Jekyll2017-11-22T06:46:34+00:00https://11011110.github.io/blog/11011110Geometry, graphs, algorithms, and moreDavid EppsteinAn uncolorable projective configuration2017-11-19T17:31:00+00:002017-11-19T17:31:00+00:00https://11011110.github.io/blog/2017/11/19/uncolorable-projective-configuration<p>An <a href="https://en.wikipedia.org/wiki/Configuration_(geometry)">projective configuration</a> is a collection of points and lines in the plane, so that each two points belong to the same number of lines and each two lines contain the same number of points.
When that same number is three for both points and lines, then the numbers of points and lines must be equal; call this number <script type="math/tex">n</script> and call the configuration an <script type="math/tex">n_3</script>-configuration. Here’s an example, with <script type="math/tex">n=13</script>:</p>
<p><img src="/blog/assets/2017/13-configuration.svg" alt="13-configuration" /></p>
<p>What makes this particular configuration interesting is a property it shares with the <a href="https://en.wikipedia.org/wiki/Fano_plane">Fano <script type="math/tex">7_3</script> configuration</a> (which I can’t draw with straight lines in the plane) but not with some other configurations that I can draw, such as the <a href="https://en.wikipedia.org/wiki/Pappus_configuration">Pappus <script type="math/tex">9_3</script> configuration</a> or <a href="https://en.wikipedia.org/wiki/Desargues_configuration">Desargues <script type="math/tex">10_3</script> configuration</a>. That property is that, if I give the points two colors, as I did in the illustration, then at least one of the lines will be monochromatic. For instance, there’s an all-red line in the illustration.
Let’s call a configuration with this property “uncolorable”, for short.</p>
<p>Uncolorability of the Fano configuration follows from the fact that every two points belong to a three-point line. If its seven points are colored with four red points and three yellow points, then each of the six pairs of red points must form a separate line with one yellow point, or else we would have a red line. But if they do, there is only one line left out of the seven that can cover all three pairs of yellow points, so we have a yellow line instead.
Similar reasoning shows that five-two and six-one splits also don’t work.</p>
<p>If you have two uncolorable <script type="math/tex">m_3</script>- and <script type="math/tex">n_3</script>-configurations, you can glue them into a single uncolorable <script type="math/tex">(m+n-1)_3</script>-configuration, as follows: delete one point from the <script type="math/tex">m_3</script>-configuration and one line from the <script type="math/tex">n_3</script>-configuration, leaving three lines and three points with too few incidences. Then, move the points and lines around (preserving their other incidences) so that the three neighboring lines from the <script type="math/tex">m_3</script> configuration each contain one of the three neighboring points from the <script type="math/tex">n_3</script> configuration.
It may not be clear how to perform this geometrically in all cases, but at least the abstract pattern of incidences that one should obtain as a result is clear.</p>
<p>If we could color the resulting glued configuration, avoiding monochromatic lines, with the three gluing points not all given the same color, then the same coloring could be used on the <script type="math/tex">n_3</script> configuration, and the deleted line would also be non-monochromatic.
On the other hand, if we could color the glued configuration, avoiding monochromatic lines and giving the three gluing points the same color as each other, then the same coloring could be used on the <script type="math/tex">m_3</script> configuration. In this case, giving the same color as the gluing points to the deleted point preserves the non-monochromatic coloring of the lines.
So if the <script type="math/tex">m_3</script>- and <script type="math/tex">n_3</script>-configurations are both uncolorable without monochromatic lines, then the glued <script type="math/tex">(m+n-1)_3</script>-configuration is also uncolorable.</p>
<p>In 1894, Steinitz showed in his dissertation that any abstract <script type="math/tex">n_3</script>-configuration can be drawn in the plane with at most one line non-straight. Using this fact, for any abstract glued configuration, one can realize the <script type="math/tex">n_3</script> configuration making the non-straight line be the deleted one. By projective duality, one can realize the <script type="math/tex">m_3</script> configuration in such a way that all its triples of lines meet at the points of the configuration except for at one point, the deleted point.
Using a projective transformation to match up the three points of one configuration with the three lines of the other shows that every glued configuration can be realized by straight lines in the plane.</p>
<p>The configuration shown in the illustration is what you get when you glue together two copies of the Fano configuration. (In moving it around to make it more legible, I lost track of which points and lines came from which side of the gluing, and I don’t know whether that can be reconstructed from the drawing or whether there are multiple valid partitions. The partition is certainly not given by the colors of the points.) Since the Fano configuration is uncolorable, so is this <script type="math/tex">13_3</script>-configuration, and so are the infinitely many configurations obtained by gluing together more than two Fanos.</p>
<p>(<a href="https://plus.google.com/100003628603413742554/posts/KeDAzkeXPVe">G+</a>)</p>David EppsteinAn projective configuration is a collection of points and lines in the plane, so that each two points belong to the same number of lines and each two lines contain the same number of points. When that same number is three for both points and lines, then the numbers of points and lines must be equal; call this number and call the configuration an -configuration. Here’s an example, with :Linkage2017-11-15T22:07:00+00:002017-11-15T22:07:00+00:00https://11011110.github.io/blog/2017/11/15/linkage<ul>
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<p><a href="https://www.wired.com/story/ai-experts-want-to-end-black-box-algorithms-in-government/">AI experts want to end ‘black box’ algorithms in government</a> (<a href="https://plus.google.com/100003628603413742554/posts/8rb7Ay7Nt64">G+</a>). As should the rest of us. As Greg Egan says in the comments, it’s not much better than haruspicy, and opening up the box to let us look at the bird guts ourselves wouldn’t be much of an improvement.</p>
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<p><a href="https://www.nytimes.com/2017/10/30/science/predatory-journals-academics.html">Predatory journals thrive because of, not in spite of, what they do for academics</a> (<a href="https://plus.google.com/100003628603413742554/posts/YSne3JWVq8c">G+</a>). So combating them means eliminating the perverse incentives that send academics to them, not merely denouncing them.</p>
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<p><a href="https://arxiv.org/abs/1512.01621">Single-exponential FPT algorithms automatically lead to better-than-brute-force exact algorithms</a> (<a href="https://plus.google.com/100003628603413742554/posts/W3gzCGatMyG">G+</a>). A STOC’16 paper by Fomin, Gaspers, Lokshtanov, and Saurabh works by a really simple trick: pick a random subset, hope it’s part of the solution, and apply the FPT algorithm to the remaining unpicked elements.</p>
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<p><a href="http://cstaecker.fairfield.edu/~cstaecker/machines/longimeter.html">The Steinhaus longimeter</a> (<a href="https://plus.google.com/100003628603413742554/posts/Ca79hCja3HH">G+</a>, <a href="https://plus.google.com/+JeffErickson/posts/M1BWFmQnwJ9">via</a>). Measure the approximate length of a curve by overlaying a transparency printed with several rotated grids and counting the number of times the curve crosses a grid line. In the G+ post I ask: Does it work to use the pinwheel tiling in place of a multigrid?</p>
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<p><a href="https://arxiv.org/abs/1711.01171">Lower bounds for <script type="math/tex">k</script>-median and <script type="math/tex">k</script>-means clustering</a> (<a href="https://plus.google.com/100003628603413742554/posts/TfTGwt4uaMT">G+</a>). A new SODA’18 paper by Cohen-Addad, de Mesmay, Rotenberg, and Roytman uses the <a href="https://en.wikipedia.org/wiki/Moment_curve">moment curve</a> to show that, under the <a href="https://en.wikipedia.org/wiki/Exponential_time_hypothesis">exponential time hypothesis</a> you can’t do much better than the trivial try-all-solutions algorithm in dimensions four or higher. But you can do better in 2d. So what about 3d?</p>
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<p><a href="https://sites.google.com/site/awmmath/awm-fellows">AWM Fellows</a> (<a href="https://plus.google.com/100003628603413742554/posts/WAV4T2JaFNB">G+</a>). The Association for Women in Mathematics honors people (not all of them women) “who have demonstrated a sustained commitment to the support and advancement of women in the mathematical sciences”.</p>
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<p><a href="http://retractionwatch.com/2017/11/07/17-johns-hopkins-researchers-resign-protest-ed-board-nature-journal/"><em>Nature</em> makes a mess of open access</a> (<a href="https://plus.google.com/100003628603413742554/posts/bddfPVzfiuX">G+</a>). Rather than retracting a plagiarized article they gave the author a slap on the wrist, so 22 editors of their journal resigned.</p>
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<p><a href="https://www.washingtonpost.com/local/women-flocking-to-statistics-the-new-hot-high-tech-field-of-data-science/2014/12/19/f3e2e486-62ed-11e4-9fdc-d43b053ecb4d_story.html">Women are going into statistics at much higher rates than into mathematics or computer science</a> (<a href="https://plus.google.com/100003628603413742554/posts/UJBUTpBcxs5">G+</a>). The article cites the teamwork, welcoming environment, and plentiful role models as partial explanations.</p>
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<p><a href="https://www.improbable.com/2017/11/09/74873/">Shape representation by zippable ribbons</a> (<a href="https://plus.google.com/100003628603413742554/posts/JWV7vTSdBsH">G+</a>). <em>Improbable Research</em> discovers some graphics people rediscovering <a href="http://erikdemaine.org/zippers/">the work of Erik Demaine, Anna Lubiw, and their parents and children</a>.</p>
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<p><a href="https://plus.google.com/+TimothyGowers0/posts/VywqstorkYa">Tiled tilings from glyptodon carapaces</a> (<a href="https://plus.google.com/100003628603413742554/posts/RwFXP8gpq1n">G+</a>). Glyptodons, an extinct family of mammals, were covered by a regularly-spaced array of skin-plates called scutes, each with the same pattern of bumps, which combined to make their skin look like it was covered irregularly by bumps. But if you look more closely the regularity of the pattern shows through. See links in G+ post for more explanation.</p>
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<p><a href="http://www.sciencemag.org/news/2017/08/science-suffers-china-s-internet-censors-plug-holes-great-firewall">Science suffers as China’s internet censors plug holes in Great Firewall</a> (<a href="https://plus.google.com/100003628603413742554/posts/797jbTj96dY">G+</a>). Many academic services that we might think of as essential, like Google Scholar, are unavailable there. On the other hand, it’s at least creating pressure for viable alternatives.</p>
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<p><a href="https://www.youtube.com/watch?v=U8sq3BplCfI">Counting intersections of flats by tiled triangles</a> (<a href="https://plus.google.com/100003628603413742554/posts/etnf56by9La">G+</a>). I’ve been mostly avoiding the <em>PBS Infinite Series</em> videos because I find the presentation style too dry, and this one isn’t really an exception, but it talks about some neat math I didn’t already know about: a way to solve problems like counting lines that can touch all of four given lines in 3d, by finding 2d tilings using equilateral triangles and rhombi.</p>
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<p><a href="https://cra.org/cra-statement-us-news-world-report-rankings-computer-science-universities/">The Computing Research Association denounces the <em>US News</em> computer science rankings as nonsense</a> (<a href="https://plus.google.com/100003628603413742554/posts/cWFTXqtrELF">G+</a>). The problem is that by using only journal publications in a field that mostly publishes in conferences, <em>US News</em> misses most research productivity, and severely distorts the rest.</p>
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<p><a href="http://www.mathpages.com/home/kmath621/kmath621.htm">An old link on generating Penrose tilings by planar duality from infinite line arrangements</a> (<a href="https://plus.google.com/100003628603413742554/posts/gskmYE7NuuA">G+</a>), brought to mind by <a href="https://plus.google.com/+TimothyGowers0/posts/1Bv9X6TCY9A">a recent artwork by François Morellet depicting the same kind of line arrangement</a>. (These line arrangements are also what the longimeter link above uses.)</p>
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</ul>David EppsteinAI experts want to end ‘black box’ algorithms in government (G+). As should the rest of us. As Greg Egan says in the comments, it’s not much better than haruspicy, and opening up the box to let us look at the bird guts ourselves wouldn’t be much of an improvement.Eulerian partitions2017-11-14T14:51:00+00:002017-11-14T14:51:00+00:00https://11011110.github.io/blog/2017/11/14/eulerian-partitions<p>Did you know that, for every embedded planar graph, it’s possible to partition the edges into two subsets, one of which forms an Eulerian subgraph of the given graph and the other of which forms an Eulerian subgraph of the dual?</p>
<p>Here by “Eulerian” I mean that all vertex degrees are even, but I allow disconnected subgraphs and vertices of degree zero. In fact, by Euler’s formula, there must be at least two degree-zero vertices somewhere (both in the primal, both in the dual, or split between the two graphs). The dual might have self-loops or parallel edges, but the result still holds as long as we count the degree of a self-loop as being two. I also allow either of the two sets of edges in the partition to be empty.</p>
<p>For example, the graph of the dodecahedron can be covered by three cycles, two of length five on two opposite faces and one of length ten between them. These three cycles form an Eulerian subgraph, and the remaining ten edges give an Eulerian subgraph of the dual consisting of two 5-cycles and two isolated vertices.</p>
<p style="text-align:center"><img src="/blog/assets/2017/dodec-edge-partition.svg" alt="Partition of the edges of a dodecahedron into Eulerian subgraphs in the primal and dual graph" /></p>
<p>Probably this is an exercise in a graph theory textbook somewhere, but I don’t know the reference. I also don’t know a direct proof for this fact. Instead, I can show that it follows from a related Eulerian partition problem on arbitrary graphs. This time, we partition vertices instead of edges. Every graph has a partition of its vertices into two subsets such that the two subgraphs induced by these two sets are both Eulerian. For instance, here’s a partition of the dodecahedron into an induced pair of 5-cycles (two opposite faces of the dodecahedron), and a complementary induced 10-cycle (the equator midway between these two faces):</p>
<p style="text-align:center"><img src="/blog/assets/2017/dodec-vertex-partition.svg" alt="Partition of the vertices of a dodecahedron into Eulerian induced subgraphs" /></p>
<p>You can go from a vertex partition of a planar graph to an edge partition by letting the primal Eulerian subgraph of the edge partition be the union of the two induced subgraphs in the vertex partition. As you go around any face of the planar graph, the vertices must alternate between the two sides of the vertex partition, implying that the remaining edges (the ones not part of either induced subgraph) must have an even number around every face, and form an Eulerian subgraph of the dual. In the other direction, suppose you have a partition of the edges of a planar graph into Eulerian and dual-Eulerian subgraphs. Then you can 2-color the faces of the dual-Eulerian subgraph, and this 2-coloring gives you a partition of the vertices of the given graph which partitions the Eulerian side into two induced subgraphs.</p>
<p>Because it doesn’t talk about dual graphs, the vertex partition formulation makes sense for any graph. Most small graphs are pretty easy, but it makes an amusing puzzle to find a partition into two even-degree induced subgraphs for the <a href="https://en.wikipedia.org/wiki/Clebsch_graph">Clebsch graph</a>, the graph you get from a four-dimensional hypercube by adding its long diagonals.</p>
<p style="text-align:center"><img src="/blog/assets/2017/Clebsch-hypercube.svg" alt="Construction of the Clebsch graph from a hypercube" /></p>
<p>Here’s an algorithm for finding Eulerian vertex partitions, and an algorithmic proof that these partitions always exist.</p>
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<p>While the graph contains an odd-degree vertex <script type="math/tex">v</script>, complement the neighborhood of <script type="math/tex">v</script> (adding an edge between every two non-adjacent neighbors, and removing the edge between every two adjacent neighbors) and then remove <script type="math/tex">v</script> from the graph.</p>
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<p>Once all remaining vertices have even degree, use the trivial vertex partition on the remaining Eulerian graph: put all vertices into one side of the partition, and make the other side be the empty set.</p>
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<p>Add back the vertices that were removed, in the reverse of the removal order. For each such vertex <script type="math/tex">v</script>, after adding it back, complement the neighborhood of <script type="math/tex">v</script> again (restoring the graph to the state it was in just prior to the removal of <script type="math/tex">v</script>) and then look at how the neighbors of <script type="math/tex">v</script> are partitioned. Because <script type="math/tex">v</script> has an odd total number of neighbors, one side of the partition must have an odd number of neighbors and the other side must have an even number. Place <script type="math/tex">v</script> on the side of the partition that has an even number of neighbors.</p>
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<p>Consider what happens to the degree of a vertex <script type="math/tex">w</script> in its induced subgraph, when a vertex <script type="math/tex">v</script> that has <script type="math/tex">w</script> as its neighbor is added back to the graph. If <script type="math/tex">w</script> is on the odd side of the partition of the neighbors of <script type="math/tex">v</script>, then complementing the neighborhood of <script type="math/tex">v</script> will cause an even number of vertices (the other ones on the same side of the partition) to change between being induced neighbors and non-neighbors of <script type="math/tex">w</script>. Since <script type="math/tex">w</script> had an even number of induced neighbors before this complement operation, it continues to have an even number of them after the operation. On the other hand, if <script type="math/tex">w</script> is on the even side, complementing the neighborhood of <script type="math/tex">v</script> will cause an odd number of vertices to change between being induced neighbors and non-neighbors of <script type="math/tex">w</script>, changing the degree of <script type="math/tex">w</script> in its induced subgraph to change from even to odd. But then, <script type="math/tex">v</script> will be added to the induced subgraph as a neighbor of <script type="math/tex">w</script>, causing its degree to become even again. And of course, since we added <script type="math/tex">v</script> to the even subset of its neighbors, its own degree in the induced subgraph also becomes even. Because we have a valid Eulerian partition after step 2 and the partition remains valid after each successive step, the algorithm terminates with a valid partition of the whole graph.</p>
<p>This is vaguely reminiscent of <a href="http://www.ics.uci.edu/~eppstein/pubs/Epp-TR-96-14.pdf">an unpublished paper on counting spanning trees</a> that I wrote some years ago. In that paper, I connected the spanning tree counting problem to the existence of a different kind of Eulerian vertex partition: a partition such that the subgraph of edges that connect one side of the partition to the other should have even degree everywhere. These kinds of partitions don’t always exist, but the paper showed that they exist if and only if the graph has an even number of spanning trees. It makes me wonder whether there’s a stronger connection here than just vague resemblance, and whether there’s some interesting structure to the set of all Eulerian partitions of a graph.</p>
<p>(<a href="https://plus.google.com/100003628603413742554/posts/9gLQatdTreG">G+</a>)</p>David EppsteinDid you know that, for every embedded planar graph, it’s possible to partition the edges into two subsets, one of which forms an Eulerian subgraph of the given graph and the other of which forms an Eulerian subgraph of the dual?Linkage2017-10-31T17:43:00+00:002017-10-31T17:43:00+00:00https://11011110.github.io/blog/2017/10/31/linkage<ul>
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<p><a href="https://www.franceinter.fr/culture/wikipedia-la-revanche-des-femmes">Closing the gap between male and female biographies on the French Wikipedia</a> (In French; <a href="https://plus.google.com/100003628603413742554/posts/SJzqXz9fpbx">G+</a>). The corresponding project on the English Wikipedia is <a href="https://en.wikipedia.org/wiki/Wikipedia:WikiProject_Women_in_Red">Women in Red</a>.</p>
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<p><a href="http://www.npr.org/sections/parallels/2017/10/10/553484924/in-china-scholars-are-being-punished-amid-growing-squeeze-on-public-expression">China is cracking down on non-party-line political expression by academics</a> both in their publications and in social media (<a href="https://plus.google.com/100003628603413742554/posts/gGWF4woTY9y">G+</a>).</p>
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<p><a href="http://www.thisiscolossal.com/2017/10/an-exquisite-collection-of-paper-pop-ups-designed-by-peter-dahmen/">Video of paper artist Peter Dahmen’s best pop-up creations</a> (<a href="https://plus.google.com/100003628603413742554/posts/JHNfMX8scF8">G+</a>).</p>
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<p><a href="http://www.asanet.org/news-events/asa-news/asa-president-eduardo-bonilla-silva-responds-chief-justice-john-roberts">“But apart from that, what have the Romans done for us?”</a> (<a href="https://plus.google.com/100003628603413742554/posts/DQ2VQcJqGhG">G+</a>). A leading sociologist responds to Chief Justice Roberts’ dismissal of all sociological research as “gobbledygook”. See also <a href="https://fivethirtyeight.com/features/the-supreme-court-is-allergic-to-math/">538 on the Supreme Court’s distaste for quantitative reasoning</a>, via <a href="https://plus.google.com/+JeffErickson/posts/Y93fpLXNSfc">+JeffE</a> and <a href="https://mathstodon.xyz/web/statuses/612291">@jhertzli</a>.</p>
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<p><a href="http://www.metafilter.com/169982/Bishops-Hat-Fan-Bird-of-Paradise-Crown">Napkin folding</a> (<a href="https://plus.google.com/100003628603413742554/posts/MjiJAFJJGqX">G+</a>). Like origami but with really floppy paper.</p>
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<p><a href="https://www.youtube.com/watch?v=3P6DWAwwViU">Numberphile video on the very big number TREE(3)</a> (<a href="https://plus.google.com/100003628603413742554/posts/QLb1VmcoTcm">G+</a>) derived from <a href="https://en.wikipedia.org/wiki/Kruskal%27s_tree_theorem">Kruskal’s tree theorem</a>. This is a key precursor to the graph minor theorem, and in the G+ comments David Roberts points to <a href="https://en.wikipedia.org/wiki/Friedman%27s_SSCG_function">Friedman’s SSCG function</a>, an even more ridiculously-quickly-growing function derived in the same way from graph minors.</p>
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<p><a href="https://www.nytimes.com/2017/10/22/climate/epa-scientists.html">“Blatant censorship”</a> (<a href="https://plus.google.com/100003628603413742554/posts/YspVj9yurgM">G+</a>). The EPA has cancelled three scheduled conference talks by its scientists about climate change.</p>
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<p><a href="https://www.eff.org/deeplinks/2017/10/portugal-bans-use-drm-limit-access-public-domain-works">Portugal takes some positive steps on digital rights management</a> (<a href="https://plus.google.com/100003628603413742554/posts/bJ2vvoXEMGy">G+</a>). The use of DRM on public domain works is forbidden, and cracking DRM to achieve fair use purposes is allowed.</p>
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<p><a href="https://plus.google.com/100749485701818304238/posts/7Uk3B8oxcVm">Gerard Westendorp’s paper flexible inside-out quartic</a> (<a href="https://plus.google.com/100003628603413742554/posts/Ra22D1WcQKx">G+</a>), based on <a href="http://www.gregegan.net/SCIENCE/KleinQuartic/KleinQuartic.html">an earlier description by Greg Egan</a>. Westendorp subsequently posted <a href="https://plus.google.com/100749485701818304238/posts/JATZhM3kdGH">another version with assembly instructions</a>. The accordion folds he uses for these models cannot actually be folded as <a href="https://en.wikipedia.org/wiki/Rigid_origami">rigid origami</a>, and it’s not clear whether there is a rigid origami version of the same shape that can be turned inside out in the same way.</p>
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<p><a href="https://academia.stackexchange.com/questions/67295/google-scholar-sort-by-date-returns-articles-from-last-year-only">Google scholar’s “sort by date” option does not sort by publication date</a> (<a href="https://plus.google.com/100003628603413742554/posts/APLv7kb9VFt">G+</a>). Instead it shows you the citations that have been added to the database in the last year, sorted by when they were added, <a href="https://plus.google.com/+google/posts/9EDeLFEijiG">just as Google intended it to do</a>.</p>
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<p><a href="https://mathoverflow.net/questions/284458/does-this-geometry-theorem-have-a-name">The angry dude theorem</a> (<a href="https://plus.google.com/100003628603413742554/posts/Rj7cxXga6nk">G+</a>): if two disjoint circles are circumscribed by a third circle, the outer bitangents of the two circles are parallel to the chords of the outer circle formed by its crossings with the inner bitangents. Name given by a MathOverflow user named July based on the appearance of the resulting diagram.</p>
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<p><a href="http://www.mathunion.org/fileadmin/IMU/Report/140810_Evaluation_of_Individuals_WEB.pdf">The IMU’s recommendation on the evaluation of individual researchers in the mathematical sciences</a> (<a href="https://plus.google.com/100003628603413742554/posts/3WVwW8bYjp8">G+</a>). “Tools such as impact factors are clearly not helpful or relevant in the context of mathematical research.” Now, if only we had similar statements about conference acceptance rates…</p>
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<p><a href="https://origami.me/anja-markiewicz/">Anja Markiewicz’s micro origami</a> (<a href="https://plus.google.com/100003628603413742554/posts/2ZrFmCaNoW8">G+</a>). Like normal origami, but a lot smaller.</p>
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<p><a href="https://mathscinet.ams.org/mathscinet/search/publications.html?jourGroupId=5589">The <em>Journal of Computational Geometry</em> is now indexed by MathSciNet</a> (<a href="https://plus.google.com/100003628603413742554/posts/XVXHPGAEUjx">G+</a>). Actually this has been true since at least last January but the announcement on the JoCG web site is dated from September.</p>
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<p><a href="https://www.quantamagazine.org/the-atomic-theory-of-origami-20171031/">The Atomic theory of origami</a> (<a href="https://plus.google.com/100003628603413742554/posts/PBrjfEsn665">G+</a>). <em>Quanta</em> describes work by Michael Assis using statistical mechanics to understand defects in <a href="https://en.wikipedia.org/wiki/Miura_fold">Miura folds</a>.</p>
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</ul>David EppsteinClosing the gap between male and female biographies on the French Wikipedia (In French; G+). The corresponding project on the English Wikipedia is Women in Red.Half a dimension short2017-10-21T23:42:00+00:002017-10-21T23:42:00+00:00https://11011110.github.io/blog/2017/10/21/half-dimension-short<p>Google Earth (or the 3d view in Google Maps) has a topology problem.</p>
<p>It’s been noticed plenty of times before, by <a href="https://www.huffingtonpost.com/2011/03/15/google-earth-bridges-pictures_n_835328.html">the Huffington Post in 2011</a>, <a href="https://weather.com/sports-recreation/weather-ventures/news/postcards-from-google-earth-20130412">the Weather Channel in 2014</a>, and <a href="https://www.theguardian.com/artanddesign/shortcuts/2016/may/02/postcards-google-earth-artist-clement-valla">the Guardian in 2016</a>, among others. But now it’s been fixed, you say? Just look at the Golden Gate Bridge! Isn’t it so much better now than in the 2013 image from the Guardian story?</p>
<p style="text-align:center"><img src="/blog/assets/2017/GG2013.jpg" alt="Golden Gate Bridge in 2013" /></p>
<p style="text-align:center"><img src="/blog/assets/2017/GG2017.jpg" alt="Golden Gate Bridge in 2017" /></p>
<p>Yes, it’s better, enough that at first glance you’re unlikely to see any problem. And it’s not just a flat billboard or a three-dimensional solid block with a bridge painted on it: You can rotate around it in 3d, and see the appropriate background scenery through the gaps between the towers and cables. But it’s still just decoration, probably a few planes with images on them that have a transparency layer. It’s not actually part of the map.
In some sense it’s like the story of the Potemkin village: a false façade that disguises the underlying poverty of the real village.</p>
<p>How can you tell? Well, the flag saying where Google thinks the bridge should be labeled kind of gives it away.But more than that, look at the shadows. The water under the bridge is (at that scale) more or less flat, and more or less opaque. The shadow of the bridge on the water should be as straight as the bridge itself, but it’s not. And there’s a faint outline of the road itself, distorted like the shadow and right above it in the image, far from the decoration that looks like the bridge. That tells us that, underneath it all, Google maps still has a model of the earth as a <a href="https://en.wikipedia.org/wiki/2.5D">two-and-a-half-dimensional thing</a>: the two-dimensional surface of latitude-longitude pairs, plus an elevation value (or, under the bridge, depth value) for each point on that surface.</p>
<p>For even slightly-less-famous bridges, it’s even worse, because the decoration part isn’t there. Here’s the <a href="https://en.wikipedia.org/wiki/Bixby_Creek_Bridge">Bixby Creek Bridge</a> in Big Sur, in a photograph and in Google Maps.</p>
<p style="text-align:center"><img src="/blog/assets/2017/Bixby11.jpg" alt="Bixby Creek Bridge in 2011" title="CC-BY-SA image Bixby Bridge from overlook.JPG by GlennTSimmons from Wikimedia Commons" /></p>
<p style="text-align:center"><img src="/blog/assets/2017/Bixby17.jpg" alt="Bixby Creek Bridge in 2017" /></p>
<p>Replacing this with another decorative bridge-image like the one for the Golden Gate, and painting over the duplicate flattened bridge with greenery, would help. But unless Google Maps makes an expensive change to its underlying data model, it will still just be a patch that hides the problem. The actual problem is one of topology: the Earth’s surface is not topologically spherical. It has handles that stick out of it and holes you can pass through; some are man-made, like these bridges, <a href="https://en.wikipedia.org/wiki/List_of_longest_natural_arches">others natural</a>. Some parts of the earth really have two different layers, with different things that should be shown for them. (Some have a lot more than two: within Google Street View, <a href="https://en.wikipedia.org/wiki/Tokyo_Station">Tokyo Station</a> has eight levels.) And those can’t be represented by elevation data over a flat or spherical surface.</p>
<p>(<a href="https://plus.google.com/100003628603413742554/posts/9RLnUPKkFVt">G+</a>)</p>David EppsteinGoogle Earth (or the 3d view in Google Maps) has a topology problem.Linkage2017-10-15T15:21:00+00:002017-10-15T15:21:00+00:00https://11011110.github.io/blog/2017/10/15/linkage<ul>
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<p><a href="https://gilkalai.wordpress.com/2017/10/01/the-world-of-michael-burt-when-architecture-mathematics-and-art-meet/">Israeli architect Michael Burt on space-filling networks and spatial tesselations</a> (<a href="https://plus.google.com/100003628603413742554/posts/AQuvAyrtiJU">G+</a>). There’s more on <a href="http://www.professormichaelburt.com/mburt/">Burt’s own site</a> but it’s not easy to navigate or link.</p>
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<p><a href="http://www.smh.com.au/national/education/year-8-dropout-becomes-face-of-a-national-schools-campaign-20170601-gwhz7h.html">Year 8 dropout becomes face of a national schools campaign</a> (<a href="https://plus.google.com/100003628603413742554/posts/DMecVAJvQb1">G+</a>) After being pushed away from mathematics by bad advice as a teenager, Charles Gray came back, earned an honors degree in her 30s, is now working towards a doctorate, and has become the ambassador for an Australian campaign to raise awareness of STEM among 9- and 10-year-old girls.</p>
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<p><a href="https://joycegeek.com/tag/euclid/">Euclid and Joyce</a> (<a href="https://plus.google.com/100003628603413742554/posts/2iBprYoim2s">G+</a>). Why James Joyce made <em>Finnegan’s Wake</em> <script type="math/tex">2\pi</script> pages long, and placed Euclid’s Proposition I at page <script type="math/tex">\pi</script>. See also the same proposition on the cover of <a href="http://bit-player.org/">Brian Hayes’</a> new collection of essays, <a href="https://mitpress.mit.edu/books/foolproof-and-other-mathematical-meditations">“Foolproof, and Other Mathematical Meditations”</a>.</p>
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<p><a href="http://www.cambridgeblog.org/2017/04/it-is-best-to-accept-the-banach-tarski-paradox/">If all sets of reals are measurable then the reals can be partitioned into more subsets than elements</a> (<a href="https://plus.google.com/100003628603413742554/posts/Rwp94dxXp6j">G+</a>). Stan Wagon sees this as evidence that we should accept the axiom of choice as true. See his preprint with Alan Taylor, <a href="http://stanwagon.com/public/TheDivisionParadoxTaylorWagon.pdf">“A Paradox Arising from the Elimination of a Paradox”</a>, for details.</p>
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<p><a href="http://thekidshouldseethis.com/post/the-kresling-pattern-and-our-origami-world">Paris-based architect Biruta Kresling on the Kresling pattern</a> (<a href="https://plus.google.com/100003628603413742554/posts/Pw3qUufX7Vc">G+</a>). A video clip from <em>The Origami Revolution</em> showing spontaneous paper-folding pattern generation by twisting cylinders until they buckle.</p>
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<p><a href="http://www.ics.uci.edu/~eppstein/pix/twsbi/">Some new toys</a> (<a href="https://plus.google.com/100003628603413742554/posts/LjtzWUcCQ5C">G+</a>). I celebrated giving the publisher the <a href="/blog/2017/07/26/forbidden-configurations-in.html">forbidden configuration book</a> by getting a fancier fountain pen, the TWSBI Mini AL, some nice ink, and a cigar case to reduce leakage when I fly with my pens.</p>
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<p><a href="https://www.youtube.com/watch?v=btPqKAGyajM">An unexpected convergence of the Euler characteristic with sport, traffic engineering, and politics</a> (<a href="https://plus.google.com/100003628603413742554/posts/ScFTEhPybLi">G+</a>, <a href="http://aperiodical.com/2017/10/petition-to-update-uk-traffic-signs-to-use-a-geometrically-plausible-football/">via</a>). Matt Parker wants the UK to stop depicting mathematically impossible shapes in the traffic signs directing motorists to football stadiums. In the G+ comments, Ian Agol points out that it might not be impossible: maybe it’s just a hyperbolic horoball.</p>
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<p><a href="https://www.math.uci.edu/~asilverb/Adventures.html">Alice’s Adventures in Numberland</a> (<a href="https://plus.google.com/100003628603413742554/posts/4737qapXHRC">G+</a>). Anecdotes by my UCI mathematics colleague Alice Silverberg about poor treatment of women in mathematics.</p>
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<p><a href="https://www.significancemagazine.com/science/563-the-remarkable-life-of-frances-wood">Frances Wood </a> (<a href="https://plus.google.com/100003628603413742554/posts/QUEwR7oKk2x">G+</a>), short-lived but well-accomplished statistician of the early 20th century.</p>
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<p><a href="https://www.quantamagazine.org/the-math-behind-gerrymandering-and-wasted-votes-20171012"><em>Quanta</em> on gerrymandering</a> (<a href="https://plus.google.com/100003628603413742554/posts/LKtEfMtuTSB">G+</a>). With a nice clear explanation of the efficiency ratio.</p>
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<p><a href="http://conwaylife.com/forums/viewtopic.php?f=11&t=2597#p52043">Simulation of Rule 110 in B35/S236</a> (<a href="https://plus.google.com/100003628603413742554/posts/AmrSkLDLbNG">G+</a>). Peter Naszvadi shows that <a href="http://www.ics.uci.edu/~eppstein/ca/b35s236/">a cellular automaton rule I investigated earlier</a> is capable of universal computation, possibly the first Life-like rule other than Life and its close relatives for which this is known. But the proof is still somewhat unsatisfactory because it relies on Turing completeness for rule 110 which in turn involves patterns with infinitely many live cells.</p>
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<p><a href="https://cacm.acm.org/blogs/blog-cacm/221749-the-binary-system-was-created-long-before-leibniz/fulltext">Binary before Leibniz</a> (<a href="https://plus.google.com/100003628603413742554/posts/d6h5haHB8FH">G+</a>). Herbert Bruderer points to the earlier work of <a href="https://en.wikipedia.org/wiki/Thomas_Harriot">Thomas Harriot</a> and <a href="https://en.wikipedia.org/wiki/Juan_Caramuel_y_Lobkowitz">Juan Caramuel y Lobkowitz</a>, and to earlier references by 20th-century historians of mathematics pointing out this earlier work. So why does Leibniz still get all the credit?</p>
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</ul>David EppsteinIsraeli architect Michael Burt on space-filling networks and spatial tesselations (G+). There’s more on Burt’s own site but it’s not easy to navigate or link.Peeling vs shortening2017-10-11T21:04:00+00:002017-10-11T21:04:00+00:00https://11011110.github.io/blog/2017/10/11/peeling-vs-shortening<p>My latest preprint is about two processes, one continuous and one discrete, that appear to do the same thing as each other. We don’t really understand why.
The preprint is “Grid peeling and the affine curve-shortening flow”, with Sariel Har-Peled and Gabriel Nivasch (<a href="https://arxiv.org/abs/1710.03960">arXiv:1710.03960</a>) and will appear at ALENEX.</p>
<p>The discrete process is the one where you find the <a href="https://en.wikipedia.org/wiki/Convex_layers">convex layers</a> of a grid, or a subset of a grid. That is, you find the convex hull, remove its vertices, find the convex hull of the rest of the points, remove its vertices, etc. I wrote about it briefly a month ago, in a <a href="https://plus.google.com/100003628603413742554/posts/32p3w4KWLXn">G+ post on a related integer sequence</a>.
When you do this to a square grid, the layers appear to get more circular as you go inwards (until they become so small that being on a grid forces them to get more polygonal again). When you do it to a rectangular grid, they appear to get elliptical. And when you do it to a quarter-infinite grid…</p>
<div style="text-align:center"><iframe width="560" height="315" src="https://www.youtube.com/embed/rX3r8KaLHck" frameborder="0" align="middle" allowfullscreen=""></iframe></div>
<p>…you appear to get a hyperbola. What’s going on?</p>
<p>The continuous process is a variant of the <a href="https://en.wikipedia.org/wiki/Curve-shortening_flow">curve-shortening flow</a> called the affine curve-shortening flow. It operates on smooth curves in the plane, by moving all the points of the curve simultaneously. Each point moves towards its local center of curvature, at a speed proportional to the cube root of the curvature (the inverse of the cube root of the radius of curvature). If you take the “cube root” part out, you get the curve-shortening flow, under which every simple closed curve instantaneously smooths itself, then more slowly becomes convex (while avoiding self-intersections), then slowly becomes more circular (while shrinking to a point). The affine curve-shortening flow does much the same thing, more slowly, but it becomes elliptical instead of circular.</p>
<p>What we noticed was that if <script type="math/tex">C</script> is a convex curve, then affine curve-shortening on <script type="math/tex">C</script> and the convex layers of the points of a fine grid inside <script type="math/tex">C</script> appear to be much the same. Here’s an example where the initial curve is less symmetric than a square or rectangle. The left side is affine curve-shortening, the right side is grid peeling, and the middle is a much coarser grid peeling (with every fifth layer shown) to demonstrate what we’re doing. Can you tell the difference between left and right?</p>
<p style="text-align:center"><img src="/blog/assets/2017/peeling-vs-flow.png" alt="Affine curve-shortening (left) vs grid peeling (right)" /></p>
<p>Of course proof by picture is not very convincing. We have some theory to support this observation (we can prove that grid peeling moves within a constant factor of the speed of affine curve shortening) and some vague heuristic arguments for why they might be the same (grid peeling is invariant under a big subset of affine transformations, so if it’s going to act like a flow it should be a flow that’s also affine invariant, and the simplest choice is affine curve-shortening). But we did send this to a conference about experiments, so we also did some experiments to see how similar the two processes are.</p>
<p>The answer: pretty similar, similar enough to convince us that (in the limit as the grid becomes finer) grid peeling converges pointwise to affine curve-shortening. See the preprint for the details, but the basic story from the experiments is that for fine grids the Hausdorff distance between the peeled grid curves and the affinely shortened curves is small, and grows smaller with finer grids. Decreasing the grid spacing by a factor of 10 decreased the Hausdorff distance by a factor of about 2.65, so that suggests that the distance is a sublinear power of the grid spacing, but that’s a wild extrapolation from two data points so it would probably be a mistake to guess the exponent with more than one digit of precision.</p>
<p>Anyway, maybe this is easy to prove, to someone who has the right theoretical tools in hand (which is not us).</p>
<p>(<a href="https://plus.google.com/100003628603413742554/posts/H84hBAh1Vbg">G+</a>)</p>David EppsteinMy latest preprint is about two processes, one continuous and one discrete, that appear to do the same thing as each other. We don’t really understand why. The preprint is “Grid peeling and the affine curve-shortening flow”, with Sariel Har-Peled and Gabriel Nivasch (arXiv:1710.03960) and will appear at ALENEX.The pentagon in the pyramid2017-10-07T17:10:00+00:002017-10-07T17:10:00+00:00https://11011110.github.io/blog/2017/10/07/pentagon-in-pyramid<p>Sliced okra has a pentagonal cross-section:</p>
<p style="text-align:center"><img src="/blog/assets/2017/pentagon-in-pyramid/okra.jpg" alt="Pentagonal slices of okra" title="CC-BY-SA image Okra sliced.JPG by Yes.aravind from Wikimedia commons" /></p>
<p>But did you know that you can also get a regular-pentagon slice from a square pyramid? Here it is in top view. The yellow color shows the interior of the pyramid after slicing it, while the blue is the original pyramid surface. (For years, in my office, I’ve had a print by Wulf Barsch showing the pyramids of Egypt, lit blue by the night.)</p>
<p style="text-align:center"><img src="/blog/assets/2017/pentagon-in-pyramid/sliced-pyramid.svg" alt="Pyramid sliced with a regular pentagon cross-section, top view" /></p>
<p>You probably already did know that a cube or regular octahedron can be sliced perpendicular to an axis of symmetry, to give a regular hexagon. Again, in top view:</p>
<p style="text-align:center"><img src="/blog/assets/2017/pentagon-in-pyramid/hexes.svg" alt="Cube and octahedron sliced with a regular hexagon cross-section, top view" /></p>
<p>The same thing works to get a square slice from a tetrahedron, or a decagonal slice from a dodecahedron or icosahedron. But these all rely heavily on the symmetry of the underlying shape: the cube has a six-fold symmetry (simultaneously rotate by 60° around a long diagonal and reflect across a perpendicular plane) that becomes a symmetry of its hexagonal slice. The pyramid does not have a five-fold symmetry. So how do we get something more symmetric by slicing it?</p>
<p>The trick is that the pyramid is only really half of the underlying symmetric shape that you’re slicing. To get a regular pentagon, the pyramid that you should start with is not the Egyptian one, but a taller pyramid, the Johnson solid with one square face and four equilateral-triangle faces. This shape is what you get if you slice a regular octahedron in half along its equator; the equator becomes the square face.</p>
<p>The octahedron still is not pentagonally symmetric. But as Schönemann wrote in 1873, you can get the twelve vertices of a regular icosahedron by subdividing each of the twelve edges of an octahedron in the golden ratio, so that the subdivision points on each octahedron face form an equilateral triangle (the blue triangles in the image below). These eight triangles form eight of the faces of the faces of the icosahedron; the other twelve faces are interior to the octahedron. This icosahedron-in-octahedron is dual to another polyhedral pair, a cube formed by selecting eight out of the twenty vertices of a regular dodecahedron.</p>
<p style="text-align:center"><img src="/blog/assets/2017/pentagon-in-pyramid/icosahedron-in-octahedron.svg" alt="Icosahedron in an octahedron" /></p>
<p>(<a href="http://steiner.math.nthu.edu.tw/d3/d2/quick-and-dirty/Smallest%20Octahedron%20Containing%20the%20Icosahedron.html">Here’s another more three-dimensional view</a>. See also <a href="https://johncarlosbaez.wordpress.com/2012/03/15/tidbits-of-geometry/">John Baez’s tidbits of geometry</a>.)</p>
<p>The neighbors of any one vertex in an icosahedron form a regular pentagon. And although this pentagon isn’t a slice of the octahedron, it is a slice of the half-octahedron pyramid.</p>
<p>This connection between pentagons and pyramids is apparently an exercise from a 1986 Russian mathematics book by I. F. Sharygin, which asks for the height of the pyramid having a regular pentagonal cross-section. I found it in “<a href="http://forumgeom.fau.edu/FG2017volume17/FG201710index.html">Putting the Icosahedron into the Octahedron</a>”, Paris Pamfilos, <em>Forum Geometricorum</em> 2017, who replaces an algebraic calculation of the pyramid height by a couple of applications of <a href="https://en.wikipedia.org/wiki/Menelaus%27_theorem">Menelaus’s theorem</a> and from the result rediscovers Schönemann’s inscription of an icosahedron within the octahedron. But I think it’s simpler to go the other way: inscribe an icosahedron within an octahedron, and use it to find the pentagonal cross-section.</p>
<p>(See the <a href="https://plus.google.com/100003628603413742554/posts/EvRgzZgzEww">G+ post</a> for discussion, including some Blender renders by Kram Einsnulldreizwei and a paper model by Gerard Westendorp)</p>David EppsteinSliced okra has a pentagonal cross-section:Square contact graphs2017-10-03T21:48:00+00:002017-10-03T21:48:00+00:00https://11011110.github.io/blog/2017/10/03/square-contact-graphs<p>I’ve <a href="/blog/2014/02/23/schramms-monster-packing.html">already written here about contact graphs of squares</a>, the graphs you get from non-overlapping squares in the plane by making a vertex per square and an edge when two squares share pieces of their edges. But now I have a new preprint on the subject: “Square-Contact Representations of Partial 2-Trees and Triconnected Simply-Nested Graphs” (with Da Lozzo, Devanny, and Johnson, <a href="https://arxiv.org/abs/1710.00426">arXiv:1710.00426</a>, to appear at <a href="https://saki.siit.tu.ac.th/isaac2017/">ISAAC 2017</a>).</p>
<p>Here’s an example, of a different kind of graph than the ones in the preprint. The squares in the image below (four of which are too big to fit and are visible near the frame of the image)…</p>
<p style="text-align:center"><img src="/blog/assets/2017/gcsq.svg" alt="Contact representation of Greek cross heptacube" /></p>
<p>…have the same pattern of contacts as the adjacencies of the following graph, a heptacube shaped like a three-dimensional Greek cross:</p>
<p style="text-align:center"><img src="/blog/assets/2017/gc7cube.svg" alt="Greek cross heptacube" /></p>
<p>By being more careful to keep each gap between squares close to square itself (filling such gaps with four-tuples of squares that are only slightly bigger than a third of the width of the gap) it is possible to repeat this construction for any number of additional levels, representing any <a href="https://en.wikipedia.org/wiki/Polycube">polycube</a> in which the connections between cubes are tree-like.</p>
<p>One thing that makes these particular graphs easier than some to represent is that they have no triangles. Triangles can be a problem with square contact graphs, because the three squares representing the vertices of the triangle have to touch each other snugly, with no space in between them. That makes the packing of squares more highly constrained, but it also means there is nowhere to put more squares that might represent vertices inside the triangle. So planar graphs that have a triangle with something inside it (either a separating triangle, or a triangle as the outer face).</p>
<p>That turns out to be the only obstacle for representing some other tree-like planar graphs, the ones in the preprint. Series-parallel graphs can be represented by square contacts unless they contain a subgraph <script type="math/tex">K_{1,1,3}</script>, a series-parallel graph that has a separating triangle in all of its planar embeddings. And Halin graphs can be represented unless they equal <script type="math/tex">K_4</script>, a Halin graph that always has a triangle as its outer face. Proving that these conditions are necessary is easy, but proving that they are sufficient is a nasty induction.</p>
<p>Unfortunately, for less-restricted classes of graphs, triangles aren’t the only obstacle.
Separating four-cycles can also be a problem, even though the polycubes have lots of them. <a href="/blog/2014/02/23/schramms-monster-packing.html">Schramm’s monster theorem</a>, the subject of my previous post, shows that every maximal planar graph without separating four-cycles has an improper square contact representation, one where sometimes adjacent squares touch at their corners rather than at their edges. But when there’s a separating four-cycle, the representation that you get from the monster theorem might collapse that cycle to four squares all meeting at their corners (like the <a href="https://en.wikipedia.org/wiki/Four_Corners">Four Corners</a>, a spot in the US where four states meet) and everything inside the cycle gets squished to a point.</p>
<p>The simplest example that I know of for this phenomenon is the <a href="https://en.wikipedia.org/wiki/Square_antiprism">square antiprism</a>. It’s a planar graph with two quadrilateral faces and eight triangles. To represent it by square contacts, of course you have to put one of the quadrilaterals as the outside face, because a triangle as the outside face would already leave no space for anything else. But if the outside face is a quadrilateral, the gap inside its four squares must be square, and the only way to pack the inner four squares with the right adjacency is to make an improper representation with the four corners touching. If you augment the graph with one more vertex in its inner quadrilateral, there is nowhere for the corresponding square to go and no representation at all.</p>
<p style="text-align:center"><img src="/blog/assets/2017/nested-quads.svg" alt="The improper representation of a square antiprism, and a graph with no square contact representation" /></p>
<p>Our preprint only handles some special classes of graphs. We have identified both nonempty triangles and separating antiprisms as obstacles to representation, but they seem unlikely to be the only ones. As we write, “a characterization of the graphs having proper contact representations by squares remains elusive”.</p>
<p>(<a href="https://plus.google.com/100003628603413742554/posts/9rfSXaeC55s">G+</a>)</p>David EppsteinI’ve already written here about contact graphs of squares, the graphs you get from non-overlapping squares in the plane by making a vertex per square and an edge when two squares share pieces of their edges. But now I have a new preprint on the subject: “Square-Contact Representations of Partial 2-Trees and Triconnected Simply-Nested Graphs” (with Da Lozzo, Devanny, and Johnson, arXiv:1710.00426, to appear at ISAAC 2017).Linkage2017-09-30T20:02:00+00:002017-09-30T20:02:00+00:00https://11011110.github.io/blog/2017/09/30/linkage<ul>
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<p><a href="http://www.graphdrawing.de/contest2017/challenge.html">This year’s Graph Drawing contest rules and graphs</a> (<a href="https://plus.google.com/100003628603413742554/posts/5y2icm5XQLx">G+</a>). This year’s challenge is over but the graphs are probably still interesting as test data. The G+ post includes an argument that some graphs require sharp crossing angles (this year’s optimization criterion).</p>
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<p><a href="http://ralphvanraaij.blogspot.nl/2017/04/borromeaanse-ringen-borromean-rings.html">Borromeaanse ringen / Borromean Rings</a> (<a href="https://plus.google.com/100003628603413742554/posts/EyR9FCmgsxV">G+</a>). One of many imagined mathematical cityscapes by Ralph van Raaij.</p>
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<p><a href="https://arxiv.org/help/eess/announcement">ArXiv adds three electrical engineering topic areas</a> (<a href="https://plus.google.com/100003628603413742554/posts/5VzFDkmDuH6">G+</a>), and a few days later another for econometrics.</p>
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<p><a href="https://en.wikipedia.org/wiki/Tur%C3%A1n%27s_brick_factory_problem">Turán’s brick factory problem</a> (<a href="https://plus.google.com/100003628603413742554/posts/Y7H4Yc53h1f">G+</a>). The problem that started the study of crossing numbers of graphs. Now a Good Article on Wikipedia.</p>
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<p><a href="https://www.newscientist.com/article/2148074-infamous-three-body-problem-has-over-a-thousand-new-solutions/">Thousands of new solutions to the three-body problem</a> (<a href="https://plus.google.com/100003628603413742554/posts/EjcB8P75NtP">G+</a>). It appears from <a href="https://arxiv.org/abs/1709.04775">the preprint</a> that these supposed new periodic orbits have only been determined numerically, and have not been verified to exist with rigorous mathematics.</p>
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<p><a href="https://www.forbes.com/sites/hilarybrueck/2017/09/20/grand-central-ceiling-women-in-science-and-stem-ge-millie-dresselhaus/">The female scientists and technologists on the ceiling of Grand Central Station</a> (<a href="https://plus.google.com/100003628603413742554/posts/iWqyhaYsQ6G">G+</a>).</p>
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<p><a href="https://mathwithbaddrawings.com/2017/09/20/the-state-of-being-stuck/">Accepting the state of being stuck</a> (<a href="https://plus.google.com/100003628603413742554/posts/DT3GzMfcquu">G+</a>). Andrew Wiles’ advice for success in mathematical research.</p>
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<p><a href="https://tesselace.com/gallery/">Algorithmically designed bobbin lace by Veronika Irvine</a> (<a href="https://plus.google.com/100003628603413742554/posts/RZRdKFaafPv">G+</a>). See <a href="http://arxiv.org/abs/1708.09778">her Graph Drawing paper with Therese Biedl</a> for more on the mathematics behind this mathematical textile art.</p>
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<p><a href="http://www.ics.uci.edu/~eppstein/pix/vmdog/index.html">View-Master Dog</a> (<a href="https://plus.google.com/100003628603413742554/posts/1D2kkHcvKZr">G+</a>), the unofficial mascot of Graph Drawing 2017.</p>
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<p><a href="https://plus.google.com/100372006947364662364/posts/5VV3yrD4SBb">Graph Drawing contest winners</a> (<a href="https://plus.google.com/100003628603413742554/posts/ZfiGvFKkuhZ">G+</a>). The link describes an interactive visualization of a citation network. But the outcome of the online part of the contest is that optimizing crossing angles above all else leads to bad drawings.</p>
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<p><a href="https://blogs.harvard.edu/pamphlet/2012/03/06/an-efficient-journal/">The economics of diamond open access</a> (<a href="https://plus.google.com/100003628603413742554/posts/QVax5u2km26">G+</a>). With ineffective paid-publisher shill in the comments.</p>
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<p><a href="http://www.thehindu.com/sci-tech/science/ugcs-approved-journal-list-has-111-more-predatory-journals/article19677830.ece">When open-access whitelists fail</a> (<a href="https://plus.google.com/100003628603413742554/posts/LivfL6iELXo">G+</a>, <a href="http://retractionwatch.com/2017/09/23/weekend-reads-sexual-harassment-scientific-misconduct-says-one-society-favorite-plagiarism-excuses/">via</a>). Somehow over a hundred predatory journals made their way onto an Indian approved-journal list. But even DOAJ-listed journals accepted a bad test paper in nearly half of the instances of one test.</p>
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</ul>David EppsteinThis year’s Graph Drawing contest rules and graphs (G+). This year’s challenge is over but the graphs are probably still interesting as test data. The G+ post includes an argument that some graphs require sharp crossing angles (this year’s optimization criterion).