2020/guided-by-open-access-principles-mit-ends-elsevier-negotiations-0611">MIT gives up on trying to get an equitable subscription deal from Elsevier, ends negotiations</a> (<a href="https://mathstodon.xyz/@11011110/104326335350221116"><script type="math/tex">\mathbb{M}</script></a>, <a href="https://news.ycombinator.com/item?id=23489068">via</a>).</p>
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<p><em><a href="https://drops.dagstuhl.de/opus/portals/lipics/index.php?semnr=16150">The Proceedings of the 17th Scandinavian Symposium and Workshops on Algorithm Theory (SWAT 2020)</a></em> (<a href="https://mathstodon.xyz/@11011110/104334978432657954"><script type="math/tex">\mathbb{M}</script></a>), newly published open-access through LIPIcs. Sadly, the conference will be online rather than in the Faroe Islands as originally planned. <em><a href="https://drops.dagstuhl.de/opus/portals/lipics/index.php?semnr=16149">The Proceedings of the 36th International Symposium on Computational Geometry (SoCG 2020)</a></em> is also now out.</p>
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<p><a href="https://www.win.tue.nl/~kbuchin/proj/ruler/art/">Illuminate</a> (<a href="https://mathstodon.xyz/@11011110/104340239573347579"><script type="math/tex">\mathbb{M}</script></a>). An online puzzle based on the art gallery theorem, part of the media exposition of this year’s Symposium on Computational Geometry. See also the <a href="https://doi.org/10.4230/LIPIcs.SoCG.2020.80">theoretical writeup</a>.</p>
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<p><a href="https://link.springer.com/book/10.1007/978-1-84800-070-4">Skiena’s <em>Algorithm Design Manual</em></a> (<a href="https://mathstodon.xyz/@11011110/104349420443124730"><script type="math/tex">\mathbb{M}</script></a>, <a href="https://news.ycombinator.com/item?id=23529759">via</a>, <a href="https://news.ycombinator.com/item?id=23055340">via2</a>, <a href="https://www.metafilter.com/187489/Free-Textbooks">see also</a>), one of 500 Springer textbooks still available for free download from the publisher.</p>
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</ul>David EppsteinGraduata data structures online (), finally done and graded. Warning: dry voice-over-slides videos, and some mistakes, because I didn’t have time to put together anything more sophisticated or edit more carefully.Infinite threshold graphs, four different ways2020-06-14T15:50:00+00:002020-06-14T15:50:00+00:00https://11011110.github.io/blog/2020/06/14/infinite-threshold-graphs<p>One of the difficulties of extending results from finite graphs to infinite ones is that it is not always obvious how to extend the definitions. A single class of finite graphs may correspond, in the infinite graph world, to several different natural classes of infinite graphs. One of the ways this can happen is through orderings: if a class of graphs has a natural ordering on its vertices (say, through a construction in which graphs in this class are built up by adding one vertex at a time) then we might get several classes of infinite graphs with different ways of restricting or not restricting this vertex ordering.</p>
<p style="text-align:center"><img src="/blog/assets/2020/threshold.svg" alt="Threshold graph" /></p>
<p>As an example of this phenomenon, consider the <a href="https://en.wikipedia.org/wiki/Threshold_graph">threshold graphs</a>, one of the simplest classes of finite graphs. An example is shown above. These can be defined in multiple equivalent ways:</p>
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<p>The finite threshold graphs are the graphs that can be built up by repeatedly adding either a universal vertex or an isolated vertex to a smaller graph. In the example, the vertices have been added in left-to-right order, with isolated vertices depicted in yellow and universal vertices in blue. This can be formalized in a way that extends to infinite graphs by saying that they are the graphs in which every nonempty induced subgraph contains either a universal vertex or an isolated vertex. Let’s call these graphs the “inductive threshold graphs”.</p>
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<p>We can also construct graphs in the reverse ordering, by repeatedly adding a vertex whose neighbors form a clique and whose non-neighbors form an independent set. More concisely, the vertex is both simplicial and cosimplicial. The example above can be constructed in this way by adding the vertices in right-to-left order, with the blue neighbors of each added vertex forming a clique and the yellow neighbors forming an independent set. This can be formalized in a way that extends to infinite graphs by requiring that every induced subgraph has a vertex that is simplicial and cosimplicial. Let’s call a graph with this property a “coinductive threshold graph”.</p>
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<p>The finite threshold graphs are the <script type="math/tex">(P_4,C_4,2K_2)</script>-free finite graphs, meaning that no four vertices form an induced subgraph that is a path, cycle, or perfect matching.</p>
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<p>The finite threshold graphs get their name from the following property: they are the graphs that we can generate by assigning weights in the interval <script type="math/tex">[0,1]</script> to the vertices and connecting two vertices by an edge whenever their sum of weights is at least one. Let’s call a graph with this property a “real threshold graph”.</p>
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</ul>
<p>All four of these properties are <a href="https://en.wikipedia.org/wiki/Hereditary_property">hereditary</a>: if a graph has the property, so do all its induced subgraphs. Because the three forbidden subgraphs <script type="math/tex">P_4</script>, <script type="math/tex">C_4</script>, and <script type="math/tex">2K_2</script> have no universal or isolated vertex, have no simplicial and cosimplicial vertex, and have no valid weight assignment, the inductive threshold graphs, coinductive threshold graphs, and real threshold graphs are all subclasses of the <script type="math/tex">(P_4,C_4,2K_2)</script>-free graphs.</p>
<p>If a graph is <script type="math/tex">(P_4,C_4,2K_2)</script>-free, we can define a relation <script type="math/tex">\le</script> on its vertices by saying that <script type="math/tex">u\le v</script> if there is no vertex <script type="math/tex">w</script> with <script type="math/tex">vuw</script> forming an induced path or complement of a path. It follows from the nonexistence of the forbidden subgraphs that this is a total preorder, that every vertex <script type="math/tex">v</script> is universal or isolated among the vertices <script type="math/tex">\{u\mid u\le v\}</script>, and that every vertex <script type="math/tex">v</script> is simplicial and cosimplicial among the vertices <script type="math/tex">\{w\mid v\le w\}</script>. In the example above, two vertices are in the same equivalence class of the ordering if they are in a contiguous block of vertices with the same color, and otherwise their ordering according to <script type="math/tex">\le</script> is the same as their left-to-right ordering. Because every finite total preorder has a minimal and a maximal element, every finite <script type="math/tex">(P_4,C_4,2K_2)</script>-free graph is an inductive threshold graph and a coinductive threshold graph.</p>
<p>However, these properties differ for infinite graphs. In an infinite inductive threshold graph, the total preorder must obey the <a href="https://en.wikipedia.org/wiki/Ascending_chain_condition">ascending chain condition</a> that there be no strictly-increasing infinite sequence of vertices, for the subgraph induced by the vertices of such a sequence would have no isolated or universal vertex. Conversely, if the order does obey the ascending chain condition, one could find an isolated or universal vertex in any subgraph by starting from an arbitrary vertex and repeatedly moving upwards in the order until getting stuck. So the inductive threshold graphs are exactly the ones whose order obeys the ascending chain condition. Similarly, the coinductive threshold graphs are exactly the ones whose order obeys the descending chain condition. But it is easy to construct orders that violate one or both of these conditions. A graph can only be a real threshold graph if the total order on the equivalence classes of its preorder can be embedded into <script type="math/tex">[0,1]</script>, and again this is not true of all total orders.</p>
<p>One consequence of this difference between classes of infinite graphs is the construction of natural statements in the first-order <a href="https://en.wikipedia.org/wiki/Logic_of_graphs">logic of graphs</a> that are true for all finite graphs but untrue for some infinite graphs. For instance, the statements that a <script type="math/tex">(P_4,C_4,2K_2)</script>-free graph has an isolated or universal vertex, and that a <script type="math/tex">(P_4,C_4,2K_2)</script>-free graph has a simplicial and cosimplicial vertex, are both true of all finite graphs, but untrue of some infinite graphs.</p>
<p>(<a href="https://mathstodon.xyz/@11011110/104345188939710302">Discuss on Mastodon</a>)</p>David EppsteinOne of the difficulties of extending results from finite graphs to infinite ones is that it is not always obvious how to extend the definitions. A single class of finite graphs may correspond, in the infinite graph world, to several different natural classes of infinite graphs. One of the ways this can happen is through orderings: if a class of graphs has a natural ordering on its vertices (say, through a construction in which graphs in this class are built up by adding one vertex at a time) then we might get several classes of infinite graphs with different ways of restricting or not restricting this vertex ordering.Linkage2020-05-31T16:08:00+00:002020-05-31T16:08:00+00:00https://11011110.github.io/blog/2020/05/31/linkage<ul>
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<p><a href="https://blogs.scientificamerican.com/roots-of-unity/diana-davis-beautiful-pentagons/">Diana Davis’s beautiful pentagons</a> (<a href="https://mathstodon.xyz/@11011110/104182521046531449"><script type="math/tex">\mathbb{M}</script></a>). I briefly mentioned her regular-pentagon billiards-trajectory art in <a href="/blog/2019/02/15/linkage.html">an earlier post</a> but now Evelyn Lamb has a much more detailed column on her and her work.</p>
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<p><a href="https://inference-review.com/article/points-and-lines">Points and lines</a> (<a href="https://mathstodon.xyz/@11011110/104190473820256744"><script type="math/tex">\mathbb{M}</script></a>). A new review of my book <em><a href="https://www.ics.uci.edu/~eppstein/forbidden/">Forbidden Configurations in Discrete Geometry</a></em>, by Daniel Kleitman, in <em>Inference</em>.</p>
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<p><a href="https://www.mathi.uni-heidelberg.de/~roquette/noetherphoto-engl.pdf">About photos of Emmy Noether</a> (<a href="https://mathstodon.xyz/@11011110/104193371530927420"><script type="math/tex">\mathbb{M}</script></a>), in which Peter Roquette apologizes for having indirectly caused <a href="https://books.google.com/books/about/Emmy_Noether.html?id=IePqBgAAQBAJ">Margaret Tent’s young-adult historical-fiction about Noether</a> to have a photo of someone else on its cover. Via MarkH<sub>21</sub> on Wikipedia, in the context of <a href="https://commons.wikimedia.org/wiki/Commons:Deletion_requests/File:Noether.jpg">a discussion of the provenance of a different photo of Noether</a>.</p>
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<p><a href="https://doi.org/10.4007/annals.2020.191.2.5">The Conway knot is not slice</a> (<a href="https://mathstodon.xyz/@11011110/104196369094406901"><script type="math/tex">\mathbb{M}</script></a>). Newly published result by Lisa Piccirillo in <em>Ann. Math.</em>, with <a href="https://www.quantamagazine.org/graduate-student-solves-decades-old-conway-knot-problem-20200519/">an overview of the significance of the result (if not much of its detail) in <em>Quanta</em></a>.</p>
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<p><a href="https://gowers.wordpress.com/2020/05/20/mathematical-research-reports-a-new-mathematics-journal-is-launched/">Tim Gowers relates the convoluted history of a mathematical announcements journal</a> (<a href="https://mathstodon.xyz/@11011110/104205046534922072"><script type="math/tex">\mathbb{M}</script></a>). <em>Electronic Research Announcements of the AMS</em> (founded 1995) moved to the American Inst. of Mathematical Sciences in 2007, but recent heavyhanded moves by the publisher led the editorial board to quit, and its name changed to <em>Electronic Research Archive</em>. In the meantime the old editorial board have a new journal: <em><a href="https://mrr.centre-mersenne.org/">Mathematical Research Reports</a></em>. See the Gowers link for details.</p>
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<p>In preparation for time travel week in graduate data structures, here’s <a href="http://www.tasteofcinema.com/2014/the-20-best-time-travel-movies-of-all-time/">a listicle of 20 time travel movies</a> (<a href="https://mathstodon.xyz/@11011110/104208167329418315"><script type="math/tex">\mathbb{M}</script></a>). There are any number of these lists but for me they should include at least <em>12 Monkeys</em>, <em>Primer</em>, <em>Safety Not Guaranteed</em>, <em>Donnie Darko</em>, and <em>Time Bandits</em>. This one adds <em>The Girl Who Leapt Through Time</em>, also good. I’d have thrown in <em>The Infinite Man</em> but it’s obscure enough that I’m not offended by its absence. The relevant one for my class, <em>Retroactive</em>, can stay off.</p>
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<p><a href="https://www.thisiscolossal.com/2020/05/xavier-puente-vilardell-wood-sculpture/">Spirals and loops twist through wooden sculptures by Xavier Puente Vilardell</a> (<a href="https://mathstodon.xyz/@11011110/104216153044887740"><script type="math/tex">\mathbb{M}</script></a>).</p>
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<p><a href="https://mathoverflow.net/a/361166/440">Dmitri Panov constructs infinitely many convex 4-polytopes with no triangle or quadrilateral 2-faces</a> (<a href="https://mathstodon.xyz/@11011110/104222101065596394"><script type="math/tex">\mathbb{M}</script></a>). The construction is pretty: take some 120-cells (all 2-faces regular pentagons and all 3-faces regular dodecahedra), embedded into hyperbolic space so all dihedrals are right angles, and glue them together on shared facets. If at most two touch at each ridge, the result is convex. Some pairs of pentagons merge into hexagons, but you still have no triangles or quads.</p>
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<p><a href="https://cp4space.wordpress.com/2013/09/06/ten-things-you-possibly-didnt-know-about-the-petersen-graph/">Ten things you (possibly) didn’t know about the Petersen graph</a> (<a href="https://mathstodon.xyz/@11011110/104226455130356772"><script type="math/tex">\mathbb{M}</script></a>). Found while making <a href="https://en.wikipedia.org/wiki/The_Petersen_Graph">a Wikipedia article on the book <em>The Petersen Graph</em></a> (for the graph itself, see <a href="https://en.wikipedia.org/wiki/Petersen_graph">its Wikipedia article</a>).</p>
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<p><a href="https://www.taylorfrancis.com/books/9781351247771">Godfried Toussaint’s book <em>The Geometry of Musical Rhythm:
What Makes a “Good” Rhythm Good?</em>, on the mathematical analysis of drumming patterns, has a new expanded posthumous 2nd edition</a> (<a href="https://mathstodon.xyz/@11011110/104233265094045761"><script type="math/tex">\mathbb{M}</script></a>). I was able to download free from there but that may be via a campus subscription; your access may vary. For a description of the 1st edition of the book (probably undetailed enough that it can describe the 2nd as well) see <a href="https://en.wikipedia.org/wiki/The_Geometry_of_Musical_Rhythm">its Wikipedia article</a>.</p>
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<p><a href="https://www.youtube.com/watch?v=CfRSVPhzN5M">French video about self-replicating patterns in cellular automata, with English subtitles</a> (<a href="https://mathstodon.xyz/@11011110/104239007149697974"><script type="math/tex">\mathbb{M}</script></a>, <a href="https://cp4space.wordpress.com/2019/06/11/self-replicator-caught-on-video/">via</a>).</p>
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<p><a href="https://mathstodon.xyz/@erou/104245194532017940">@erou visualizes the complexity of Karatsuba’s algorithm for integer multiplication as a Sierpiński triangle</a>, inside a square, with a number of dark pixels proportional to the steps of the algorithm. The square itself counts in the same way the complexity of the naive algorithm.</p>
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<p><a href="https://www.insidehighered.com/quicktakes/2020/05/28/proposed-legislation-would-bar-chinese-stem-graduate-students">Republicans propose legislation to bar Chinese from science</a> (<a href="https://mathstodon.xyz/@11011110/104248070190238732"><script type="math/tex">\mathbb{M}</script></a>). I’m having difficulty distinguishing this sort of move from “<a href="https://en.wikipedia.org/wiki/Nuremberg_Laws">Nazis propose legislation to bar Jews from science</a>”.</p>
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<p><a href="https://wrog.dreamwidth.org/63735.html">Arthur C. Clarke and the projective plane</a> (<a href="https://mathstodon.xyz/@ColinTheMathmo/104253017415627794"><script type="math/tex">\mathbb{M}</script></a>). Wrog beanplates Clarke’s “The Wall of Darkness” (1949).</p>
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<p><a href="https://doi.org/10.1112/blms/16.3.278">Volumes of projections of unit cubes</a> (<a href="https://mathstodon.xyz/@11011110/104261517104618440"><script type="math/tex">\mathbb{M}</script></a>), Peter McMullen, <em>Bull LMS</em> 1984. A cute theorem that deserves to be better known: if you hold a unit cube in the noonday sun, at any angle, its shadow’s area equals its height (elevation difference between lowest and highest point). It follows immediately that the biggest possible shadow is a hexagon with area = long diagonal length = <script type="math/tex">\sqrt{3}</script>, and the smallest shadow is a unit square. Similar things happen in higher dimensions.</p>
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<p><a href="http://roberthodgin.com/project/meander">Meander, a procedural system for generating historical maps of rivers that never existed</a> (<a href="https://mathstodon.xyz/@11011110/104265448606504741"><script type="math/tex">\mathbb{M}</script></a>, <a href="https://www.metafilter.com/187292/Meander-generating-historical-maps-of-rivers-that-never-existed">via</a>). The way this system models the motion of river beds over time looks a lot like the time-reversal of the curve-shortening flow, but with added tangential motion that causes bends to flow downstream (and maybe helps maintain smoothness) and with shortcutting of oxbows.</p>
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</ul>David EppsteinDiana Davis’s beautiful pentagons (). I briefly mentioned her regular-pentagon billiards-trajectory art in an earlier post but now Evelyn Lamb has a much more detailed column on her and her work.Linkage2020-05-15T16:42:00+00:002020-05-15T16:42:00+00:00https://11011110.github.io/blog/2020/05/15/linkage<ul>
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<p><a href="https://www.nytimes.com/2020/04/30/books/celebrity-bookshelves-tv-coronavirus.html">Famous people’s bookshelves, visible as they teleconference from home</a> (<a href="https://mathstodon.xyz/@11011110/104097353853629567"><script type="math/tex">\mathbb{M}</script></a>, <a href="https://www.metafilter.com/186820/Famous-Peoples-Bookshelves">via</a>). The story calls this inadvertent, but that seems dubious. Much of my teleconferencing has books behind me, deliberately, mostly as a clean background in a part of the house where it’s convenient to sit and lighting is good, but also because very few views in my house avoid books. On the other hand, at least one colleague has substituted a fake background from a library…</p>
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<p><a href="http://faculty.smcm.edu/sgoldstine/pinecones.html">Susan Goldstine paints numbers onto the scales of pinecones</a> (<a href="https://mathstodon.xyz/@11011110/104102680813443034"><script type="math/tex">\mathbb{M}</script></a>) to show how phyllotaxis causes the Fibonacci numbers to line up.</p>
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<p>In a post on <a href="https://www.flyingcoloursmaths.co.uk/dictionary-of-mathematical-eponymy-the-quine-mccluskey-algorithm/">the Quine–McCluskey algorithm for minimizing Boolean functions</a> (<a href="https://mathstodon.xyz/@11011110/104114018052115751"><script type="math/tex">\mathbb{M}</script></a>), Colin Beveridge suggests that all mathematical Q-eponyms are named for Dan Quillen or Willard Quine. But even if you discount Quasi-abelian categories, Quasi-Hopf algebras, and Quasi-Newton methods (named after Quasi-Abel, Quasi-Hopf, and Quasi-Newton) there’s also <a href="https://en.wikipedia.org/wiki/Qvist%27s_theorem">Qvist’s theorem</a> in finite geometry, from Bertil Qvist.</p>
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<p><a href="https://www.math.columbia.edu/~woit/wordpress/?p=11723">Why the Szpiro conjecture is still a conjecture</a> (<a href="https://mathstodon.xyz/@11011110/104120078920669171"><script type="math/tex">\mathbb{M}</script></a>). See especially the linked collection of comments from the preceding post (with significant contributions from Fields medalist Peter Scholze) for details of why Scholze thinks Mochizuki’s claimed proof not only doesn’t work, but can’t work.</p>
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<p>Colin (not to be confused with Colin, above) <a href="https://mathstodon.xyz/@ColinTheMathmo/104123191826114771">asks for self-intersecting polyhedra with only hexagonal faces</a>. I described <a href="/blog/2009/09/18/not-nauru-graph.html">a non-self-intersecting toroidal polyhedron with L-shaped hexagonal faces</a> in an earlier post, but I think the <a href="https://en.wikipedia.org/wiki/Small_triambic_icosahedron">small triambic icosahedron</a> (<a href="https://digital.lib.washington.edu/researchworks/handle/1773/4593">conjectured by Grünbaum to be the only face-symmetric polyhedron with more than five sides per face</a>) is closer to what he is asking for.</p>
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<p><a href="https://blog.computationalcomplexity.org/2020/05/why-is-there-no-dn-grid-for-hilberts.html">Bill Gasarch asks: why is there no grid for Hilbert’s 10th?</a> (<a href="https://mathstodon.xyz/@11011110/104131498905150965"><script type="math/tex">\mathbb{M}</script></a>). What he wants to know is, for which pairs <script type="math/tex">(d,n)</script> can we algorithmically find integer solutions to degree-<script type="math/tex">d</script> <script type="math/tex">n</script>-variable polynomial equations, and for which pairs is it undecidable? The answer seems to be: we can solve them when <script type="math/tex">d\le 2</script>, we can’t solve them for some pairs of larger numbers, and there’s a big gap of unknown pairs.</p>
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<p><a href="https://euro-math-soc.eu/news/20/05/8/prize-winners-announced">EMS Prize, Klein Prize, and Neugebauer Prize</a> (<a href="https://mathstodon.xyz/@11011110/104133715419191450"><script type="math/tex">\mathbb{M}</script></a>). Winners: Karim Adiprasito, Ana Caraiani, Alexander Efimov, Simion Filip, Aleksandr Logunov, Kaisa Matomäki, Phan Thành Nam, Joaquim Serra, Jack Thorne, and Maryna Viazovska; Arnulf Jentzen; Karine Chemla.</p>
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<p><a href="https://cstheory.stackexchange.com/q/46746/95">On the complexity of a “list” datastructure in the RAM model</a> (<a href="https://mathstodon.xyz/@11011110/104142162759402806"><script type="math/tex">\mathbb{M}</script></a>). Linked lists can insert and delete at arbitrary positions. Arrays can find the value at position <script type="math/tex">i</script>. Binary trees can do both, but with log time per operation. Combining methods for maintaining order in a list and integer ranking/unranking instead gives <script type="math/tex">O(\log n/\log\log n)</script>. This showed up just in time for me to add some of the ideas to the syllabus for my ongoing online graduate data structures class.</p>
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<p>Some reading on “git pull –rebase”: <a href="https://medium.com/@DGabeau/git-pull-rebase-vs-git-pull-c2b352fe53aa">Gabeau/medium</a>, <a href="https://coderwall.com/p/7aymfa/please-oh-please-use-git-pull-rebase">Hasiński/coderwall</a>, <a href="https://stackoverflow.com/questions/2472254/when-should-i-use-git-pull-rebase">Shved & Mortensen/stackoverflow</a> <span style="white-space:nowrap">(<a href="https://mathstodon.xyz/@11011110/104145964667479373"><script type="math/tex">\mathbb{M}</script></a>).</span> I have been using this in situations with multiple authors simultaneously working on a paper, to keep my contributions to the edit history linear (avoiding edit conflicts and merge bubbles in the history). It’s been working well for me but I realize it might be controversial…</p>
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<p><a href="https://en.wikipedia.org/wiki/Klaus_Roth">Klaus Roth (1925–2015)</a>, Fields medalist who made important contributions to Diophantine approximation, arithmetic combinatorics, and discrepancy theory (<a href="https://mathstodon.xyz/@11011110/104151822927869863"><script type="math/tex">\mathbb{M}</script></a>). Now a Good Article on Wikipedia.</p>
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<p><a href="https://www.scottaaronson.com/blog/?p=4794">Scott Aaronson discusses four new preprints in quantum computing</a> (<a href="https://mathstodon.xyz/@11011110/104164952895881843"><script type="math/tex">\mathbb{M}</script></a>).</p>
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<p><a href="https://thmatters.wordpress.com/2020/05/14/call-for-nominations-for-talg-new-editor-in-chief/">Call for nominations for new editor-in-chief of <em>ACM Transactions on Algorithms</em></a> (<a href="https://mathstodon.xyz/@11011110/104168138945005653"><script type="math/tex">\mathbb{M}</script></a>). Nominations are due <span style="white-space:nowrap">June 8.</span></p>
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<p>My campus computing support people have decided that the middle of a very work-intensive term is the time to make me choose between stopping making illustrations or giving up access to the scientific literature (<a href="https://mathstodon.xyz/@11011110/104174466523993151"><script type="math/tex">\mathbb{M}</script></a>). The connection is the VPN I use to access campus journal and database subscriptions. They want to upgrade to a version incompatible with OS X 10.12. Upgrading OS X would be incompatible with my purchased Adobe software and make me pay money I don’t have for a subscription instead.</p>
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</ul>David EppsteinFamous people’s bookshelves, visible as they teleconference from home (, via). The story calls this inadvertent, but that seems dubious. Much of my teleconferencing has books behind me, deliberately, mostly as a clean background in a part of the house where it’s convenient to sit and lighting is good, but also because very few views in my house avoid books. On the other hand, at least one colleague has substituted a fake background from a library…The inbox of a triangle2020-05-12T00:24:00+00:002020-05-12T00:24:00+00:00https://11011110.github.io/blog/2020/05/12/inbox-of-triangle<p>In affine geometry, the minimum-area ellipse surrounding a given triangle and the maximum-area ellipse within it (its two <a href="https://en.wikipedia.org/wiki/Steiner_inellipse">Steiner ellipses</a>) are concentric and similar. This can be seen easily by performing an affine transformation to an equilateral triangle, observing that in this case these ellipses are concentric circles (the circumcircle and incircle), and that the extreme ellipses of the transformed shape are the transforms of the extreme ellipses of the original shape. In <a href="/blog/2020/04/28/cartesian-triangle-centers.html">Cartesian geometry</a>, where the Cartesian coordinates can be independently linearly transformed or swapped, something similar turns out to happen. In this case, the minimum-area axis-parallel rectangle surrounding a given triangle (its bounding box) and the maximum-area rectangle within it (let’s call it the <em>inbox</em>) are always similar, though not concentric.</p>
<p style="text-align:center"><img src="/blog/assets/2020/inbox.svg" alt="Bounding boxes and inboxes of two triangles" /></p>
<p>The inbox always touches all three sides of its triangle. For if it missed any one of the sides, it could be made larger, while staying within the triangle, by a dilation centered at the opposite vertex. It turns out that for triangles in general position (no two vertex coordinates equal) there are two cases, shown above. If the two median coordinates belong to different triangle vertices (as is true for all acute triangles) then the inbox touches all three sides at separate points, and has a fourth vertex interior to the triangle, as shown on the left. If there is one triangle vertex with both median coordinates (necessarily an obtuse vertex), then the inbox touches that vertex and the opposite side, with two other vertices free, as shown on the right.
Let’s call these the acute and obtuse cases, even though the triangle in the acute case might actually be obtuse.</p>
<p>In the obtuse case, we can apply a Cartesian transformation to make the bounding box square. The inbox lies within an isosceles right triangle formed by axis-parallel lines through the obtuse vertex and by the opposite side of the triangle. By the symmetry of this isosceles right triangle, the inbox is also square. Just as in the Steiner ellipse argument, this implies that the inbox of the untransformed triangle is similar to its bounding box.</p>
<p>In the acute case, let’s again draw axis-parallel lines through the median coordinates. Each line crosses one triangle side, and the line segment between these two crossings cuts the triangle into a smaller triangle (light blue below) and an <a href="https://en.wikipedia.org/wiki/Orthodiagonal_quadrilateral">orthodiagonal quadrilateral</a> (dark blue), with the property that any rectangle that touches the three sides of the triangle also touches the fourth side of the quadrilateral. In this case, the rectangle formed by the four midpoints of the quadrilateral (its <a href="https://en.wikipedia.org/wiki/Varignon%27s_theorem">Varignon rectangle</a>) must be the one with maximum area. For, this rectangle has exactly half the area of the orthodiagonal quadrilateral, as the four triangular flaps surrounding it can be folded over to exactly cover the rectangle. Any other rectangle would have two longer flaps and two shorter flaps, which when folded would overlap near the middle of the rectangle, showing that the total flap area is larger than the rectangle area.</p>
<p style="text-align:center"><img src="/blog/assets/2020/envelope.svg" alt="Construction of an orthodiagonal quadrilateral in which the inbox is inscribed in the acute case" /></p>
<p>The inbox has half the width and half the height of the bounding box of the orthodiagonal quadrilateral, so they both are similar. I have a coordinate-based proof that they are also both similar to the bounding box of the original triangle, but not a nice synthetic proof. Suppose we perform a Cartesian transformation of the triangle so that its cut-off vertex (the one not part of the orthodiagonal quadrilateral) is at the origin and so that its bounding box is the unit square. Let the median coordinates of the triangle vertices be <script type="math/tex">p</script> and <script type="math/tex">q</script>. Then the vertex of the bounding box of the orthodiagonal quadrilateral that is closest to the origin has coordinates <script type="math/tex">(pq,pq)</script>, on the diagonal of the unit square, and its farthest vertex has coordinates <script type="math/tex">(1,1)</script>, so it is a square too.</p>
<p>The center of the Steiner ellipses is the centroid of the triangle, the only affine-equivariant triangle center. Both the center of the inbox and the center of its similarity with the bounding box appear to be Cartesian triangle centers in the sense of <a href="/blog/2020/04/28/cartesian-triangle-centers.html">my previous post</a>: they are continuous Cartesian-equivariant functionals from triangles to points. Unlike the centroid or the bounding-box center, they cannot be calculated separately in the two coordinates, as can be seen from the first example, where the median <script type="math/tex">y</script>-coordinate is halfway between the other two but the inbox center and center of similarity <script type="math/tex">y</script>-coordinates are not. However, they differ from the equal-box-area center of my previous post, which is outside the triangle in the obtuse case while these centers are always inside.</p>
<p>(<a href="https://mathstodon.xyz/@11011110/104156395479629486">Discuss on Mastodon</a>)</p>David EppsteinIn affine geometry, the minimum-area ellipse surrounding a given triangle and the maximum-area ellipse within it (its two Steiner ellipses) are concentric and similar. This can be seen easily by performing an affine transformation to an equilateral triangle, observing that in this case these ellipses are concentric circles (the circumcircle and incircle), and that the extreme ellipses of the transformed shape are the transforms of the extreme ellipses of the original shape. In Cartesian geometry, where the Cartesian coordinates can be independently linearly transformed or swapped, something similar turns out to happen. In this case, the minimum-area axis-parallel rectangle surrounding a given triangle (its bounding box) and the maximum-area rectangle within it (let’s call it the inbox) are always similar, though not concentric.Hanoi vs Sierpiński2020-05-03T22:03:00+00:002020-05-03T22:03:00+00:00https://11011110.github.io/blog/2020/05/03/hanoi-vs-sierpinski<p>The Hanoi graphs and Sierpiński graphs both look like the <a href="https://en.wikipedia.org/wiki/Sierpi%C5%84ski_triangle">Sierpiński triangle</a>, and have a very similar recursive construction from triples of smaller graphs of the same type, but they are not quite the same graphs as each other. The Sierpiński graphs (left, below) are the graphs of the vertices and boundary edges of partially-constructed Sierpiński triangles; they can also be formed from three smaller Sierpiński graphs by identifying pairs of extreme vertices (the vertices of degree two at the three corners of the triangular layout). The <a href="https://en.wikipedia.org/wiki/Hanoi_graph">Hanoi graphs</a> (right, below) are the state spaces of the tower of Hanoi puzzle, in which rings of different size are moved one at a time between three pegs, only allowing moves that keep the rings sorted on each peg. They also have a construction from three smaller Hanoi graphs, but where the pairs of extreme vertices are connected by an edge rather than identified.</p>
<p style="text-align:center"><img src="/blog/assets/2020/hanoi-vs-sierpinski.svg" alt="Hanoi graphs and Sierpiński graphs" /></p>
<p>The difference between them comes out much more strongly when you generalize them to higher dimensions. The Sierpiński triangle generalizes to tetrahedra (a popular shape for kites) and higher-dimensional simplices; <a href="https://commons.wikimedia.org/wiki/File:Alexander_Graham_Bell_facing_his_wife,_Mabel_Hubbard_Gardiner_Bell,_who_is_standing_in_a_tetrahedral_kite,_Baddeck,_Nova_Scotia.tif">the photo below</a> shows <a href="https://en.wikipedia.org/wiki/Mabel_Gardiner_Hubbard">Mabel Bell</a> and <a href="https://en.wikipedia.org/wiki/Alexander_Graham_Bell">Alexander Graham Bell</a>, seemingly about to kiss, in a three-dimensional Sierpiński graph, the framework for a kite.</p>
<p style="text-align:center"><img src="/blog/assets/2020/bell-kite-kiss.jpg" alt="Mabel Bell and Alexander Graham Bell kissing in a Sierpiński tetrahedron kite frame, from https://commons.wikimedia.org/wiki/File:Alexander_Graham_Bell_facing_his_wife,_Mabel_Hubbard_Gardiner_Bell,_who_is_standing_in_a_tetrahedral_kite,_Baddeck,_Nova_Scotia.tif" /></p>
<p>Again, the <script type="math/tex">d</script>-dimensional Sierpiński graph has a recursive construction from <script type="math/tex">d+1</script> smaller graphs of the same type, identified at extreme vertices (the vertices of degree <script type="math/tex">d</script> at the <script type="math/tex">d+1</script> corners of the layout). Because the number of vertices separating the subgraphs at each level of the recursion is so small, these graphs have bounded <a href="https://en.wikipedia.org/wiki/Treewidth">treewidth</a>, and a few years ago on the TCS stackexchange <a href="https://cstheory.stackexchange.com/q/36542">I calculated the treewidth of the Sierpiński triangle graphs explicitly as being four</a>. The same bound transfers easily enough to the three-peg Hanoi graphs.</p>
<p>The analogue of higher dimensions for the Hanoi graphs is to use more pegs. The Hanoi graph with <script type="math/tex">p</script> pegs and <script type="math/tex">r</script> rings has <script type="math/tex">p^r</script> states, more or less the same as the Sierpiński graph for <script type="math/tex">(d-1)</script>-dimensional Sierpiński fractals with <script type="math/tex">r</script> levels of recursion. Here’s the one with two rings; each state is described by a pair of letters, using a capital letter for the peg holding the larger ring and a lowercase letter for the peg holding the smaller ring.</p>
<p style="text-align:center"><img src="/blog/assets/2020/Hanoi-4-2.svg" alt="Hanoi graph for two rings on four pegs" /></p>
<p>The recursive construction for these graphs combines <script type="math/tex">p</script> copies of a smaller graph of the same type: one copy for each position where the largest ring can be placed, and a smaller graph describing the placements of the smaller rings once the largest ring has been placed. These copies of the smaller graph are connected together by edges describing the movements of the largest ring. But I’ve only drawn an example for two rings because these graphs get messy and hard to draw very quickly. The reason is not the exponential number of total vertices, but the large number of connections from one recursive subgraph to another. Two recursive subgraphs are connected whenever the largest ring can move from its peg in one subgraph to its peg in the other, and this is allowed whenever these two pegs have no smaller rings on them. So in a Hanoi graph with <script type="math/tex">p</script> pegs, <script type="math/tex">r</script> rings, and <script type="math/tex">p^r</script> vertices, each pair of recursive subgraphs has <script type="math/tex">(p-2)^{r-1}</script> edges between them, one for each placement of the smaller rings on the <script type="math/tex">p-2</script> remaining pegs.</p>
<p>The recursive subdivision with <script type="math/tex">(p-2)^{r-1}</script> edges between subgraphs leads to a tree-decomposition with treewidth <script type="math/tex">O\bigl((p-2)^r\bigr)</script>, and this naturally raises the question of whether this is tight or whether some other less-intuitive recursive decomposition has smaller cuts between its recursive subgraphs. This is the question studied in my newest preprint, “On the treewidth of Hanoi graphs” (<a href="https://arxiv.org/abs/2005.00179">arXiv:2005.00179</a>), with UCI student Daniel Frishberg and Oregon State student Will Maxwell, to appear at <a href="https://sites.google.com/view/fun2020/home">FUN 2020</a> (supposedly to be held in person in September in Italy after being rescheduled from June, but I’m not holding my breath). We don’t get a precise answer, but we succeed in proving bounds on the treewidth of the form <span style="white-space:nowrap"><script type="math/tex">\Omega\bigl((p-2)^r/r^{O(1)}\bigr)</script>.</span> This is nearly tight for fixed <script type="math/tex">p</script> and variable <script type="math/tex">r</script>: we get the same exponential function of <script type="math/tex">r</script> as the upper bound, and are smaller than the upper bound by only a much lower-order polynomial factor. But the exact treewidth remains elusive.</p>
<p>To put it in a possibly more familiar form, when one of these graphs (for a fixed number of pegs and variable number of rings) has <script type="math/tex">n</script> vertices, it has separators of size <script type="math/tex">O(n^c)</script>, where <script type="math/tex">c=\log_p (p-2)</script>. For the four-peg Hanoi graphs, this means separators of size <script type="math/tex">O(\sqrt{n})</script>, more or less the same as for planar graphs (although these graphs seem far from planar). But that nice exponent is just a coincidence caused by the fact that <script type="math/tex">p=4</script> is a power of <script type="math/tex">p-2=2</script>. For other choices of <script type="math/tex">p</script>, that doesn’t happen and we get a transcendental exponent <script type="math/tex">c</script>. So these graphs don’t even act like <script type="math/tex">(p-1)</script>-dimensional graphs, for which a reasonable separator exponent might be the rational number <script type="math/tex">(p-2)/(p-1)</script>. And they certainly don’t act like the Sierpiński graphs, for which the exponent is zero.</p>
<p>(<a href="https://mathstodon.xyz/@11011110/104108481482094736">Discuss on Mastodon</a>)</p>David EppsteinThe Hanoi graphs and Sierpiński graphs both look like the Sierpiński triangle, and have a very similar recursive construction from triples of smaller graphs of the same type, but they are not quite the same graphs as each other. The Sierpiński graphs (left, below) are the graphs of the vertices and boundary edges of partially-constructed Sierpiński triangles; they can also be formed from three smaller Sierpiński graphs by identifying pairs of extreme vertices (the vertices of degree two at the three corners of the triangular layout). The Hanoi graphs (right, below) are the state spaces of the tower of Hanoi puzzle, in which rings of different size are moved one at a time between three pegs, only allowing moves that keep the rings sorted on each peg. They also have a construction from three smaller Hanoi graphs, but where the pairs of extreme vertices are connected by an edge rather than identified.