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0xDE - LiveJournal.comWed, 01 Mar 2017 05:35:59 GMTLiveJournal / LiveJournal.com110111107784841personalhttp://l-userpic.livejournal.com/32934265/77848410xDE
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100100http://11011110.livejournal.com/342630.htmlWed, 01 Mar 2017 05:35:59 GMTLinkage
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<ul><li><a href="http://www.thisiscolossal.com/2017/02/forest-of-60000-rainbow-numbers/">Emmanuelle Moureaux's walk-through forest of colored digits</a> reminds me of some <a href="https://en.wikipedia.org/wiki/I_Saw_the_Figure_5_in_Gold">earlier digital art</a> (<a href="https://plus.google.com/100003628603413742554/posts/LepqHiHNhvV">G+</a>)</li><br /><li><a href="https://plus.google.com/101113174615409489753/posts/NpmFMDyCKai">Sariel peels huge grids</a>, with the result appearing to approximate the affine curve-shortening flow (<a href="https://plus.google.com/100003628603413742554/posts/BU3stE1JWU9">G+</a>)</li><br /><li><a href="https://www.brainpickings.org/2012/12/03/berenice-abbott-documenting-science/">Berenice Abbott's vintage scientific photography</a> (<a href="https://plus.google.com/100003628603413742554/posts/5xBE1oMghJM">G+</a>)</li><br /><li><a href="https://twitter.com/MikeTaylor/status/832973591847202816/photo/1">The growth of perverse incentives in academia</a> or <a href="https://en.wikipedia.org/wiki/Campbell%27s_law">Campbell's law</a> in action (<a href="https://plus.google.com/100003628603413742554/posts/cH5mmTTNoMm">G+</a>)</li><br /><li><a href="http://www.chronicle.com/article/Meet-the-Math-Professor/239260">Moon Duchin fights gerrymandering with geometry</a> (<a href="https://plus.google.com/100003628603413742554/posts/6x5KnZKYqYd">G+</a>)</li><br /><li><a href="https://doi.org/10.1137/1.9781611974782.122">Orlin and Sedeño-Noda find shortest cycles</a> using an elegant new algorithm resembling Johnson's (<a href="https://plus.google.com/100003628603413742554/posts/QrQPH9cFspV">G+</a>)</li><br /><li><a href="https://plus.google.com/113086553300459368002/posts/eecry6w2hbq">Quasiperiodic girih zoom</a> by Greg Egan (<a href="https://plus.google.com/100003628603413742554/posts/PaaL4xDEJTU">G+</a>)</li><br /><li><a href="http://blog.plover.com/lang/anagram-scoring-2.html">Scoring anagram quality</a> by using maximum independent sets in a graph of grid points (<a href="https://plus.google.com/100003628603413742554/posts/3xefYk8uoLG">G+</a>)</li><br /><li><a href="http://laughingsquid.com/an-origami-inspired-ballistic-shield-designed-to-stop-bullets-and-protect-law-enforcement/">Origami police siege sheild</a> (<a href="https://plus.google.com/100003628603413742554/posts/AF9FrXF3gQS">G+</a>)</li><br /><li><a href="https://ecotonemagazine.org/map/mystic-island/">Interdigitating trees of land and water in an NJ subdivision</a> (<a href="https://plus.google.com/100003628603413742554/posts/dRfrQcqfRr3">G+</a>)</li><br /><li><a href="http://mathoverflow.net/q/263370/440">Are all Dehn invariants achievable?</a> (<a href="https://plus.google.com/100003628603413742554/posts/aVX15bs7NQk">G+</a>)</li></ul>http://11011110.livejournal.com/342630.htmlvotingtilingphotographypolytopesgraph algorithmstreesartgeometryacademiaorigamipublic0http://11011110.livejournal.com/342371.htmlMon, 20 Feb 2017 03:28:32 GMTTriangle-free penny graphs are 2-degenerate
http://11011110.livejournal.com/342371.html
A <a href="https://en.wikipedia.org/wiki/Penny_graph">penny graph</a> (the topic of today's new Wikipedia article) is what you get from pennies by shoving them together on a tabletop, keeping them only one layer thick, and looking at which pairs of pennies are touching each other. In a 2009 paper, Konrad Swanepoel suggested that, when there are no three mutually-touching pennies, the number of adjacencies should be at most 2<i>n</i> − 2√<i>n</i>. I don't know how to prove that, and as far as I know the problem remains open. But here's an argument for why the number of edges should be strictly less than 2<i>n</i> − 4, the upper bound given by Swanepoel.<br /><br />The basic idea is that, in any set of triangle-free pennies, some penny is touched at most twice. Or, in graph theoretic terms, a triangle-free penny graph must be <a href="https://en.wikipedia.org/wiki/Degeneracy_(graph_theory)">2-degenerate</a>. For, suppose to the contrary that all pennies had at least three neighbors, and consider the outer face of the planar graph formed by the pennies (marked by the blue polyline below). For each penny on the outer face, draw a line through its center and the center of a neighboring penny that is not one of its two outer-face neighbors (the red lines in the figure).<br /><br /><div align="center"><img src="http://www.ics.uci.edu/~eppstein/0xDE/triangle-free-penny-graph.jpg" width="640" height="486"></div><br /><br />As you walk around the outer face (clockwise, say) these red lines would always have to point consistently inwards and outwards, so they would have to turn with you, eventually making a complete circuit in the clockwise direction. But they can only turn counterclockwise! If you follow an outer face edge whose adjacent inner face is a quadrilateral, the red lines stay parallel (as they do in most of the picture) and otherwise they turn the wrong way. The impossibility of the red lines making a complete clockwise turn while only turning counterclockwise at each step shows that our assumption, that all pennies have three neighbors, cannot be correct. Therefore, there is a penny on the outer face with at most two neighbors.<br /><br />Five-vertex triangle-free penny graphs have at most five edges, so by induction <i>n</i>-vertex triangle-free penny graphs have at most 2<i>n</i> − 5 edges, very slightly edging out an upper bound of 2<i>n</i> − 4 given by Swanepoel based on Euler's formula.<br /><br />The fact that penny graphs are 3-degenerate is a standard exercise, part of an easy proof of the four-color theorem for these graphs. Similarly, the 2-degeneracy of triangle-free penny graphs leads to an easy proof of <a href="https://en.wikipedia.org/wiki/Gr%C3%B6tzsch%27s_theorem">Grötzsch's theorem</a> for them. So I would be surprised if the 2-degeneracy of the triangle-free penny graphs is new, but I didn't see it when I was researching the Wikipedia article. Does anyone know of a reference?http://11011110.livejournal.com/342371.htmlcirclesgeometrygraph theorypublic7http://11011110.livejournal.com/342045.htmlFri, 17 Feb 2017 02:31:51 GMTAn unsuccessful attempt to use CairoSVG to generate small vector-graphics PDF files
http://11011110.livejournal.com/342045.html
The following image shows the onion layers of a 6 × 6 grid (see <a href="https://plus.google.com/u/0/101113174615409489753/posts/PdoVfoHY6bJ">Sariel's post for context</a>). It consists of 42 circles, 36 with a fill and a stroke and 6 with only a stroke.<br /><br /><div align="center"><img src="http://www.ics.uci.edu/~eppstein/0xDE/grid-onion.svg" style="background-color:white;"></div><br />My usual workflow is to draw this sort of thing in Adobe Illustrator, but one drawback of doing things this way is huge files: as a PDF file suitable for inclusion in LaTeX, these 42 circles take up 155k bytes. I've recently started cutting my file size by opening the Illustrator PDF in the Apple Preview application, and then using Preview's "export" command with the "reduce file size" filter. This cuts it down to 41k, still nothing impressive but a lot better. It does have the drawback that going between Illustrator and Preview multiple times has sometimes messed up my fonts, so I think it's best viewed as a one-way filter: if I want to re-edit the same file, I need to keep the original.<br /><br />But if I save as an SVG file from Illustrator (like the one above) it's only 3.3K, a reasonable size for such a simple image. Reading the SVG back into Illustrator and saving as PDF blows it back up to 88k, which is still way too big. So I thought: maybe there's a good command line tool for converting SVG to PDF? I could have a workflow where I use Illustrator only to read and edit SVG files (keeping them around so that I can re-edit them if I want, without as much confusion as keeping two different PDF versions of the same file) and use a different program to convert them to small PDFs.<br /><br />After some searching, I found CairoSVG, which (after a lot of hassle installing, see below) seemed to work well: it produces a 3.5k byte PDF file from the same image.<br /><br />The installation process was a bit of a mess. Very different from typical OS X download-and-run installation processes:<ol><li>Install pip from <a href='https://pip.pypa.io/en/stable/installing/'>https://pip.pypa.io/en/stable/installing/</a></li><li>Use pip to install CairoSvg per <a href='http://cairosvg.org/'>http://cairosvg.org/</a></li><li>Try running cairosvg from the command line, but it doesn't work because it is unable to load the cairo library. After much searching, discover what the CairoSVG web site never tells you: that cairo is a separate piece of software that you also need to install.</li><li>Install MacPorts from <a href='https://www.macports.org/install.php'>https://www.macports.org/install.php</a></li><li>Try using MacPorts to install cairo, but discover that it requires the App version of Xcode and not just the command-line developer tools I already had installed.</li><li>Install Xcode from the Apple App store</li><li>Try MacPorts again, but discover that the App store install is not the real install.</li><li>Run Xcode to install it for real.</li><li>Use MacPorts to install cairo per <a href='https://www.cairographics.org/'>https://www.cairographics.org/</a></li><li>Try running cairosvg from the command line, but it still can't find the library.</li><li>Searching the net for the error message eventually leads to <a href='https://github.com/Kozea/WeasyPrint/issues/79'>https://github.com/Kozea/WeasyPrint/issues/79</a> which advises setting the environment variable DYLD_FALLBACK_LIBRARY_PATH to /opt/local/lib</li><li>It used to be the case that setting environment variables globally was done by editing the file ~/.MacOSX/environment.plist but that no longer works — see <a href='http://stackoverflow.com/questions/603785/environment-variables-in-mac-os-x/'>http://stackoverflow.com/questions/603785/environment-variables-in-mac-os-x/</a> — so instead I've been putting them in my .login file.</li><li>After editing .login to set the environment variable I need to make a new terminal window because the old ones won't have it set yet.</li></ol>After all this, the command-line cairosvg runs, and the command line "cairosvg grid-onion.svg -o grid-onion.pdf" produces the small PDF file reported above.<br /><br />But we're not done...<br /><br />It turns out that the resulting PDF has a different image size than the original file. After some more groveling on the net I discovered that it's using a default value of 96 dots per inch rather than respecting the 72dpi value used by Illustrator. So when I view the resulting pdf files in Illustrator, Preview, or TeX, they don't have the same scale as when I drew them. This is a problem, because I've been using relative image sizes (the scale= parameter of graphicx) when including images in some of my LaTeX documents.<br /><br />The cairosvg command line program has two options for changing the scale. We can either set the program to use 72 dpi, or we can tell it to scale the image by a factor of 1.33. Both generate images that have the correct size. But now the bounding box of the image has been destroyed, and it's instead using a bounding box that's way too big. This turns out to be true for svg to png conversion as well: if I try to use cairosvg to make a png from my svg file, it also loses the bounding box. There are no options for telling cairosvg to avoid destroying the bounding box, and no suggestion on the cairosvg issue tracking site that anyone knows or cares about this problem.<br /><br />After all this, I'm ready to give up on cairosvg as being worth what I've paid for it (nothing). Does anyone know of an svg-to-pdf conversion pipeline that produces small pdf files but actually works?http://11011110.livejournal.com/342045.htmltoolspublic26http://11011110.livejournal.com/341841.htmlThu, 16 Feb 2017 06:12:57 GMTLinkage
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<ul><li><a href="http://weburbanist.com/2017/01/30/telemetry-tessellated-paper-sculptures-marry-art-and-engineering/">Telemetry: Tessellated paper sculptures by Matthew Shlian</a> (<a href="https://plus.google.com/100003628603413742554/posts/EHqR9h1Foyv">G+</a>)</li><br /><li><a href="https://twitter.com/wattenberg/status/826937390531018752">Polygonal billiard paths</a>, after the work of Iranian-American Fields medalist Maryam Mirzakhani (<a href="https://plus.google.com/100003628603413742554/posts/TiPwCy69vbg">G+</a>)</li><br /><li><a href="http://blog.computationalcomplexity.org/2017/02/we-are-all-iranians.html">We are all Iranians</a>: Lance Fortnow speaks out about the travel ban and its affect on his colleagues (<a href="https://plus.google.com/100003628603413742554/posts/C3MNu8wPWbH">G+</a>)</li><br /><li><a href="https://plus.google.com/109619272370691037001/posts/SMhikhisRnG">David Wood shares the saga of the journal publication of one of his papers</a>, highlighting the long and unnecessary delays in the system (<a href="https://plus.google.com/100003628603413742554/posts/2B6Jap3JDj6">G+</a>)</li><br /><li><a href="http://bit-player.org/2017/a-tantonalizing-problem">A tantonalizing problem</a>: Brian Hayes tracks down the original proof of the non-integrality of the partial sums of the harmonic series (<a href="https://plus.google.com/100003628603413742554/posts/C6p8tBxPgHR">G+</a>)</li><br /><li><a href="http://mathematica.stackexchange.com/questions/136974/code-that-generates-a-mandala">Code that generates a mandala</a> (<a href="https://plus.google.com/100003628603413742554/posts/XLqTzzJYTCK">G+</a>)</li><br /><li><a href="https://terrytao.wordpress.com/2017/02/05/a-bound-on-partitioning-clusters/">Terence Tao on a problem resembling 3SUM, but with disjoint set union instead of addition</a> (<a href="https://plus.google.com/100003628603413742554/posts/NBv9ryVYT2E">G+</a>)</li><br /><li><a href="http://physics.aps.org/story/v25/st4">How dynamic demands cause tree veins not to form trees</a> (<a href="https://plus.google.com/100003628603413742554/posts/jQwmPXJ7Nw7">G+</a>)</li><br /><li><a href="http://boingboing.net/2017/02/09/wikipedia-policy-declares-the.html"><i>Daily Mail</i> officially declared non-reliable</a> (<a href="https://plus.google.com/100003628603413742554/posts/azjfowmPz8s">G+</a>)</li><br /><li><a href="https://archive.org/stream/ioanniskepplerih00kepl#page/26/mode/2up">The Calkin–Wilf tree in the works of Kepler</a> (<a href="https://plus.google.com/100003628603413742554/posts/68sjkx8mjGC">G+</a>)</li><br /><li><a href="http://naukas.com/2017/02/11/feliz-dia-de-la-mujer-y-la-nina-en-ciencia/">Notable women in science</a> (in Spanish) for the <a href="http://www.un.org/en/events/women-and-girls-in-science-day/">International Day of Women and Girls in Science</a> (<a href="https://plus.google.com/100003628603413742554/posts/PvCRnA9sBZL">G+</a>)</li><br /><li><a href="https://youtu.be/7s-YM-kcKME">How to apportion the rent for unequal rooms in a shared apartment, using combinatorics</a> (<a href="https://plus.google.com/100003628603413742554/posts/J1RiaXMmB12">G+</a>)</li><br /><li><a href="https://www.wired.com/2017/02/want-secure-elections-maybe-dont-cut-security-funding/">Congressional Republicans zero out the budget of the office that tests and certifies the security of voting machines</a> (<a href="https://plus.google.com/100003628603413742554/posts/VPWFiBVhPM4">G+</a>)</li><br /><li><a href="http://boingboing.net/2017/02/13/us-born-nasa-scientist-was-det.html">Don't travel internationally with electronic devices unless you want to give the feds all your passwords</a> (<a href="https://plus.google.com/100003628603413742554/posts/VgcY41z8i2G">G+</a>)</li></ul>http://11011110.livejournal.com/341841.htmlfeminismcombinatoricswikipediasecuritytreesartnumber theoryacademiaorigamipoliticspublic0http://11011110.livejournal.com/341698.htmlWed, 15 Feb 2017 02:27:22 GMTComplete bipartite polyhedra
http://11011110.livejournal.com/341698.html
The tetrahedron and the <a href="https://en.wikipedia.org/wiki/Cs%C3%A1sz%C3%A1r_polyhedron">Császár polyhedron</a> are polyhedra whose edges and vertices form complete graphs; it is unknown whether any others exist. At the tutorial session for Graph Drawing 2015, I asked whether there are any complete bipartite polyhedra.<br /><br />As the definition of non-convex polyhedra can be the subject of heated debate (check out <a href="https://en.wikipedia.org/wiki/Talk:Polyhedron">Wikipedia:Talk:Polyhedron</a> if you dare), I should clarify that the kind of polyhedron I am looking for is a piecewise-linearly-embedded manifold whose maximal linear pieces (the faces) are simple polygons. This is all true of the Császár polyhedron, for instance. For non-geometric definitions of polyhedra, the existence of complete bipartite polyhedra is already known; for instance Archdeacon ("a survey of self-dual polyhedra", 1992) describes a polyhedral embedding of <i>K</i><sub>6,5</sub> on a topological surface with 17 crosscaps, dual to <i>K</i><sub>9</sub> on that surface.<br /><br />A near-miss geometric complete bipartite polyhedron is the complete bipartite graph <i>K</i><sub>2,2</sub>, which forms the graph of a <a href="https://en.wikipedia.org/wiki/Dihedron">doubly covered square</a>. This is in some sense a Euclidean convex polyhedron; at least, it satisfies the criteria of <a href="https://en.wikipedia.org/wiki/Alexandrov%27s_uniqueness_theorem">Alexandrov's characterization of convex polyhedra by their surface metrics</a>. But its two faces coincide, rather than intersecting only at their shared vertices and edges. The regular octahedron forms the graph of a complete tripartite graph <i>K</i><sub>2,2,2</sub>, and the cube forms a subgraph of <i>K</i><sub>4,4</sub> but is missing some edges (the long diagonals) needed to form the whole complete bipartite graph.<br /><br />In a complete bipartite graph, represented in this way, each face must be a quadrilateral, because higher order faces would have diagonals that form edges of other faces, not possible in a manifold. This puts some constraints on which bipartite graphs we can have: the number of edges must be divisible by two (so that there is an integer number of quadrilaterals) and in addition the number of edges must be 2 (mod 4) if the number of vertices is odd, or 0 (mod 4) if the number of vertices is even, so that the Euler characteristic will be even. (Odd Euler characteristics only happen for non-oriented manifolds, which cannot be embedded without crossings in Euclidean space.) And <i>K</i><sub>2,<i>n</i></sub> is never possible, because it has vertices of degree two which must come from coplanar faces or collinear edges, neither of which is allowed by the specific definition of polyhedra chosen above.<br /><br />Based on this calculation, the simplest complete bipartite graphs that could work are <i>K</i><sub>4,4</sub> and <i>K</i><sub>3,6</sub>. Both of these form nice polyhedral subdivisions of a torus:<br /><br /><div align="center"><img src="http://www.ics.uci.edu/~eppstein/0xDE/complete-bipartite-tori.svg"></div><br /><br />I have been unable to find manifold embeddings of these tori, but I also don't have a proof that they're impossible. I did, however, find a realization of <i>K</i><sub>4,4</sub> as a crossed polyhedron (meaning: a subdivision of a topological manifold, with its vertices mapped to distinct points of Euclidean space in such a way that the faces are mapped to distinct planes, but allowing edge and face crossings):<br /><br /><div align="center"><img src="http://www.ics.uci.edu/~eppstein/0xDE/crossed-K44.svg"></div><br /><br />It's based on a <a href="https://en.wikipedia.org/wiki/Complete_quadrangle">complete quadrilateral</a> in the <i>z</i> = 0 plane (the black lines) with two of its six points lifted and lowered in <i>z</i> to form pairs of points. In this arrangement of points, <i>A</i> and <i>C</i> form dart-shaped quadrilaterals with the blue points on the line <i>x</i> = 0 and with the other two blue points. Similarly, <i>B</i> and <i>D</i> form two darts with pairs of the two blue points. <i>A</i> and <i>D</i> form a tilted dart with the two blue points on the diagonal <i>x</i> = <i>y</i>, and <i>B</i> and <i>C</i> also form a tilted dart with the same two diagonal points. Finally, <i>A</i> and <i>B</i> form a crossed quadrilateral (an <a href="https://en.wikipedia.org/wiki/Antiparallelogram">antiparallelogram</a>) with the two blue points on the line <i>x</i> + <i>y</i> = 3, and <i>B</i> and <i>D</i> form another antiparallelogram with the same two blue points. So each edge of the complete bipartite graph is used twice, giving us a topological manifold.<br /><br />This is not the torus above, because the face pairings are different. It turns out to be a different torus:<br /><br /><div align="center"><img src="http://www.ics.uci.edu/~eppstein/0xDE/brick-wall-K44.svg"></div><br /><br />So, this does show that <i>K</i><sub>4,4</sub> can be mapped to space so that the points form eight planes of four points, intersecting in the desired edges, and forming a configuration of eight points and eight planes with four points per plane and four planes per point. But it doesn't really solve the original problem. Can it be untangled to form an embedded torus with the same connectivity?<br /><br />ETA 2/16: <i>K</i><sub>3,6</sub> is also easy to realize as a crossed polyhedron. Place <i>A</i>, <i>B</i>, and <i>C</i> on a triangle, choose three planes through each pair of these points, and place the other six points where triples of these planes meet. But be careful to choose the correct triples of planes: if you place <i>A</i>, <i>B</i>, and <i>C</i> on an equilateral triangle, and the other six points at equal distances above and below the midpoints, you can realize <i>K</i><sub>3,6</sub> as a crossed polyhedron with three square sides and six antiparallelogram sides, but it's a Klein bottle, not a torus.http://11011110.livejournal.com/341698.htmlpolytopesconfigurationspublic0http://11011110.livejournal.com/341333.htmlTue, 31 Jan 2017 20:35:05 GMTLinkage from out of the country
http://11011110.livejournal.com/341333.html
I'm in Barbados for a week, but that isn't helping me avoid seeing the circus our head bozo has made.<br /><ul><li><a href="http://boingboing.net/2017/01/16/manifold-a-pad-a-100-origami.html">Pad of origami puzzles</a> (<a href="https://plus.google.com/100003628603413742554/posts/jWxd2caH5sz">G+</a>)</li><br /><li><a href="https://www.insidehighered.com/news/2017/01/18/librarians-list-predatory-journals-reportedly-removed-due-threats-and-politics">Beall's list of predatory journal publishers taken down</a> — grab an archived copy while you can (<a href="https://plus.google.com/100003628603413742554/posts/iac5yr8TZwp">G+</a>)</li><br /><li><a href="http://lemire.me/blog/2017/01/20/how-quickly-can-you-remove-spaces-from-a-string/">How quickly can you remove spaces from a string?</a> A demonstration that bit-parallel programming techniques really work (<a href="https://plus.google.com/100003628603413742554/posts/FRTxnt9bSS2">G+</a>)</li><br /><li><a href="http://www.cut-the-knot.org/blue/DistanceToManyPoints.shtml">Average distance to points on a circle</a> (with better solutions on <a href="https://plus.google.com/100003628603413742554/posts/16REKXmrHAP">G+</a>)</li><br /><li><a href="http://www.mfo.de/about-the-institute/news/cancellation-of-the-current-elsevier-contract-as-of-31-december-2016">Oberwolfach (and the other German University libraries) no longer have access to Elsevier journals</a> (<a href="https://plus.google.com/100003628603413742554/posts/77VmrhnZA24">G+</a>)</li><br /><li><a href="http://fpt.wikidot.com/">Call for nominations — Nerode Prize in multivariate algorithmics and parameterized complexity — due by email March 1</a> (<a href="https://plus.google.com/100003628603413742554/posts/7KSBr9ceQ7u">G+</a>)</li><br /><li><a href="https://www.theguardian.com/media/2017/jan/24/journalists-charged-felonies-trump-inauguration-unrest">Trump's disrespect for free speech is showing: four journalists charged with felonies for covering anti-Trump protests</a> (<a href="https://plus.google.com/100003628603413742554/posts/Ax4ZXM2nKsE">G+</a>)</li><br /><li><a href="http://www.scottaaronson.com/blog/?p=3167">First they came for the Iranians</a>, Scott A. on the immigration shutdown (<a href="https://plus.google.com/100003628603413742554/posts/WoC9qtU48at">G+</a>)</li><br /><li><a href="https://gieseanw.wordpress.com/2012/10/21/a-comprehensive-step-by-step-tutorial-to-computing-dubins-paths/">Dubins paths</a>, or how to park your car when it won't go into reverse (<a href="https://plus.google.com/100003628603413742554/posts/b3EkXiNkgiR">G+</a>)</li><br /><li><a href="https://www.wired.com/2017/01/kim-albrecht-trump-data-viz/">Complex visualization untangles Trump’s business ties</a> (<a href="https://plus.google.com/100003628603413742554/posts/92e2PmnXoKt">G+</a>)</li><br /><li><a href="http://www.thisiscolossal.com/2017/01/marc-fornes-under-magnitude-installation/">Giant aluminum coral reef sculpture by Marc Fornes & Very Many</a> (<a href="https://plus.google.com/100003628603413742554/posts/Xmat6D2mKcU">G+</a>)</li><br /><li><a href="https://plus.google.com/+RoiceNelson/posts/U5AQ46zq3XZ">Roice Nelson untangles the structure of the Borromean rings complement from <i>Not Knot</i></a> (<a href="https://plus.google.com/100003628603413742554/posts/KX78GcDXAtZ">G+</a>)</li><br /><li><a href="https://www.youtube.com/watch?v=gB9n2gHsHN4">Self-similar things are often fractals, but things with fractal dimensions are not all self-similar</a>, and allowing some non-self-similarity is essential in modeling nature by fractals (<a href="https://plus.google.com/100003628603413742554/posts/Gy78GbZDp1S">G+</a>)</li></ul>http://11011110.livejournal.com/341333.htmlfractalsbit parallelismfree speechcorporatizationgraph drawingbibliographycirclestopologyartorigamipoliticspublic0http://11011110.livejournal.com/340999.htmlMon, 23 Jan 2017 03:31:38 GMTRainy day
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The view out my back door this afternoon:<br /><br /><div align="center"><img src="http://www.ics.uci.edu/~eppstein/pix/rainytrellis/RainyTrellis-m.jpg" border="2" style="border-color:black;" /></div><br /><br />I'm pretty sure the white hair-like things in the background are raindrops, motion-blurred into streaks.<br /><br />I just replaced my laptop (the old one's keyboard and trackpad were dying), so if the colors are off or different from before, that's why. Also much fussing with Python ensued to get my old gallery-creation script running again, since the brain transplant didn't include any libraries I might have been depending on, and the versions I could figure out how to install in their place had a slightly different API. I'm just glad my old copy of Photoshop still runs and I don't have to switch to the subscription model or find an alternative.http://11011110.livejournal.com/340999.htmlpythonphotographypublic0http://11011110.livejournal.com/340796.htmlThu, 19 Jan 2017 01:24:03 GMTWeak moves in high-dimensional tic-tac-toe
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Everyone who plays tic-tac-toe quickly learns that the nine squares have different strengths. The center square is best, because it belongs to four lines. The corners, on three lines each, are next. And weakest are the squares in the middle of a side of the tic-tac-toe grid, which are only on two lines each. What about higher dimensions? (Let's ignore the fact that tic-tac-toe on a 3×3×... grid is an easy win in more than two dimensions.) Which are the weak cells, and which are the strong ones?<br /><br />For instance, the image below shows a four-dimensional tic-tac-toe grid, projected to the plane. Each cell of the grid becomes a colored dot; the five colors distinguish the cells by the dimension of the hypercube face that they're the midpoint of. We have vertices (blue), edges (yellow), squares (red), cubes (white), and the center of the hypercube itself (black). But I've only shown a subset of the lines, four through each point. If I included all the lines, which colors would be more powerful (on more lines), and which less?<br /><br /><div align="center"><img src="http://www.ics.uci.edu/~eppstein/0xDE/ttt4d.png"></div><br /><br />It is not hard to work out the formula for this, if we give each of these points coordinates that are <i>d</i>-tuples of the numbers in the set {−1,0,1}. If <i>k</i> of these coordinates are equal to 0, then the given point is the middle point of (3<sup><i>k</i></sup> − 1)/2 lines. These lines can be described by replacing each zero by one of three possibilities: 0, <i>x</i>, or −<i>x</i>, and then plugging in 1, 0, and −1 as the value of <i>x</i>. There are 3<sup><i>k</i></sup> choices for how to replace each zero, one of which doesn't give a line: if we leave all the zeros as they are, then all three choices of <i>x</i> give the same point. But for any other set of replacements, we get a line. Each line gets represented twice, because we can negate all the replaced coordinates and get the same line. That double-counting explains the division by two in the formula.<br /><br />These aren't the only lines, though. If there are <i>k</i> zeros in the coordinates of a point, there are <i>d</i> − <i>k</i> nonzeros. These, also, can be replaced, either keeping the same value in place, replacing a −1 with −<i>x</i>, or replacing a 1 by <i>x</i>. When we plug in 1, 0, and −1 as the value of <i>x</i>, we get a line on which the given point is the first point. This time there is no double-counting, but there are only two choices per coordinate, so we get 2<sup><i>d</i> − <i>k</i></sup> − 1 lines.<br /><br />The center point (the one that in this system has all-zero coordinates) is always the winner, the most powerful point. It has the maximum number of lines possible, because they cover all of the other points in pairs.<br /><br />To find the least powerful point, we need to minimize (3<sup><i>k</i></sup> − 1)/2 + 2<sup><i>d</i> − <i>k</i></sup> − 1. This can easily be done numerically, showing for instance that in our 4d picture the black point is on 40 lines, the white points are on 14, the red points are on 7, the yellow are on 8, and the blue points are on 15. The red points, the middle-dimensional ones, are the losers in this case.<br /><br />However, in higher dimensions, the weakest cells are not the middle-dimensional ones. To minimize the total number of lines, we want the two types of lines (the ones where the point is in the middle, and the ones where the point is first) to be roughly balanced in number. Ignoring the subtraction of one and division by two in the formulas (as these do not significantly affect the result when the dimension is high) this balanced is achieved for points whose number of zero coordinates is approximately (log<sub>6</sub>2)<i>d</i>, and whose number of nonzero coordinates is approximately (log<sub>6</sub>3)<i>d</i>. These, then, are the weakest points in high-dimensional tic-tac-toe boards.<br /><br />This also shows that, in high dimensions, no point is extremely weak. They are all on a number of lines that grows as a polynomial of the size of the whole board. If there are <i>n</i> cells on the whole board, then the number of lines through even the weakest cell is proportional to <i>n</i><sup>log<sub>6</sub>2</sup> ≈ <i>n</i><sup>0.387</sup>.<a name='cutid1-end'></a>http://11011110.livejournal.com/340796.htmlhypercubegame theorypublic0http://11011110.livejournal.com/340516.htmlWed, 18 Jan 2017 03:20:08 GMTCourse prerequisites are not DAGs
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When I cover directed acyclic graphs for my algorithms classes, I usually mention as an example (familiar to the students) a DAG with university courses as its vertices and with prerequisites as its edges.<br /><br />Today I learned that this is a lie, or maybe I should say more of a lie than I already thought it was. It turns out that, in <a href="http://catalogue.uci.edu/donaldbrenschoolofinformationandcomputersciences/#courseinventory">our course catalog</a>, prerequisites are not just lists of courses that all must be taken first. Instead, even if we ignore the grade thresholds in some of them, they are complicated and-or formulas describing the combinations of courses that are allowed to be used as preparation. Some of them are written in conjunctive normal form (ors of ands), some are in disjunctive normal form (ands of ors), and at least one (our information retrieval course) requires multiple levels of parentheses in its formula.<br /><br />I had thought that this was mostly because some courses that have different numbers really have the same content, and should be treated as equivalent for the purposes of the prerequisite ordering. In order-theoretic terms, that would mean that we have a quasi-order rather than a partial order. For instance, we went through a refactoring of our curriculum a few years back and our prerequisite listings still include the numbers for both our old sophomore data structures course and our new one. But that's just to avoid blocking the leftover students who took it before we changed; you wouldn't want to take both classes. Or, we have three different ways you can learn linear algebra: by doing lots of hand calculations (the way the engineers want), by learning about vector spaces as abstract coordinate-free things (the way the mathematicians want), or by interacting with MATLAB or the equivalent (the way the computer scientists, and especially the machine learning peopl want). But you would only take one of these, not all three. So my feeling was that, at least when you restricted the set of classes to the ones taken by any individual student, you would get a partial order.<br /><br />But even that is not true. Take a look early in the linked catalog page, at our course CS 113, computer game development. Its prerequisites are one of computer graphics, AI, software design, critical analysis of games, graph algorithms, or game design. Why that list? I don't know, it's not my course. But it's completely reasonable to take more than one of these courses, in various different orderings, and the fact that it's an "or" of these courses rather than an "and" means that we're not treating this list the way we would for topological ordering of DAGs.<br /><br />So, if not a DAG, what is the prerequisite structure? An <a href="https://en.wikipedia.org/wiki/Antimatroid">antimatroid</a>! Or almost. Once you've fulfilled the prerequisites for a course, you can take the course in any later term that it's offered — they don't go stale. More abstractly, an element, once available to be included in an ordering, remains available until it is used. And this is the defining property of an antimatroid. The "almost" is because there are also some constraints that you can't take both of some pairs of classes for credit, but this only applies if you look at the whole course catalog at once. If you look at what any individual student takes, I think it really is an antimatroid. Of course, I may not have examined the catalog closely enough to find the exceptions...http://11011110.livejournal.com/340516.htmlantimatroidspublic0http://11011110.livejournal.com/340317.htmlMon, 16 Jan 2017 07:51:12 GMTLinkage
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<ul><li><a href="https://chroniclevitae.com/news/1535-nobody-cares-about-your-edu-anymore">Why you need a non-university-run email account</a> (<a href="https://plus.google.com/100003628603413742554/posts/aSFx8L8wG7X">G+</a>)</li><br /><li><a href="http://skrekkogle.com/postpost.html">Let your post office make your folded art for you</a> (<a href="https://plus.google.com/100003628603413742554/posts/hEmBN5zSiwM">G+</a>)</li><br /><li><a href="https://www.quantamagazine.org/20170103-fractal-dynamics-from-3d-julia-sets">Convex shapes that are curved only at a fractal boundary</a> (<a href="https://plus.google.com/100003628603413742554/posts/1mcATNJphKr">G+</a>)</li><br /><li><a href="http://people.cs.uchicago.edu/~laci/update.html">Babai fixes a bug</a> (<a href="https://plus.google.com/100003628603413742554/posts/SQRwXHye5VL">G+</a>)</li><br /><li><a href="http://www.latimes.com/local/lanow/la-me-ln-harvey-mudd-tech-women-adv-snap-story.html">How Harvey Mudd is making CS majority-female</a> (<a href="https://plus.google.com/100003628603413742554/posts/dv7xPBSau2k">G+</a>)</li><br /><li><a href="https://blog.coralproject.net/the-real-name-fallacy/">Real name policies still no good against trolls</a> (<a href="https://plus.google.com/100003628603413742554/posts/UXb29H5wk1C">G+</a>)</li><br /><li><a href="https://mathwithbaddrawings.com/2017/01/04/1-2-trillion-ways-to-play-the-same-sudoku/">How to permute a Sudoku puzzle</a> (<a href="https://plus.google.com/100003628603413742554/posts/J1RDt4KokE7">G+</a>)</li><br /><li><span class="ljuser i-ljuser i-ljuser-type-Y " data-ljuser="brianhayes" lj:user="brianhayes" ><a href="http://brianhayes.livejournal.com/profile" target="_self" class="i-ljuser-profile" ><img class="i-ljuser-userhead" src="http://l-stat.livejournal.net/img/syndicated.svg?v=6283?v=145.3" /></a><a href="http://brianhayes.livejournal.com/" class="i-ljuser-username" target="_self" ><b>brianhayes</b></a></span>'s <a href="http://bit-player.org/2017/joint-mathematics-morsels">JMM morsels</a> including the statistics of stable populations, navigating traffic lights in a Manhattan grid, and a stochastic traffic-model cellular automaton (<a href="https://plus.google.com/100003628603413742554/posts/X6hVmtY516T">G+</a>)</li><br /><li><a href="http://jdh.hamkins.org/no-regular-polygons-in-the-integer-lattice/">The only integer-coordinate regular polygons are squares</a> (<a href="https://plus.google.com/100003628603413742554/posts/ZQtpGoUHrRC">G+</a>)</li><br /><li><a href="https://vimeo.com/198605915">Strobed and spun sculptures look like they're wiggling and fractaling</a> (<a href="https://plus.google.com/100003628603413742554/posts/a4W4Ug5Nxq4">G+</a>)</li><br /><li><a href="https://mobile.twitter.com/PaulGowder/status/819285619411140609">Iowa moves to ban tenure...</a> (<a href="https://plus.google.com/100003628603413742554/posts/WES5VNLJ15w">G+</a>)</li><br /><li><a href="http://www.chronicle.com/article/Missouri-Lawmaker-Who-Wants-to/238886">...and so does Missouri</a> (<a href="https://plus.google.com/100003628603413742554/posts/HmnvAHaZAL3">G+</a>)</li><br /><li><a href="https://en.wikipedia.org/wiki/Clique_problem">Clique problem</a>, now a Good Article on Wikipedia (<a href="https://plus.google.com/u/0/100003628603413742554/posts/1dNHMvtK8j3">G+</a>)</li><br /><li><a href="https://www.washingtonpost.com/news/energy-environment/wp/2017/01/11/on-the-eve-of-trump-obamas-energy-dept-leaves-behind-a-new-scientific-integrity-policy/">DOE preemptively bulwarks the academic freedom of its scientists</a> (<a href="https://plus.google.com/100003628603413742554/posts/84gMjaZ6JBi">G+</a>)</li><br /><li><a href="https://youtu.be/hTWIokjzeH0">Square tile in a seemingly-wrongly-sized envelope</a> (<a href="https://plus.google.com/100003628603413742554/posts/K1gSzAyHurQ">G+</a>)</li></ul>http://11011110.livejournal.com/340317.htmlfractalsfeminismcellular automatawikipediasudokuemailgraph algorithmsanonymitycliquesartacademiaorigamipoliticspublic0http://11011110.livejournal.com/340001.htmlSun, 01 Jan 2017 04:03:34 GMTLinkage for the end of an arbitrary temporal measurement unit
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<ul><li><a href="http://graphdrawing.de/contest2017/topics.html">This year's graph drawing contest</a> (<a href="https://plus.google.com/100003628603413742554/posts/VK2MLFTFZaJ">G+</a>)</li><br /><li><a href="https://www.ias.edu/ideas/2016/agol-hyperbolic-link-complements">Ian Agol reviews knot and link volumes</a> (<a href="https://plus.google.com/100003628603413742554/posts/hspKNYhMjTj">G+</a>)</li><br /><li><a href="https://www.youtube.com/watch?v=EtmeTKPLcpM">How to turn crossing surfaces into interwoven meshes</a> (<a href="https://plus.google.com/100003628603413742554/posts/JWZqhAokWFJ">G+</a>)</li><br /><li><a href="http://www.medievalists.net/2016/12/leonardo-rapunzel-mathematics-hair/">Leonardo, Rapunzel and the mathematics of hair</a> (<a href="https://plus.google.com/100003628603413742554/posts/XBkf1c3o1Xh">G+</a>)</li><br /><li><a href="http://regolo54.tumblr.com/post/154499795732/impossible">Impossible-tribar torus link</a> (<a href="https://plus.google.com/100003628603413742554/posts/CVqkLDhyhyb">G+</a>)</li><br /><li><a href="https://commons.wikimedia.org/wiki/File:World_of_Wikipedia_by_Jon_Robson.png">World of Wikipedia fantasy map by Jon Robson</a> (<a href="https://plus.google.com/100003628603413742554/posts/gDtgAZE9yZs">G+</a>)</li><br /><li><a href="http://blog.computationalcomplexity.org/2016/12/moneyball-for-academics.html">Moneyball for academics</a> (<a href="https://plus.google.com/100003628603413742554/posts/jdpivxFd1ba">G+</a>)</li><br /><li><a href="https://www.theguardian.com/science/alexs-adventures-in-numberland/2016/oct/03/meet-the-mathekniticians-and-their-amazing-woolly-maths-creations">Greco-Latin squares, integer divisibility, the binary number system, and more, represented in kniitting</a> (<a href="https://plus.google.com/100003628603413742554/posts/ZRWAEf6sWRm">G+</a>)</li><br /><li><a href="https://arxiv.org/abs/1612.06373">Free book on the singularities of real algebraic curves by Étienne Ghys</a> (<a href="https://plus.google.com/100003628603413742554/posts/FYUi1r6yShM">G+</a>)</li><br /><li><a href="https://plus.google.com/109724473260510367225/posts/NoRWJcvk8U3">Maarten Löffler's holiday puzzle, involving coloring arrangements of circles</a> (<a href="https://plus.google.com/100003628603413742554/posts/NF5F3wTs75N">G+</a>)</li><br /><li><a href="https://twitter.com/alanferrier/status/808764211648200706">Humorous image captions in Wikipedia: not allowed</a> (<a href="https://plus.google.com/100003628603413742554/posts/VFKF26WsBPV">G+</a>)</li><br /><li><a href="https://www.sfmoma.org/exhibition/tomas-saraceno/">Tomás Saraceno's cloud cities (polyhedral wireframe art) at SFMOMA</a> (<a href="https://plus.google.com/100003628603413742554/posts/YoSeTwzxDgf">G+</a>)</li><br /><li><a href="http://www.metafilter.com/164293/LiveJournal-represents-social-media-without-borders">Remnant LiveJournalers panic over server move to Russia</a> (<a href="https://plus.google.com/100003628603413742554/posts/ViXCWGzov65">G+</a>)</li><br /><li><a href="https://www.youtube.com/watch?v=6-4SCQpingg">Sidedness is a property of embeddings, not of surfaces</a> (<a href="https://plus.google.com/100003628603413742554/posts/gyFYqvCANVN">G+</a>)</li></ul>http://11011110.livejournal.com/340001.htmltopologygraph drawingwikipediaacademiaartpublic0http://11011110.livejournal.com/339847.htmlFri, 30 Dec 2016 05:54:36 GMTFort Bragg
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<a href="http://www.ics.uci.edu/~eppstein/pix/fortbragg/">A few street scenes from Fort Bragg</a>.<br /><br /><div align="center"><img src="http://www.ics.uci.edu/~eppstein/pix/fortbragg/1-m.jpg" border="2" style="border-color:black;" /></div>http://11011110.livejournal.com/339847.htmlmendocinophotographypublic0http://11011110.livejournal.com/339591.htmlWed, 28 Dec 2016 23:43:11 GMTArpeggio V, Bruce Beasley
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Street art in front of the Mitchell Park Library, Palo Alto.<br /><br /><div align="center"><img src="http://www.ics.uci.edu/~eppstein/pix/paloalto/BruceBeasleyArpeggioV-m.jpg" border="2" style="border-color:black;" /></div><br /><br /><a href="http://brucebeasley.com/">The artist's gallery has many additional interesting geometric sculptures</a>.<br /><br /><a href="http://www.ics.uci.edu/~eppstein/pix/paloalto/">A couple of other unrelated photos from Palo Alto</a>.http://11011110.livejournal.com/339591.htmlartphotographypublic0http://11011110.livejournal.com/339209.htmlFri, 16 Dec 2016 05:20:57 GMTLinkage
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<ul><li><a href="http://www.ics.uci.edu/~tebradle/drawings.html">A crash course in self-balancing trees</a>, cartoon by UCI grad student Tatiana Bradley (<a href="https://plus.google.com/100003628603413742554/posts/4ducsREJDtQ">G+</a>)</li><br /><li><a href="http://gallery.bridgesmathart.org/exhibitions/2015-joint-mathematics-meetings/smbelcas">Knitted graph theory and knot theory</a>, by sarah-marie belcastro (<a href="https://plus.google.com/100003628603413742554/posts/Fzf6dAo86Vd">G+</a>)</li><br /><li><a href="http://fortune.com/2016/12/03/tech-muslim-tracking/">IBM helped the Nazis track down the Jews, so why should the tech industry behave any better this time around?</a> (<a href="https://plus.google.com/100003628603413742554/posts/QuGiZxPqFtq">G+</a>)</li><br /><li><a href="http://www.spoon-tamago.com/2012/11/14/kohei-nawas-new-work-merges-3d-scanning-and-texture-mapping/">Sculpture that overlays texture maps and human forms</a>, by Kohei Nawa (<a href="https://plus.google.com/u/0/100003628603413742554/posts/DNzyMdnDxH9">G+</a>)</li><br /><li><a href="https://www.youtube.com/watch?v=qpTuWLF2TKA">Light-painted photos of dancers that look more like drawings</a>, by Echo Lew (<a href="https://plus.google.com/100003628603413742554/posts/C7zjBuEXEbX">G+</a>)</li><br /><li><a href="http://www.math.rutgers.edu/~zeilberg/Opinion104.html">Doron Zeilberger on the Balkanization of mathematics conferences</a> (<a href="https://plus.google.com/100003628603413742554/posts/C88iwxgUGPH">G+</a>)</li><br /><li><a href="http://tex.stackexchange.com/questions/18987/how-to-make-the-pdfs-produced-by-pdflatex-smaller">How to shrink PDF files</a>, useful for online rec letter sites with small file size limits (<a href="https://plus.google.com/100003628603413742554/posts/1de7E6Cuk9G">G+</a>)</li><br /><li><a href="http://stackoverflow.com/questions/41031106/which-general-purpose-sorting-algorithm-does-swift-use-it-does-not-perform-well/41070802#41070802">Apple's Swift sorting library falls down when its input is already sorted</a> (<a href="https://plus.google.com/100003628603413742554/posts/YUmcgTckevo">G+</a>)</li><br /><li><a href="http://www.thisiscolossal.com/2016/12/tessellated-origami-sculptures-by-goran-konjevod/">Origami with layered textures</a>, by Goran Konjevod (<a href="">G+</a>)</li><br /><li><a href="https://plus.google.com/117663015413546257905/posts/U83MSyf4yGA">Trump's moves to completely eliminate climate research and roll back progress on global warming</a> (<a href="https://plus.google.com/100003628603413742554/posts/9duThne7TJe">G+</a>)</li><br /><li><a href="http://smf.emath.fr/content/prix-salem-2016">Maryna Viazovska wins the Salem Prize for using modular forms to solve sphere packing</a> (<a href="https://plus.google.com/100003628603413742554/posts/5LDAWFBbrVd">G+</a>)</li><br /><li><a href="https://www.newscientist.com/article/2115570-third-ever-natural-quasicrystal-found-in-siberian-meteorite/">A new naturally-occurring quasicrystal</a> (<a href="https://plus.google.com/100003628603413742554/posts/cee8GFdNUVU">G+</a>)</li><br /><li><a href="http://www.thisiscolossal.com/2016/12/stephen-orlando-light-paintings/">Stephen Orlando follows the chronophotography tradition of Marey</a> (<a href="https://plus.google.com/100003628603413742554/posts/YCYh7Wa7fMq">G+</a>)</li><br /><li><a href="https://en.wikipedia.org/wiki/Pythagorean_tiling">Pythagorean tiling</a>, now a Wikipedia good article (<a href="https://plus.google.com/100003628603413742554/posts/gENzqvayJqF">G+</a>)</li></ul>http://11011110.livejournal.com/339209.htmlpolar bearstoolssortingcorporatizationtilingconferencesgraph theorydata structuresphotographyknot theoryartpoliticsorigamipublic0http://11011110.livejournal.com/339105.htmlThu, 08 Dec 2016 05:11:21 GMTMany points in small boxes
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My latest arXiv preprint is "Covering many points with a small-area box", <a href="https://arxiv.org/abs/1612.02149">arXiv:1612.02149</a>, with de Berg, Cabello, Cheong, and Knauer. It is about finding an axis-parallel rectangle that covers <i>k</i> out of <i>n</i> points in the plane and has as small area as possible. Here, both <i>k</i> and <i>n</i> are variables, but <i>k</i> is expected to be significantly smaller than <i>n</i>. So, in finding a fast algorithm, the goal is first to minimize the exponent of <i>n</i> in the time bound, and only after that to minimize the dependence on <i>k</i>. We achieve a bound of <i>O</i>(<i>nk</i><sup>2</sup>log <i>n</i>). There are also some related algorithms for approximating the dual problem where you are given a fixed area and want to maximize the number of points that are covered.<br /><br />The result has a bit of a curious history. A group of us worked on it, and came up with the best algorithms we could, thinking it was a new and unsolved problem. Then, we did a more careful literature search and found that it was actually a known problem, and worse than that, that the time bounds we had achieved were all known or improved, so that we had no result. But then we looked again and discovered that the supposedly known bounds aren't actually true.<br /><br />What happened was that there was a line of research (that I was part of) in the early 1990s on finding "small" subsets of a given number of points. A sequence of papers on this subtopic identified many different definitions of what it meant to be small (like having low circumradius, diameter, etc) that could all be solved by similar algorithms. These algorithms worked by covering the input by clusters of <i>O</i>(<i>k</i>) points, one of which could be guaranteed to include the optimal solution, and then worked separately within each cluster. A simple version of this is to choose <i>O</i>(<i>k</i>) nearest neighbors to each point and to use these <i>n</i> sets of near neighbors as the clusters; later refinements used fewer and easier-to-find clusters. The algorithms of this type varied by what they did inside the clusters, but this general method ensured that the dependence on <i>n</i> would be linear. And one of the problems solved using this approach was almost the same as the one in our new paper, but with axis-parallel rectangle perimeter in place of rectangle area.<br /><br />Later, another paper studied the problem of finding smallest rectangles covering <i>k</i> points, but with <i>k</i> assumed to be very close to <i>n</i> rather than much smaller than <i>n</i>, so that factors of <i>k</i> in the runtime are bad and factors of (<i>n</i> − <i>k</i>) are good. It solved both the minimum-perimeter and minimum-area problems for the large-<i>k</i> case. Of course, it quoted the past work on the minimum perimeter problem for small <i>k</i>, but it didn't make clear that this past work solved only the case of perimeter minimization and not area minimization. And so, other papers, quoting that one, started also saying that the minimum-area <i>k</i>-point rectangle problem (for the small-<i>k</i> case) had also already been solved.<br /><br />It hadn't, but now it has.http://11011110.livejournal.com/339105.htmlcomputational geometryrectanglespaperspublic0http://11011110.livejournal.com/338897.htmlThu, 01 Dec 2016 04:30:27 GMTLinkage
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<ul><li><a href="http://rin.io/im-not-good-at-math/">"I'm not good at math",</a> car conversation with a 10-year-old girl about what math really is (illustrated, but with dubiously-reliable image hosting; <a href="https://plus.google.com/100003628603413742554/posts/iwVqxLz8zb9">G+</a>)</li><br /><li><a href="https://www.youtube.com/watch?v=ixduANVe0gg">Video on sphere eversion (based on a double-Klein-bottle trick) with a paperfolding challenge at the end</a> (<a href="https://plus.google.com/100003628603413742554/posts/T2eVjGWc5zE">G+</a>)</li><br /><li><a href="http://www.spoon-tamago.com/2016/11/14/graphic-wire-sculpture-mitsuru-koga/">Minimalist wire sculptures of 3d shapes by Mitsuru Koga</a> (<a href="https://plus.google.com/100003628603413742554/posts/GByPKZ7iF8a">G+</a>)</li><br /><li><a href="https://www.youtube.com/attribution_link?a=NPulBOW7ppI&u=%2Fwatch%3Fv%3DciM6wigZK0w%26feature%3Dshare">Video on hypersphere packing, and the strange behavior of hyperspheres inside hypercubes</a> (<a href="https://plus.google.com/100003628603413742554/posts/C3JnVvDPAHz">G+</a>)</li><br /><li><a href="http://www.forbes.com/sites/quora/2016/11/22/a-lesson-from-particle-physics-on-why-things-around-us-dont-start-randomly-liquefying/#2de554604309">What it takes to get systems of solid particles to act like a liquid: polishing them helps, but isn't enough by itself.</a> (<a href="https://plus.google.com/100003628603413742554/posts/cmrcL7pAXDH">G+</a>)</li><br /><li><a href="https://www.newscientist.com/article/2113283-crowdsourced-prime-number-could-help-solve-a-50-year-old-problem/">One of the six remaining candidates for the smallest odd k such that no number of the form k 2^n + 1 is prime has been knocked out</a> (<a href="https://plus.google.com/100003628603413742554/posts/HQqntgB9bCw">G+</a>)</li><br /><li><a href="https://plus.google.com/102991298250177867939/posts/RP4ZuNB9PyP">Mathematician Michel Deza has died</a> (<a href="https://plus.google.com/100003628603413742554/posts/SDK1QpLAHEH">G+</a>)</li><br /><li><a href="https://blog.wikimedia.org/2016/11/25/wiki-loves-earth-2016-winners/">The top fifteen winning photos from Wiki Loves Earth</a> (<a href="https://plus.google.com/100003628603413742554/posts/F2qwY955PWh">G+</a>)</li><br /><li><a href="http://aperiodical.com/2013/10/an-enneahedron-for-herschel/">The Herschel enneahedron</a>, a nice symmetric geometric realization of the simplest non-Hamiltonian polyhedron (<a href="https://plus.google.com/100003628603413742554/posts/C7PQvUReGrS">G+</a>)</li><br /><li><a href="https://www.youtube.com/watch?v=lUONqAJu8EQ">Antiparallelogram-based linkage traces out a three-lobed curve</a> (more linkage curves linked from <a href="https://plus.google.com/100003628603413742554/posts/5tDmEkjSAdL">G+</a>)</li><br /><li><a href="https://gowers.wordpress.com/2016/11/29/time-for-elsexit/">Secret deal for funneling UK university money to Elsevier</a> (<a href="https://plus.google.com/100003628603413742554/posts/BCfZzF4USSu">G+</a>)</li><br /><li><a href="https://arxiv.org/abs/1611.08740">On the number of ordinary lines determined by sets in complex space</a> (<a href="https://plus.google.com/100003628603413742554/posts/9PvAvuwbE11">G+</a>)</li></ul>http://11011110.livejournal.com/338897.htmllinkagescorporatizationnumber theorymathematicsgeometryartphotographylandscapepublic1http://11011110.livejournal.com/338464.htmlThu, 24 Nov 2016 21:59:33 GMTHappy Thanksgiving
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Happy Thanksgiving to all. Here's a photo from a walk I took this morning at the San Joaquin Wildlife Sanctuary.<br /><br /><div align="center"><img src="http://www.ics.uci.edu/~eppstein/pix/sjws7/GreenStream-m.jpg" border="2" style="border-color:black;" /></div><br /><br /><a href="http://www.ics.uci.edu/~eppstein/pix/sjws7/">The gallery has a few more</a>.http://11011110.livejournal.com/338464.htmlphotographylandscapepublic3http://11011110.livejournal.com/338196.htmlSun, 20 Nov 2016 23:32:06 GMTEgyptian fractions with practical denominators
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While looking something else up on <a href="https://oeis.org/">OEIS</a> I ran across <a href="http://maths.nju.edu.cn/~zwsun/UnitFraction.pdf">a conjecture by Zhi-Wei Sun from September 2015</a> that every positive rational number has an Egyptian fraction representation in which every denominator is a <a href="https://oeis.org/A005153">practical number</a>. The conjecture turns out to be true; here's a proof.<br /><br />First, some background. An <a href="https://en.wikipedia.org/wiki/Egyptian_fraction">Egyptian fraction</a> is a representation of a given number as a sum of distinct unit fractions. And a <a href="https://en.wikipedia.org/wiki/Practical_number">practical number</a> is a number <i>n</i> such that all other numbers up to <i>n</i> can be represented by sums of distinct divisors of <i>n</i>. These two concepts were already connected: if <i>n</i> is practical then all fractions <i>m</i>/<i>n</i> have Egyptian fractions formed by representing <i>m</i> as a sum of divisors of <i>n</i> and then dividing each term of the sum by <i>n</i>. If <i>n</i> is not itself practical you can apply the same trick to <i>mp</i>/<i>np</i> where <i>p</i> is a large enough practical number (large enough to make <i>np</i> also be practical). But these methods don't control the divisors of the unit fractions that they produce, so they don't answer Sun's question.<br /><br />Instead, to prove Sun's conjecture, let's first restrict our attention to rationals <i>m</i>/<i>n</i> with <i>m</i> < <i>n</i> (we'll handle the rest later), and turn to a different method for generating Egyptian fractions, the <a href="http://www.ics.uci.edu/~eppstein/numth/egypt/binary.html">binary remainder method</a>. To make a more concrete example, let's use <i>m</i>/<i>n</i> = 117/129. The binary remainder method can be thought of as greedily removing power-of-two fractions:<br /><br />117/129<br />= 1/2 + 105/(2*129)<br />= 1/2 + 1/4 + 81/(4*17)<br />= 1/2 + 1/4 + 1/8 + 33/(8*17)<br />= 1/2 + 1/4 + 1/8 + 66/(16*17)<br />= 1/2 + 1/4 + 1/8 + 1/32 + 3/(32*17)<br />= 1/2 + 1/4 + 1/8 + 1/32 + (2+1)/(32*17)<br />= 1/2 + 1/4 + 1/8 + 1/32 + 1/2064 + 1/4128<br /><br />Each step in this sequence doubles the final denominator, and replaces the final numerator by its double mod <i>n</i>. For instance, in the third line above, 81 = 2*105 (mod 129). We include another power of two in the expansion whenever doubling the previous numerator causes it to exceed <i>n</i>, causing a nontrivial reduction mod <i>n</i>. So the numerators stay below <i>n</i>. Eventually we reach a point where the binary expansion of the numerator (its representation as a sum of powers of two, here 3=2+1) uses powers of two smaller than the ones in the denominator, and we can perform this expansion and stop.<br /><br />Unfortunately, this doesn't quite work for Sun's problem. The power-of-two fractions are practical, but the ones formed at the end from the binary expansion of the numerator might not be (although in this example they are). But there's an easy fix: continue greedily taking off powers of two until reaching a denominator of the form 2<sup><i>i</i></sup><i>n</i> where 2<sup><i>i</i></sup> > <i>n</i><sup>2</sup>/2. Then, when we do the binary expansion, each denominator of one of the resulting unit fractions will itself be of the form 2<sup><i>j</i></sup><i>n</i> where <i>j</i> is still large, large enough that 2<sup><i>j</i></sup> > <i>n</i>/2. This is a large enough power of two to make the denominator 2<sup><i>j</i></sup><i>n</i> be practical.<br /><br />Let's try this on another simple example:<br /><br />14/17<br />= 1/2 + 11/(2*17)<br />= 1/2 + 1/4 + 5/(4*17)<br />= 1/2 + 1/4 + 10/(8*17)<br />= 1/2 + 1/4 + 1/16 + 3/(16*17)<br />= 1/2 + 1/4 + 1/16 + 6/(32*17)<br />= 1/2 + 1/4 + 1/16 + 12/(64*17)<br />= 1/2 + 1/4 + 1/16 + 1/128 + 7/(128*17)<br />= 1/2 + 1/4 + 1/16 + 1/128 + 14/(256*17)<br />= 1/2 + 1/4 + 1/16 + 1/128 + (8+4+2)/(256*17)<br />= 1/2 + 1/4 + 1/16 + 1/128 + 1/544 + 1/1088 + 1/2176<br /><br />We didn't stop at 5/(4*17) and expand it into (4+1)/(4*17) = 1/17 + 1/68 because those denominators are impractical. Instead we waited until reaching a big enough power of two (256 > 17<sup>2</sup>/2) to guarantee that the binary expansion would produce practical denominators.<br /><br />That method finds practical expansions for all rational numbers that are less than one. What about larger rationals? For those, we need to know that the practical numbers are distributed roughly like the prime numbers, close enough to the same distribution that the sum of inverse practical numbers diverges. So we can use a greedy algorithm that repeatedly subtracts the largest unused inverse-practical number, and this will eventually leave the remaining fraction smaller than all already-used practical numbers. At this point we can switch to the binary method. Let's try this for the rational number 19/9:<br /><br />19/9<br />= 1/1 + 10/9<br />= 1/1 + 1/2 + 11/18<br />= 1/1 + 1/2 + 1/4 + 13/36<br />= 1/1 + 1/2 + 1/4 + 1/6 + 7/36<br />= 1/1 + 1/2 + 1/4 + 1/6 + 1/8 + 5/72<br /><br />and now since 5/72 < 1/8, it's safe to switch to the binary method without worrying that it will produce any of the same fractions:<br /><br />5/72<br />= 5/(8*9)<br />= 1/16 + 1/(16*9)<br />= 1/16 + 2/(32*9)<br />= 1/16 + 4/(64*9)<br />= 1/16 + 1/144<br /><br />where, in the second-to-last line, the power of two (64) is big enough to match the half-square of the odd part (9<sup>2</sup>/2 = 40.5). Therefore,<br /><br />19/9 = 1/1 + 1/2 + 1/4 + 1/6 + 1/8 + 1/16 + 1/144<br /><br />Sun also made a similar conjecture with the numbers formed by subtracting one from a prime in place of the practical numbers. Perhaps similar methods will work there too, but this seems harder to prove. Sun offered $1000 for a proof, so it might be worthwhile to try.<a name='cutid1-end'></a>http://11011110.livejournal.com/338196.htmlegyptian fractionsnumber theorypublic0http://11011110.livejournal.com/337934.htmlWed, 16 Nov 2016 21:33:06 GMTA better-behaved subtraction game
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I used the <a href="http://11011110.livejournal.com/337587.html">sieving algorithm for subtract-a-square</a> to generate its P-positions (the ones you want to move to) in order to estimate how quickly this sequence of positions grows. Based on some assumptions that the sequence was basically random and on considerations involving the coupon collector problem, I was expecting its growth rate (the number of P-positions up to <i>n</i>) to be roughly <i>O</i>(sqrt <i>n</i> log <i>n</i>). But that seems to be inconsistent with the data. Instead (based on generating the first 350k or so of these numbers) it seems to be more like <i>O</i>(<i>n</i><sup>0.69</sup>), a surprisingly high growth rate.<br /><br />Below is a table of some P-positions (the middle column), the number of P-positions up to that value (left column), and the exponent one would infer from those two numbers:<pre>22 109 0.658881526366
39 204 0.688882847768
66 425 0.692265864743
99 827 0.684021054309
158 1625 0.684757862413
255 3209 0.686333836589
419 6414 0.688764172045
682 12812 0.689885260226
1087 25647 0.688637867852
1752 51202 0.688752703605
2821 102517 0.688593820775
4497 204817 0.687755821596
7436 409687 0.689776813171
12068 819267 0.69023196924
19659 1638632 0.690896170477
31742 3276892 0.690915578787
51489 6553775 0.691222814299
83851 13107272 0.691745659235
136858 26215012 0.692354994835
221074 52429275 0.69233584282
358248 104857942 0.692489695744</pre>In any subtraction game like this one, the growth rates of the move set and the P-position set must multiply together to at least linear. But it occurred to me to wonder: maybe they are always superlinear? That is, maybe there is some more general reason why the winning positions in subtract-a-square seem to be significantly more dense than the squares? But the answer is no. There's a different subtraction game where both number of moves up to <i>n</i> and the number of winning positions up to <i>n</i> are proportional to the square root, with some oscillation in the constant of proportionality.<br /><br />To define such a game, let the available moves be the <a href="http://oeis.org/A000695">Moser–de Bruijn sequence</a> of sums of distinct powers of 4: 0, 1, 4, 5, 16, 17, 20, 21, 64, ... Any integer <i>n</i> can be decomposed into its even part <i>n</i> & 0x55555555, a member of its sequence, and its odd part <i>n</i> & 0xaaaaaaaa, which is two times a member of the sequence.<br />The P-positions for the game defined by these moves are the ones whose even part is zero, and the unique winning move (when the even part is nonzero) is to subtract off the even part.<br /><br />Another way of looking at this, in number theoretic terms, is that the Moser–de Bruijn sequence is something like a <a href="https://en.wikipedia.org/wiki/Sidon_sequence">Sidon sequence</a>. A Sidon sequence is defined by the property that the sums <i>x</i> + <i>y</i> of pairs of numbers <i>x</i>, <i>y</i> from the sequence are all distinct. To achieve this property, the number of elements in an infinite Sidon sequence up to <i>n</i> must (for infinitely many <i>n</i>) grow strictly smaller than the square root of <i>n</i>. Instead, the Moser–de Bruijn sequence has distinct sums of the form <i>x</i> + 2<i>y</i>, and because this form is a little different it can grow proportionally to the square root.<br /><br />So, any explanation of the high growth rate of P-positions in subtract-a-square will need to depend on the actual distribution of square numbers, and not just on their frequency.<br /><br /><b>Update Nov. 18:</b> Compare <a href="http://www.ics.uci.edu/~eppstein/cgt/subsquare.html">these old messages</a> on modular behavior of the subtract-a-square positions with Ruzsa's lower bound technique in the <a href="https://en.wikipedia.org/wiki/Furstenberg%E2%80%93S%C3%A1rk%C3%B6zy_theorem">Furstenberg–Sárközy theorem</a>. Then look at the table below, of what happens when we represent subtract-a-square positions in base 5 and count digit frequencies for each digit position of these numbers (here, for positions up to 4 million):<pre>digit 0 : 15179 8 18520 36 2803
digit 1 : 7369 7280 7277 7351 7268
digit 2 : 11273 3903 10586 6161 4614
digit 3 : 7230 7303 7360 7470 7160
digit 4 : 7863 6726 7524 7453 6894
digit 5 : 7094 7278 7346 7280 7302</pre> It looks very much to me like the greedy algorithm is trying to emulate Ruzsa's strategy in base 5 (that is, to use a square-difference-free set of digits in even positions and arbitrary digits in odd positions), which would already give it approximately <i>n</i><sup>0.715338</sup> positions. If it could bring in other prime factors to the base it could do even better. Maybe it emulates Ruzsa for all bases simultaneously?<a name='cutid1-end'></a>http://11011110.livejournal.com/337934.htmlgame theorynumber theorypublic0http://11011110.livejournal.com/337735.htmlWed, 16 Nov 2016 06:54:03 GMTLinkage
http://11011110.livejournal.com/337735.html
Gah LiveJournal's new post page is broken and I can't preview. Let's hope this works.<br /><ul><li><a href="https://www.miraikan.jst.go.jp/en/exhibition/tsunagari/authagraph.html">Tetrahedral area-preserving world map projection</a> (<a href="https://plus.google.com/100003628603413742554/posts/jcr8Wj7hUmq">G+</a>)</li><br /><li><a href="http://www.kenyonreview.org/kr-online-issue/2016-fall/selections/jamie-zvirzdin-656342/">Tips for popular-audience science writing</a> (<a href="https://plus.google.com/100003628603413742554/posts/VYLrKVTUrpB">G+</a>)</li><br /><li><a href="https://www.propublica.org/article/google-has-quietly-dropped-ban-on-personally-identifiable-web-tracking">Google opts in to matching personal info to web trackers. Here's how to opt out.</a> (<a href="https://plus.google.com/100003628603413742554/posts/iTgiTPLcoeU">G+</a>)</li><br /><li><a href="https://en.wikipedia.org/wiki/Paracompact_uniform_honeycombs">New spherical renders of hyperbolic honeycombs</a> (<a href="https://plus.google.com/100003628603413742554/posts/YKwfy2hw13r">G+</a>)</li><br /><li><a href="http://www.theatlantic.com/science/archive/2016/11/math-women/506417/">How a corrosive culture keeps women out of leadership positions on math journals</a> (<a href="https://plus.google.com/100003628603413742554/posts/fwidsbqU5A8">G+</a>)</li><br /><li><a href="http://www.ams.org/profession/ams-fellows/new-fellows">New AMS Fellows</a> (<a href="https://plus.google.com/100003628603413742554/posts/jJhZHVtnLcz">G+</a>)</li><br /><li><a href="https://arxiv.org/abs/1611.01661">Spanning trees in multipartite geometric graphs</a> (<a href="https://plus.google.com/100003628603413742554/posts/ajK9WDVQRvG">G+</a>)</li><br /><li><a href="https://www.youtube.com/watch?v=ikAb-NYkseI">Inspirational speech to turn your fear and loathing into constructive energy</a> (<a href="https://plus.google.com/100003628603413742554/posts/1QvM1JSr6gJ">G+</a>)</li><br /><li><a href="https://medium.com/message/never-trust-a-corporation-to-do-a-librarys-job-f58db4673351">The failure of Google's mission to preserve the information from the past, and the alternative provided by archive.org</a> (<a href="https://plus.google.com/100003628603413742554/posts/ChWApSXsKCA">G+</a>)</li><br /><li><a href="http://blog.plover.com/2016/11/11/">Bad citation practices on Wikipedia</a> (<a href="https://plus.google.com/100003628603413742554/posts/brQc4rNfcWz">G+</a>)</li><br /><li><a href="http://www.konradvoelkel.com/2015/05/git-for-math">How and why mathematicians should use git for their papers</a> (<a href="https://plus.google.com/100003628603413742554/posts/SxTGFbHpnuC">G+</a>)</li><br /><li><a href="http://fractaldonana.blogspot.co.uk/2009/05/island-of-enmediowetlands-of-el-odiel.html">Spanish photographer Héctor Garrido's fractal landscapes</a> (<a href="https://plus.google.com/100003628603413742554/posts/gKXKHjiMSHr">G+</a>)</li><br /><li><a href="https://www.youtube.com/watch?v=49KvZrioFB0">Cute puzzle about subdividing a square into different but nearly-equal-area rectangles</a> (<a href="https://plus.google.com/100003628603413742554/posts/Dyek8ju5vsZ">G+</a>)</li></ul>http://11011110.livejournal.com/337735.htmlfeminismtoolsrectangleswikipediasecurityphotographylandscapegeometrypaperspublic0http://11011110.livejournal.com/337587.htmlTue, 15 Nov 2016 06:24:44 GMTShaving a superpolylog from subtract-a-square
http://11011110.livejournal.com/337587.html
<p>The game of <a href="https://en.wikipedia.org/wiki/Subtract_a_square">subtract-a-square</a> is played by two players with a pile of coins. They take turns taking a square number of coins until none are left. The last player to move wins. So how can we play this game optimally and efficiently? As it turns out, there's some deep number theory lurking in the answer.</p>
<p>I covered this game in my undergraduate algorithms class today, as an example of dynamic programming. It's perhaps unusual among dynamic programming algorithms as having a fractional exponent. If you define the value of a position to be +1 if the player about to move wins, and –1 otherwise, then you get the usual dynamic programming recurrence for any game: the value of any position is the maximum, over available moves, of the negated values of the positions resulting from each move. A position with <i>n</i> coins has <i>O</i>(sqrt <i>n</i>) moves available, so evaluating each position takes <i>O</i>(sqrt <i>n</i>) time. Looping over all positions and evaluating each of them in this way takes a total amount of time that is <i>O</i>(<i>n</i><sup>3/2</sup>).</p>
<p>In Python, <a href="https://oeis.org/A030193">the set of winning positions</a> (the ones you want to move to, with value –1) can be listed by the following code, which essentially follows this algorithm:</p>
<pre>def A030193():
winners = set()
def win(n):
i = 1
while i*i <= n:
if n - i*i in winners:
return False
i += 1
return True
n = 0
while True:
if win(n):
yield n
winners.add(n)
n += 1</pre>
<p>But then after the lecture I realized that there's a different way of doing this, closer in spirit to the sieve of Eratosthenes. Rather than looking backward from each position to all the numbers that are smaller than it by a square difference, let's look forward from each winning position we find to the ones that are larger by a square. After we knock all of those positions out, the remaining ones are the winners.</p>
<pre>def A030193():
walkers = {}
steps = {}
n = 0
while True:
if n not in walkers:
yield n
walkers[n] = [n]
steps[n] = 0
for w in walkers[n]:
steps[w] += 1
next = w+(steps[w]*steps[w])
if next not in walkers:
walkers[next] = [w]
else:
walkers[next].append(w)
n += 1</pre>
<p>What's the running time of this sieving algorithm? I don't know, but it seems to depend on some deep questions in number theory!</p>
<p>It's easy to see that, when generating the winning positions up to <i>n</i>, it takes time <i>O</i>(sqrt <i>n</i>) per winning position. This is an improvement on the dynamic program, which multiplied the same factor by <i>n</i> rather than by the number of winning positions. But how many winning positions are there?</p>
<p>I don't know of any actual analysis of this (beyond an observation of Golomb that shows that there must be at least sqrt <i>n</i> of them), but there has been a lot of work in number theory on square-difference-free sets: sets of numbers no two of which differ by a square. And that's true here, so we can plug in the upper bounds that are known for the more general square-difference-free problem to analyze our algorithm.</p>
<p>Unfortunately the best of these bounds that I know of, by Pintz, Steiger, and Szemerédi (see the Wikipedia link for the precise bounds and reference) are pretty weak and also tricky to prove. They show that the number of elements in any square-difference-free set is sublinear, but only by a slightly-superpolylogarithmic factor. So the sieving algorithm is faster than the dynamic programming algorithm by at least the same factor.</p>
<p>Maybe the fact that we care about a single specific square-difference-free set (the winning positions in subtract-a-square) can save us, and allow for a simpler and stronger analysis than what we have for general square-difference-free sets? Again, I don't know. The sieving idea can also be combined with FFT-based bit-vector convolution to give a provably near-linear algorithm for subtract-a-square but one that's much more complicated and less implementable. I'm more interested in learning more about the complexity of the simple sieving method implemented above.</p><a name='cutid1-end'></a>
Update: See <a href="http://11011110.livejournal.com/337934.html">a later post</a> for some experiments on the growth rate of this set.http://11011110.livejournal.com/337587.htmlpythongame theorynumber theorypublic0http://11011110.livejournal.com/337178.htmlWed, 09 Nov 2016 05:24:18 GMTThings to say goodbye to
http://11011110.livejournal.com/337178.html
Most or all of the following seem destined to be gone, in the US and (for some) the world:<br /><br />Climate change countermeasures and continued research.<br />A coordinated response to extreme climate events.<br />Regulation of carbon dioxide as a pollutant.<br />National parks and wildlife preserves.<br />Guaranteed health care.<br />State-level limits on how predatory the health care industry can be.<br />A recovering economy and continuing tech boom.<br />Internet neutrality.<br />The minimum wage.<br />Social security and medicare.<br />The Supreme Court.<br />Abortion rights and access to contraception.<br />Anti-discrimination laws and same-sex marriage.<br />The truth defense from libel.<br />No wholesale legal persecution of opposition parties.<br />Restrictions on gun ownership.<br />Religious freedom for anyone other than Christians.<br />Limits on how blatantly Christianity can be used as an excuse for bigotry.<br />Redistricting rules that prevent continued gerrymandering.<br />Anything resembling adequate polling places in poor and minority districts.<br />Universal voter registration.<br />Pressure to reduce sexual harassment and sexual assaults in our universities, or anywhere else.<br />Reduced or free college tuition for anyone.<br />Any hope of reining in the out-of-control racists in our police forces.<br />Farm workers, housecleaners, construction workers, food service workers, and all those other people who do the things nobody else wants to do.<br />The ability to bring in talented students and workers from other countries.<br />NATO and the free Baltic states.<br />Nuclear non-proliferation.<br />A president who wouldn't start a nuclear war based on a whim or personal slight.<br /><br />Clinton could still pull this out, mathematically, but barring a miraculous turnaround in Wisconsin or Arizona there doesn't seem to be any remaining path to 270 electoral votes. And with the House and Senate in Republican hands and a free slot on the court there are no checks and balances left; the voters were the last line of defense.<br /><br />Anyone in a nice safe part of the world looking for academic computer scientists? Canada? Iceland? New Zealand?http://11011110.livejournal.com/337178.htmlpoliticspublic3http://11011110.livejournal.com/336922.htmlTue, 01 Nov 2016 05:54:35 GMTLinkage for Halloween
http://11011110.livejournal.com/336922.html
<ul><li><a href="http://jjj.de/fxt/fxtpage.html">Jörg Arndt's library and book of bit-manipulation programming techniques</a> (<a href="https://plus.google.com/100003628603413742554/posts/LiFy3yVuhUY">G+</a>)</li><br /><li><a href="http://mathoverflow.net/q/240186/440">The geometric median of a solid triangle</a>, another triangle center whose lack of closed form is keeping it out of <a href="http://faculty.evansville.edu/ck6/encyclopedia/ETC.html">ETC</a> (<a href="https://plus.google.com/100003628603413742554/posts/M4ahu8Nbrn3">G+</a>)</li><br /><li><a href="http://www.thisiscolossal.com/2016/10/polygons-a-versatile-measuring-spoon-inspired-by-origami-folds/">Rigid-origami variable-capacity measuring spoon</a> (<a href="https://plus.google.com/100003628603413742554/posts/BVTwSjt8pMR">G+</a>)</li><br /><li><a href="http://www.math.uwaterloo.ca/tsp/pubs/">"Easily the largest road-distance TSP that has been solved to date", a pub-crawl with 24727 stops</a> (<a href="https://plus.google.com/100003628603413742554/posts/YZohQx3N7Lz">G+</a>)</li><br /><li><a href="https://github.com/MGunlogson/CuckooFilter4J">High performance Java implementation of a Cuckoo filter</a> (<a href="https://plus.google.com/100003628603413742554/posts/NeSGLpg1MkX">G+</a>)</li><br /><li><a href="https://www.theguardian.com/science/2016/oct/22/nonsense-paper-written-by-ios-autocomplete-accepted-for-conference#comment-85925997">OMICS conference accepts a paper written by autocomplete</a> (<a href="https://plus.google.com/100003628603413742554/posts/P9uBSYymhh3">G+</a>)</li><br /><li><a href="https://www.youtube.com/watch?v=DelH1S32dOg">Matt Parker makes mathematical sculpture from office supplies</a> (<a href="https://plus.google.com/100003628603413742554/posts/YLGNzQCz2iG">G+</a>)</li><br /><li><a href="https://en.wikipedia.org/wiki/Rule_90">The Rule 90 cellular automaton on Wikipedia</a> (<a href="https://plus.google.com/100003628603413742554/posts/gZPtrzWY9By">G+</a>)</li><br /><li><a href="http://www.cs.technion.ac.il/~gershon/personal/EscherForReal/">3d objects that look like Escher's impossible objects</a> (<a href="https://plus.google.com/100003628603413742554/posts/FZPdf91mGos">G+</a>)</li><br /><li><a href="https://en.wikipedia.org/wiki/Dual_graph">Dual graph on Wikipedia</a> (<a href="https://plus.google.com/100003628603413742554/posts/c5dPKWgvJD1">G+</a>)</li><br /><li><a href="http://finance.yahoo.com/news/turnitin-announces-content-partnership-world-174100638.html">ArXiv uses Turnitin to detect plagiarized submissions; Turnitin gains access to arXiv content to detect plagiarism elsewhere</a> (<a href="https://plus.google.com/100003628603413742554/posts/2rpDueDNXzn">G+</a>)</li><br /><li><a href="http://boingboing.net/2016/10/25/beautifully-color-coded-waters.html">Using the Strahler number to visualize watersheds</a> (<a href="https://plus.google.com/100003628603413742554/posts/99zYcFGaLzs">G+</a>)</li><br /><li><a href="https://rjlipton.wordpress.com/2016/10/29/absolute-firsts/?utm_source=twitterfeed&utm_medium=twitter">The many women in computing who have been the the first X, no qualifier</a> (<a href="https://plus.google.com/100003628603413742554/posts/XgXnsqGvGtt">G+</a>)</li><br /><li><a href="http://www.popularmechanics.com/science/g2816/5-simple-math-problems/">Five easy-to-state but hard-to-solve problems in mathematics</a> (<a href="https://plus.google.com/100003628603413742554/posts/EUoWNBK5bY7">G+</a>)</li><br /><li><a href="https://plus.google.com/+MichaelNelson/posts/H9DNomHQuvq">Symmetries of the permutohedron</a> (<a href="https://plus.google.com/100003628603413742554/posts/WQ1gvuCb4qh">G+</a>)</li></ul>http://11011110.livejournal.com/336922.htmlfeminismunsolvedcellular automatabit parallelismwikipediaconferencesgraph theorydata structuresartgeometrypermutohedronexponential algorithmsorigamiplagiarismpublic0http://11011110.livejournal.com/336685.htmlMon, 24 Oct 2016 00:51:22 GMTTridecenary frustum tree
http://11011110.livejournal.com/336685.html
<p>In <a href="https://en.wikipedia.org/wiki/Conway%27s_Game_of_Life">Conway's Game of Life</a>, and <a href="https://en.wikipedia.org/wiki/Life-like_cellular_automaton">other cellular automata with the same neighborhood structure</a>, you can determine the values of the cells in any 2<i>x</i> × 2<i>x</i> square of cells from their values <i>x</i> steps earlier in a bigger 4<i>x</i> × 4<i>x</i> square. If we view this as a time-space diagram, with two of the dimensions as the spatial dimensions of the Life grid and the third dimension as time, we can view the bigger starting square and the smaller resulting square as the bottom and top of a sort of stepped pyramid formed by stacking concentric squares. The values of the cells in each layer of the pyramid are determined by the ones in the layer below. The squares shrink inward as they rise upward, because the other cells outside the pyramid need more information (outside the base of the pyramid) to determine their values. The image below shows this construction for <i>x</i> = 1, 2, and 4, with each time-space state of a cell represented as a cube.</p>
<p align="center"><img src="http://www.ics.uci.edu/~eppstein/0xDE/Stepped-Pyramids.svg" style="background-color:white;"></p>
<p>The shape that these stepped pyramids approximate, for large values of <i>x</i>, is a square pyramid with its top point chopped off. It's called a <a href="https://en.wikipedia.org/wiki/Frustum">frustum</a> (plural frusta).</p>
<p>The <a href="https://en.wikipedia.org/wiki/Hashlife">Hashlife</a> algorithm for simulating Conway's Game of Life is often described as being based on a quadtree, a two-dimensional structure of squares divided into smaller squares, but really it's a three-dimensional recursive decomposition based on these shapes. The hash table that gives the algorithm its name stores a collection of these frusta of different sizes (all powers of two), with their initial state (the values of the cells in the bottom face) as their hash keys and with their final state (the values in the top face) as the associated hash values. With this structure you can quickly jump from a square of cells whose values form one of the keys to the resulting state some number of steps later, without having to simulate all the steps in between.</p>
<p>What about when you encounter an initial state that you don't recognize, because it's not already in the hash table? Then you need to make a new frustum and store it in the hash table. To do so, divide the top face of the new frustum into four smaller squares, and divide the bottom face into sixteen smaller squares (all smaller by a factor of two than the top square of the new frustum).
Place four overlapping small frusta under the four small squares on the top face, connecting them to nine small squares in the middle plane of the new frustum, and nine more small frusta connecting these nine squares to the base.
Then the top face values for the new frustum can be found by looking up the values of the nine smaller frusta on the bottom, and then using the results to look up the values for the four smaller frusta on top. In this way, we have decomposed one big frustum into 13 smaller frusta, which can in turn be decomposed recursively in the same way.</p>
<p align="center"><img src="http://www.ics.uci.edu/~eppstein/0xDE/13-Frusta.svg" style="background-color:white;"></p>
<p>In fact, you don't need to know the actual values of any of the cells, except in the smallest frusta (the ones with 16 inputs and 4 outputs) to make this all work. At any higher level, you can represent each square of inputs or outputs symbolically, by a pointer to the frusta that have each of its four quarters as their bottom faces. In this way, each different frustum in the hash table can be represented by only a constant number of pointers, without also needing to store any big grids of cells.</p>
<p>The conventional wisdom, I think, is that Hashlife is good only for automata whose patterns begin in a sparse state and then stay that way (with lots of repeated substructures that the hash table can take advantage of). But actually, even for an automaton whose evolution is chaotic and structureless, Hashlife can provide a speedup. If you stop building new frusta when they become too big (where too big means that the surface area is logarithmic in the size of the grid you're simulating) then the total number of distinct frusta in the hash table is linear, no bigger than the grid itself. This method lets you simulate an <i>n</i> × <i>n</i> grid by decomposing it into only <i>O</i>(<i>n</i><sup>2</sup>/log <i>n</i>) overlapping frusta, whose height (number of time steps jumped) is proportional to the square root of the logarithm. Therefore, the algorithm time per cellular automaton time step is <i>O</i>(<i>n</i><sup>2</sup>/log<sup>3/2</sup> <i>n</i>), which compares favorably to the <i>O</i>(<i>n</i><sup>2</sup>/<i>w</i>) time per step of a conventional bit-parallel simulation on a machine with <i>w</i> bits per word. Of course, the random memory access pattern of the hash table could eat up most of the savings, making the comparison less clear in practice than it is in theory...</p>
<p>This 13-way decomposition of frusta into overlapping smaller frusta seems like it could also be useful in some scientific simulation problems, when there is a fixed speed at which information from one part of the simulation can propagate to another. But I don't know of any actual uses of this structure outside of cellular automaton simulation.</p><a name='cutid1-end'></a>http://11011110.livejournal.com/336685.htmlcellular automatageometrypublic2http://11011110.livejournal.com/336554.htmlSun, 16 Oct 2016 05:18:50 GMTLinkage
http://11011110.livejournal.com/336554.html
<ul><li><a href="https://www.quantamagazine.org/20160929-ghissi-altarpiece-and-mathematics/">Restoring the St. John Altarpiece...with mathematics!</a> (<a href="https://plus.google.com/100003628603413742554/posts/GMvZq1Sz8XJ">G+</a>)</li><br /><li><a href="https://plus.google.com/+IsaacCalder/posts/KrBP46dbWXP">Nonconvex equilateral hecatontetracontahedron</a> (<a href="https://plus.google.com/100003628603413742554/posts/iHr7G1oFBUn">G+</a>)</li><br /><li><a href="http://mathtourist.blogspot.com/2007/05/integral-heptagons.html">Sets of seven non-cocircular points with integral distances</a> (<a href="https://plus.google.com/100003628603413742554/posts/SdF6QLCdy7A">G+</a>)</li><br /><li><a href="http://lathisms.org/">Biographies of prominent Latin and Hispanic mathematicians</a> (<a href="https://plus.google.com/100003628603413742554/posts/2e7gjEfzew2">G+</a>)</li><br /><li><a href="https://www.theguardian.com/uk-news/2016/oct/04/rudd-announces-crackdown-on-overseas-students-and-new-work-visas">UK cuts off the flow of foreign students</a>, because British higher education hasn't imploded quickly enough in the wake of Brexit and needs a push (<a href="https://plus.google.com/100003628603413742554/posts/jSHVx9YJcDr">G+</a>)</li><br /><li><a href="http://news.mit.edu/2016/automating-dna-origami-opens-door-many-new-uses-0526">DNA origami: programmable matter for making nano-scale shapes and structures</a> (<a href="https://plus.google.com/100003628603413742554/posts/A71t5NSxX9H">G+</a>)</li><br /><li><a href="https://medium.com/i-data/trumpwon-trend-vs-reality-16cec3badd60#.hvq0e0vyl">How political hoaxes spread on twitter</a> (<a href="https://plus.google.com/100003628603413742554/posts/VRdGXqiygEA">G+</a>)</li><br /><li><a href="http://mathoverflow.net/questions/215211/what-algebraic-structures-are-related-to-the-mcgee-graph">The McGee graph animated in 3d to show off its symmetries</a> (<a href="https://plus.google.com/100003628603413742554/posts/QBiCfQdksbm">G+</a>)</li><br /><li><a href="https://en.wikipedia.org/wiki/2-satisfiability">2-satisfiability</a>, now a Good Article on Wikipedia (<a href="https://plus.google.com/100003628603413742554/posts/W9aYxXETMPp">G+</a>)</li><br /><li><a href="https://www.thestar.com/news/world/2016/09/29/canadian-medical-journals-hijacked-for-junk-science.html">OMICS hijacks two previously-reputable publishers and their journals</a> (<a href="https://plus.google.com/100003628603413742554/posts/3j6SiJPd83P">G+</a>)</li><br /><li><a href="http://sechel.de/reuleaux.html">The Reuleaux triangle tetrahedron</a>, what you get if you glue four Reuleaux triangles edge-to-edge (<a href="https://plus.google.com/100003628603413742554/posts/MYgATQDGwcU">G+</a>)</li><br /><li><a href="http://www.computational-geometry.org/documents/2017_cfp.pdf">Symposium on Computational Geometry call for papers</a> (abstracts Nov. 28, papers Dec. 5; <a href="https://plus.google.com/100003628603413742554/posts/HWKVRMNRuES">G+</a>)</li><br /><li><a href="https://plus.google.com/117663015413546257905/posts/1M6eqp6s8k5">The McGee graph, again, as a non-rigid unit distance graph</a> (<a href="https://plus.google.com/100003628603413742554/posts/iLKAfybLG6x">G+</a>)</li></ul>http://11011110.livejournal.com/336554.htmlcomputational geometrygraph drawingwikipediaconferencessocial networksartgeometryacademiapublic0