Jekyll2021-05-11T07:36:21+00:00https://11011110.github.io/blog/feed.xml11011110Geometry, graphs, algorithms, and moreDavid EppsteinArc-triangle tilings2021-05-09T16:20:00+00:002021-05-09T16:20:00+00:00https://11011110.github.io/blog/2021/05/09/arc-triangle-tilings<p>Every triangle tiles the plane, by 180° rotations around the midpoints of each side; some triangles have other tilings as well. But if we generalize from triangles to arc-triangles (shapes bounded by three circular arcs), it is no longer true that everything tiles. Within any large region of the plane, the lengths of bulging-outward arcs of each radius must be balanced by equal lengths of bulging-inward arcs of each radius, and the only way to achieve this with a single tile shape is to keep that same balance between convex and concave length on each tile. Counting line segments as degenerate cases of circular arcs, this gives us three possibilities:</p>
<ul>
<li>
<p>Ordinary triangles, with three straight sides, which always tile in the ordinary way.</p>
<p style="text-align:center"><img src="/blog/assets/2021/ordinary-triangle-tiling.svg" alt="Tiling by ordinary triangles" /></p>
</li>
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<p>Arc-triangles with two congruent curved sides (one bulging out and one in) and one straight side. These always tile, by matching up the curved sides to form strips of triangles bounded by their straight sides. Some of these arc-triangles also have other tilings.</p>
<p style="text-align:center"><img src="/blog/assets/2021/wave-triangle-tiling.svg" alt="Tiling by arc-triangles with two curved sides" /></p>
</li>
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<p>Arc-triangles with three sides of the same curvature, the shorter two having equal total length to the longest side. The long side must bulge outwards and the other two sides must bulge inwards. Again, these always tile, although the tiling cannot be edge-to-edge.</p>
<p style="text-align:center"><img src="/blog/assets/2021/scale-triangle-tiling.svg" alt="Tiling by arc-triangles with three curved sides" /></p>
</li>
</ul>
<p>The ordinary triangles tile by translation and rotation, and the three-curved-side arc-triangles tile by translation only, without even needing rotations. However, the two-curved-side triangles generally need reflections for their tilings. If tilings by translation and rotation are desired, then only some of these tile: I think only the ones with angles of \(\pi/3\), \(\pi/2\), or \(2\pi/3\) at the vertex between the two curved sides.</p>
<p style="text-align:center"><img src="/blog/assets/2021/pinwheels.svg" alt="Tiling by arc-triangles with two curved sides, without reflection" /></p>
<p>A curious property of the arc-triangles that tile is that they all have interior angles summing to \(\pi\), something that is not true of most arc-triangles. On the other hand, it is easy to find arc-triangles with angles summing to \(\pi\) that do not tile, so the angle sum does not completely characterize the tilers among the arc-triangles.</p>
<p>(<a href="https://mathstodon.xyz/@11011110/106207851143984141">Discuss on Mastodon</a>)</p>David EppsteinEvery triangle tiles the plane, by 180° rotations around the midpoints of each side; some triangles have other tilings as well. But if we generalize from triangles to arc-triangles (shapes bounded by three circular arcs), it is no longer true that everything tiles. Within any large region of the plane, the lengths of bulging-outward arcs of each radius must be balanced by equal lengths of bulging-inward arcs of each radius, and the only way to achieve this with a single tile shape is to keep that same balance between convex and concave length on each tile. Counting line segments as degenerate cases of circular arcs, this gives us three possibilities:Congratulations, Dr. Matias!2021-05-07T14:50:00+00:002021-05-07T14:50:00+00:00https://11011110.github.io/blog/2021/05/07/congratulations-dr-matias<p><a href="https://pmatias.com/">Pedro Ascensao Ferreira Matias</a>, one of the students working with Mike Goodrich in the UC Irvine <a href="https://www.ics.uci.edu/~theory/">Center for Algorithms and Theory of Computation</a>, passed his Ph.D. defense today.</p>
<p>Pedro is Portuguese, and came to UCI after a bachelor’s degree from the University of Coimbra in Portugal and a master’s degree from Chalmers University of Technology in Sweden.</p>
<p>The general topic of Pedro’s research is “exact learning”, the inference of structured information from queries or other smaller pieces of data. I’ve written here before about my work with Matias on <a href="/blog/2019/02/21/mutual-nearest-neighbors.html">nearest-neighbor chains</a> and on <a href="/blog/2019/08/17/footprints-in-snow.html">tracking paths in planar graphs</a>, the problem of placing sensors on a small subset of vertices so that, by detecting the order in which a path reaches each sensor, you can uniquely determine the whole path. His dissertation combines the tracking paths work with a second paper on tracking paths (“How to Catch Marathon Cheaters: New Approximation Algorithms for Tracking Paths”, <a href="https://arxiv.org/abs/2104.12337">arXiv:2104.12337</a>, to appear at WADS 2021), and a paper on reconstructing periodic and near-periodic strings from sublinear numbers of queries (“Adaptive Exact Learning in a Mixed-Up World: Dealing with Periodicity, Errors and Jumbled-Index Queries in String Reconstruction”, <a href="https://arxiv.org/abs/2007.08787">arXiv:2007.08787</a>, in SPIRE 2020). He also has recent papers on reconstructing trees in SPAA 2020 and ESA 2020.</p>
<p>After finishing his doctorate, Pedro’s next position will be working for Facebook.</p>
<p>Congratulations, Pedro!</p>
<p>(<a href="https://mathstodon.xyz/@11011110/106196168129163033">Discuss on Mastodon</a>)</p>David EppsteinPedro Ascensao Ferreira Matias, one of the students working with Mike Goodrich in the UC Irvine Center for Algorithms and Theory of Computation, passed his Ph.D. defense today.Linkage2021-04-30T17:00:00+00:002021-04-30T17:00:00+00:00https://11011110.github.io/blog/2021/04/30/linkage<ul>
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<p><a href="https://www.eff.org/deeplinks/2021/03/googles-floc-terrible-idea">The EFF on FLoC</a> (<a href="https://mathstodon.xyz/@11011110/106079326968515725">\(\mathbb{M}\)</a>), Google’s plan for browsers to aggregate your browsing habits and make them public for ad-personalization. Short summary: it’s a bad idea and if you care about privacy you should switch to a non-Chrome browser. Technical summary: it’s based on <a href="https://en.wikipedia.org/wiki/K-anonymity">k-anonymity</a>, known as <a href="https://doi.org/10.1109/CASoN.2010.139">inadequate at protecting individual privacy in social networks</a>. If you use Chrome, assume all bad guys on the web can see all your browsing.</p>
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<p>Relevant to my recent post on Pick’s theorem: <a href="https://www.youtube.com/watch?v=osF2JhrVHxc">Chris Staecker on the dot planimeter</a> (<a href="https://mathstodon.xyz/@11011110/106090366361151274">\(\mathbb{M}\)</a>), a device for <a href="https://en.wikipedia.org/wiki/Dot_planimeter">approximating area by counting grid points</a>.</p>
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<p><a href="https://www.peeta.net/">Anamorphic street art by Peeta transforms building shapes into 3d geometric abstractions</a> (<a href="https://mathstodon.xyz/@11011110/106100758800338449">\(\mathbb{M}\)</a>, <a href="https://weburbanist.com/2019/07/12/anamorphic-street-art-new-abstract-murals-by-peeta-pop-off-the-wall/">via</a>).</p>
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<p><a href="https://lore.kernel.org/linux-nfs/YH+zwQgBBGUJdiVK@unreal/">Students of University of Minnesota assistant professor Kangjie Lu caught allegedly deliberately sending buggy patches to Linux kernel as some kind of breaching experiment</a>, resulting in <a href="https://lore.kernel.org/linux-nfs/YH%2FfM%2FTsbmcZzwnX@kroah.com/">the whole university being banned from Linux kernel development</a> (<a href="https://mathstodon.xyz/@11011110/106104208447478044">\(\mathbb{M}\)</a>, <a href="https://lobste.rs/s/3qgyzp/they_introduce_kernel_bugs_on_purpose">via</a>). They claim to have been declared IRB-exempt but this appears to be a mistake by the IRB. See also <a href="https://cse.umn.edu/cs/statement-cse-linux-kernel-research-april-21-2021">department reaction</a> and <a href="https://www.metafilter.com/191207/How-to-get-your-University-banned-in-1-easy-step">metafilter discussion</a>.</p>
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<p><a href="https://mastodon.social/@joshmillard/106109652170356976">Josh Millard plays with algorithmically-generated pen-plotter art</a>; <a href="https://www.patreon.com/posts/50677186">more</a>.</p>
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<p><a href="https://www.flyingcoloursmaths.co.uk/a-pretty-puzzle/">You could prove that the number of integer solutions to \(x^2+xy+y^2=a\) is a multiple of six for positive \(a\) by finding a hidden group structure</a> (<a href="https://mathstodon.xyz/@11011110/106113101236984677">\(\mathbb{M}\)</a>). Or, you could recognize that it’s the norm of the Eisenstein integers under a small change of basis from the usual one and that they have six-fold rotational symmetry.</p>
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<p><a href="https://www.origamitessellations.com/2018/03/paper-engineering-from-the-bauhaus-josef-albers-to-the-modern-day/">Paper engineering from the Bauhaus</a> and <a href="https://www.origamitessellations.com/2018/04/reverse-engineering-bauhaus-paper-designs-part-two/">reverse-engineering Bauhaus paper designs</a> (<a href="https://mathstodon.xyz/@11011110/106118829181433597">\(\mathbb{M}\)</a>). These designs are more curved kirigami than origami, producing smooth-looking 3d shapes from cut sheets of flat paper.</p>
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<p><a href="https://www.scientificamerican.com/article/the-art-of-mathematics-in-chalk/">The art of mathematics in chalk</a> (<a href="https://mathstodon.xyz/@11011110/106124866673951925">\(\mathbb{M}\)</a>, <a href="https://whatsonmyblackboard.wordpress.com/">see also</a>). Teaser for the forthcoming book <em>Do Not Erase: Mathematicians and Their Chalkboards</em>, featuring several photographic spreads of chalkboard illustrations and formulas and their explanations. They appear to be mostly set-ups rather than captured from active research, but still pretty and interesting. <a href="/blog/2019/09/30/linkage.html">I linked an earlier post on this in 2019</a> but with fewer photos and no explanations.</p>
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<p><a href="https://coq.discourse.group/t/renaming-coq/1264">The Coq theorem prover brainstorms a name change</a> (<a href="https://mathstodon.xyz/@11011110/106127971601789143">\(\mathbb{M}\)</a>, <a href="https://www.metafilter.com/191240/Not-every-woman-is-offended-by-this-name-but-enough-people-are">via</a>), after too many women get harrassed for saying they work on Coq.</p>
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<p><a href="https://en.wikipedia.org/wiki/Tetrad_(geometry_puzzle)">Tetrad puzzle</a> (<a href="https://mathstodon.xyz/@11011110/106136227486814000">\(\mathbb{M}\)</a>). It’s possible to arrange four congruent hexagons so they tile a disk with each pair sharing a length of boundary, but the known pentagons with four pairwise-touching copies leave a hole in the region they tile. Is the hole necessary?</p>
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<p><a href="https://github.andrewt.net/mercator-rotator/">Mercator Rotator</a> (<a href="https://mastodon.social/@andrewt/105950778379233419">\(\mathbb{M}\)</a>), a tool for drawing Mercator-projection world maps with different viewpoints than the usual one. Set the pole on a place you don’t like to see the map of a world without it.</p>
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<p><a href="https://mathoverflow.net/q/138752/440">Tetrahedra passing through a hole</a> (<a href="https://mathstodon.xyz/@11011110/106152970915208525">\(\mathbb{M}\)</a>). This is from eight years ago, but was active again recently. The question is: what’s the smallest-area hole in a plane through which you can push a unit tetrahedron? DPKR has a very pretty animated answer, but sadly it’s not optimal: there’s a triangular hole with smaller area \(1/\sqrt{8}\), known <a href="https://doi.org/10.1016/j.comgeo.2011.07.004">minimal for translational motion</a>. The problem for more general motion seems to be still open.</p>
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<p><a href="http://arxiv.org/abs/1112.4205v2">Lagarias’s survey on the Takagi Function</a> and <a href="https://www.jstor.org/stable/2324028">Mallows’ survey on Conway’s $10,000 sequence</a> (<a href="https://mathstodon.xyz/@11011110/106153079186577081">\(\mathbb{M}\)</a>, <a href="https://mathstodon.xyz/@esoterica/106152848486538030">via</a>, <a href="https://en.wikipedia.org/wiki/Blancmange_curve">see also</a>) have very similar-looking figures, but little or no overlap in references. Maybe someone knows of an explanation for the similarity?</p>
</li>
</ul>David EppsteinThe EFF on FLoC (\(\mathbb{M}\)), Google’s plan for browsers to aggregate your browsing habits and make them public for ad-personalization. Short summary: it’s a bad idea and if you care about privacy you should switch to a non-Chrome browser. Technical summary: it’s based on k-anonymity, known as inadequate at protecting individual privacy in social networks. If you use Chrome, assume all bad guys on the web can see all your browsing.How good is greed for the no-three-in-line problem?2021-04-28T18:23:00+00:002021-04-28T18:23:00+00:00https://11011110.github.io/blog/2021/04/28/how-good-greed<p>The <a href="http://eurocg21.spbu.ru/">37th European Workshop on Computational Geometry (EuroCG 2021)</a> was earlier this month, but its <a href="http://eurocg21.spbu.ru/wp-content/uploads/2021/04/proceedings.pdf">book of abstracts</a> remains online. This has an odd position in the world of academic publishing: the “abstracts” are really short papers, so it looks a lot like a published conference proceedings. However, it declares that you should really pretend that it’s not a proceedings, in order to allow the same work to go on to another conference with a published proceedings, getting around the usual prohibitions on double publication. Instead, its papers “should be considered a preprint rather than a formally reviewed paper”. But I think that doesn’t preclude citing them, with care, just as you might occasionally cite arXiv preprints. The workshop’s lack of peer review and selectivity is actually a useful feature, allowing it to act as an outlet for works that are too small or preliminary for publication elsewhere. In North America, the <a href="http://cccg.ca/">Canadian Conference on Computational Geometry</a> performs much the same role, but does publish a proceedings; its <a href="https://projects.cs.dal.ca/cccg2021/the-call-for-papers-is-out/">submission deadline</a> is rapidly approaching.</p>
<p>Anyway, one of the EuroCG not-really-a-published-paper things is mine: “Geometric dominating sets – A minimum version of the no-three-in-line problem”, with Oswin Aichholzer and Eva-Maria Hainzl. As the title suggests, it’s related to the <a href="https://en.wikipedia.org/wiki/No-three-in-line_problem">no-three-in-line problem</a>, in which one must place as many points as possible in a grid so that no three are collinear. I’ve written about the same problem here <a href="/blog/2018/11/10/random-no-three.html">several</a> <a href="/blog/2018/11/12/gurobi-vs-no.html">times</a> <a href="/blog/2018/12/08/general-position-hypercube.html">already</a>. On an \(n\times n\) grid, there’s an easy upper bound of \(2n\) on the number of points, but it’s widely conjectured that the actual number is a smaller linear function of \(n\). It was a big step forward when Erdős showed that \(n\bigl(1-o(1)\bigr)\) points can be placed, and this was later improved to \(\tfrac{3}{2}n\bigl(1-o(1)\bigr)\).</p>
<p>These big no-three-in-line sets are constructed algebraically, but what if we try something simpler, a greedy algorithm that just adds points one by one (in a random or systematic order) until getting stuck? This question was already asked in the 1970s by Martin Gardner, and studied by several other authors since. But it is, if anything, even more frustratingly unknown than the no-three-in-line problem itself. We don’t know whether, in general, it’s possible to get stuck with fewer points than the maximum solution to the no-three-in-line problem, or even whether it’s possible to get stuck with fewer than \(2n\) points for infinitely many values of \(n\). For some values of \(n\) we do know smaller stuck solutions, though: for instance, here’s one with \(28\) points on a \(36\times 36\) grid.</p>
<p style="text-align:center"><img src="/blog/assets/2021/greedy-no3-36x36.svg" alt="A 28-point greedy solution to the no-three-in-line problem on a 36x36 grid" /></p>
<p>It was known that greedy solutions always have \(\Omega(\sqrt{n})\) points, and one of our main results is to improve this bound to \(\Omega(n^{2/3})\). The known \(\Omega(\sqrt{n})\) lower bound is easy to see: A single line through two selected points can cover at most \(n\) other grid points, so you need \(n\) lines to cover the whole grid, and you need \(\Omega(\sqrt{n})\) points to determine this many lines. With fewer points, there won’t be enough lines through your points to cover the whole grid, and your greedy solution won’t be stuck. Our new \(\Omega(n^{2/3})\) bound looks more carefully at the tradeoff between numbers of lines and numbers of points per line. It can be divided into two cases:</p>
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<p>Suppose, first, that the selected point set has the property that, for any selected point \(p\), the lines through \(p\) cover fewer than \(n^{4/3}\) grid points. Because each selected point covers few grid points, we need to select many points to cover the whole grid: at least \(\Omega(n^{2/3})\) points.</p>
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<p>Suppose on the other hand that the lines through some point \(p\) cover at least \(n^{4/3}\) grid points. Parameterize these lines by the \(L_\infty\) distance to the closest grid point (regardless of whether that point is one of the selected ones). Then there are \(O(k)\) lines with parameter \(k\), each of which covers \(O(n/k)\) grid points. Summing over small values of \(k\) shows that, even if we use lines that cover as many grid points as possible, we need \(\Omega(n^{2/3})\) lines through \(p\) to cover this many grid points. Each of these lines is determined by another selected point, so we need \(\Omega(n^{2/3})\) selected points.</p>
</li>
</ul>
<p>The actual proof in the paper takes into account that not all the grid points near \(p\) are the nearest on their line, and does the summation over small values of \(k\) more carefully, to get more precise constant factors in the bounds. Our paper also includes another variation of the problem in which we allow our selected points to be collinear but require the lines through them to cover all unselected points. There, we can make a little progress: we show that \(n\) points, or in some cases slightly fewer than \(n\) points, are sufficient. The same \(\Omega(n^{2/3})\) lower bound is still valid for this case, but there’s still a big gap between the lower bound and the upper bound.</p>
<p>(<a href="https://mathstodon.xyz/@11011110/106146843522019241">Discuss on Mastodon</a>)</p>David EppsteinThe 37th European Workshop on Computational Geometry (EuroCG 2021) was earlier this month, but its book of abstracts remains online. This has an odd position in the world of academic publishing: the “abstracts” are really short papers, so it looks a lot like a published conference proceedings. However, it declares that you should really pretend that it’s not a proceedings, in order to allow the same work to go on to another conference with a published proceedings, getting around the usual prohibitions on double publication. Instead, its papers “should be considered a preprint rather than a formally reviewed paper”. But I think that doesn’t preclude citing them, with care, just as you might occasionally cite arXiv preprints. The workshop’s lack of peer review and selectivity is actually a useful feature, allowing it to act as an outlet for works that are too small or preliminary for publication elsewhere. In North America, the Canadian Conference on Computational Geometry performs much the same role, but does publish a proceedings; its submission deadline is rapidly approaching.Pick’s shoelaces2021-04-17T23:23:00+00:002021-04-17T23:23:00+00:00https://11011110.github.io/blog/2021/04/17/picks-shoelaces<p>Two important methods for computing area of polygons in the plane are <a href="https://en.wikipedia.org/wiki/Pick%27s_theorem">Pick’s theorem</a> and the <a href="https://en.wikipedia.org/wiki/Shoelace_formula">shoelace formula</a>. For a simple lattice polygon (a polygon with a single non-crossing boundary cycle, all of whose vertex coordinates are integers) with \(i\) integer points in its interior and \(b\) on the boundary, Pick’s theorem computes the area as</p>
\[A=i+b/2-1.\]
<p>The shoelace formula has various formulations, but in the version I’m going to use, it is a sum over oriented edges of the polygon. Let \((x,y)\to (x',y')\) denote an edge of the polygon that connects the two points \((x,y)\) and \((x',y')\), oriented in the clockwise direction around the polygon so that if you travel from \((x,y)\) to \((x',y')\) along this line segment, then the interior of the polygon will be on your right. Then, according to the shoelace formula, the area is</p>
\[A= \frac{1}{2}\sum_{(x,y)\to (x',y')}(x'-x)(y'+y).\]
<p>The shoelace formula looks messier than Pick’s formula but it’s much easier for computers to evaluate, both because polygons are generally represented in computers by their boundary rather than by the points they contain, and because there are often many fewer boundary edges than interior lattice points. It also doesn’t need the coordinates to be integers. But since these two formulas produce the same value, it would be interesting to see a more direct relation between them, explaining one formula in terms of the other.</p>
<h1 id="proof-of-the-shoelace-formula">Proof of the shoelace formula</h1>
<p>It is convenient to shift the polygon to lie entirely above the \(x\)-axis, without changing its area or number of grid points. After this shift, the nonzero terms \(\tfrac{1}{2}(x'-x)(y'+y)\) of the shoelace formula can be interpreted as the signed areas of trapezoids, extending vertically down from each non-vertical edge \((x,y)\to (x',y')\) to the \(x\)-axis. The reason for doing this shift is to orient all of the trapezoids the same way, downwards from their edges, and to avoid complications with edges that cross the \(x\)-axis.</p>
<p style="text-align:center"><img src="/blog/assets/2021/pick-trapezoids.svg" alt="Trapezoids extending downward from each edge of a polygon" /></p>
<p>Now consider what you would see from a generic point in the upper half-plane, if you looked straight upwards. (By “generic”, I mean that the point isn’t on an edge or directly below a vertex, because these would introduce additional and unimportant cases to our analysis.) Your line of site would pass through a sequence of zero or more polygon edges, whose trapezoids have positive sign when the line passes from inside the polygon to outside and negative sign when the line passes from outside to inside. Because of this alternation between inside and outside, between positive and negative, a point outside the polygon sees equally many edges of each sign, but a point in the polygon sees one more positive edge than negative.</p>
<p>Another way of expressing the same counts of signed edges above each point involves the <em>characteristic functions</em> of the polygon and the trapezoids, functions that are 1 inside each shape and zero outside it. The area of a shape is just the integral of its characteristic function. We’ll leave these functions undefined on the boundary of the shapes, but that’s ok because the boundary points contribute nothing to the total area. Then because of the cancellation between positive and negative signs, at the generic points where all of these functions are defined, the characteristic function of the polygon equals the signed sum of characteristic functions of trapezoids. By the sum rule for integrals, the area of the polygon equals the signed sum of trapezoid areas. With a little more complication, the same argument can also be made to work directly for unshifted polygons.</p>
<h1 id="counting-lattice-points-in-trapezoids">Counting lattice points in trapezoids</h1>
<p>Can we use the same argument to prove Pick’s theorem, in the form that the shoelace formula equals Pick’s formula? We’d like to interpret each term of the shoelace formula as a count over grid points, decompose the polygon in the same way into a sum of trapezoids, and argue that counting points in each trapezoid and then summing produces the same result as summing the trapezoids and then counting points. The difficulty is that now we can’t ignore the points on the axis or on boundary of the trapezoids, because their contribution to the count is nonzero.</p>
<p>First, let’s see how we can interpret each shoelace term as a grid point count.
Rotating a trapezoid around the midpoint of its defining edge produces another trapezoid whose union with the first trapezoid is an axis-aligned rectangle. Pick’s theorem is easy to see for these rectangles, in the simplified form that the area is the sum of one unit for integer points inside the rectangle, half a unit for integer points on its edges, and a quarter of a unit for integer points at its vertices. These fractional numbers of units are merely the amounts of rectangle area nearest to each point.</p>
<p style="text-align:center"><img src="/blog/assets/2021/pick-rectangle.svg" alt="The area of a lattice trapezoid equals the sum of units of lattice points: a whole unit for points in the trapezoid, half a unit on the boundary, and a quarter unit for the four corners" /></p>
<p>For the trapezoids we will use the same assignment of units to lattice points: one unit for points in the trapezoid, half on its boundary, and a quarter at its vertices, regardless of the actual trapezoid angles at those vertices. The figure above shows each point decorated in blue with its number of units.
The equality of area with total units still holds true, because both the area of the trapezoid and the total number of units for its integer points are half that of the rectangle. Each point off the edge that contributes its units to the trapezoid has a reflection that does not contribute its units. And each point on the edge contributes half its units to the trapezoid and half to the reflection. So the same terms \(\tfrac{1}{2}(x'-x)(y'+y)\) of the shoelace formula also count the contributions of units in each trapezoid to Pick’s formula.</p>
<h1 id="picks-formula-from-sums-of-trapezoids">Pick’s formula from sums of trapezoids</h1>
<p>We know how much each lattice point contributes to Pick’s formula: one unit if it is inside the polygon, half if it is on the boundary, or zero if it is outside. We also know how much each lattice point contributes to each trapezoid of the shoelace formula: one unit if it is interior to the trapezoid, half if it is on an edge, a quarter if it is on a corner, and none if it is exterior. In order to prove that Pick’s formula and the shoelace formula are equal, we want to show that all points make equal contributions. And for most points, this turns out to be true.</p>
<p>The vertical ray from any point \(p\) may pass through the boundary of the polygon in multiple ways: \(p\) itself may lie on the boundary, the ray may cross an edge of the boundary, it may pass through a vertex of the boundary that has edges to its left and right, or it may brush past the boundary at a vertex that has edges only to the left or only to the right. We can skip over these last “brush past” cases, because the two trapezoids from these edges have opposite signs and cancel each other in the total trapezoid contribution of \(p\). If \(p\) lies on an edge, then it gets a (positive or negative) contribution of \(1/2\), and when \(p\) is a vertex of the polygon with edges extending left and right, then it gets the same contribution in two quarters. The remaining cases make a contribution of \(\pm 1\), and as for the area calculation they alternate in sign. Canceling these alternating contributions shows that \(p\) always has a total contribution from its trapezoids equal to its contribution to Pick’s formula.</p>
<p>I thought at first that the points on the \(x\)-axis might need a different calculation, because they get only half as much from each trapezoid. But half zero is still zero; they contribute nothing to the sum of trapezoids and nothing to the Pick formula. The points that do need special treatment are the polygon vertices at which the two incident edges do not extend to the left and right, either because one is vertical or because both extend in the same direction. The contribution to Pick’s formula for these vertices is still \(1/2\), but the total contribution from the trapezoids is either an integer (if the incident edges extend in the same direction) or a quarter-integer (if one edges is vertical). So we’ll have to count how much we’ve missed at those points.</p>
<p>Consider driving around the polygon clockwise, in the same direction that we oriented the edges. As you drive, you can make a right turn (from rightward to vertically down, vertically down to leftward, etc), a double-right u-turn, a left turn, or a double-left turn. Points at which the polygon makes an angle but continues in the same general left-to-right or right-to-left direction don’t count as turns. Then at each right turn Pick’s formula will run a deficit of 1/4 unit in total contributions, compared to the sum of trapezoids. A double-right u-turn gives a deficit of 1/2 unit, the same as two right turns. A left turn gives 1/4 more unit to Pick than to the sum of trapezoids, and a double-left gives 1/2 more unit. To return to your starting direction, you must make four more right turns than left, so the total difference in contributions from these terms comes out to exactly the \(-1\) correction term in Pick’s formula.</p>
<p style="text-align:center"><img src="/blog/assets/2021/pick-deficits.svg" alt="Differences in contributions at turning points of the e polygon" /></p>
<p>In summary, when we sum contributions over all points, the points inside the polygon contribute one unit to Pick’s formula and one unit to the sum of trapezoids. The points outside the polygon contribute zero to both. The points on the boundary that are not turns contribute \(1/2\) to both. And the points on the boundary that are turns contribute different amounts to Pick’s formula and to the sum of trapezoids, with the total of these differences equalling the \(-1\) correction term in Pick’s formula. Therefore, the overall value of Pick’s formula equals the sum of point contributions in trapezoids, which equals the signed sum of trapezoid areas, which equals the polygon area.</p>
<h1 id="generalizations-of-picks-formula">Generalizations of Pick’s formula</h1>
<p>The same idea lets us generalize Pick’s formula to a polygon with holes, defined as a connected region of the plane whose boundary is a disjoint union of simple polygons. The shoelace formula for area in the version we’re using here, and its proof, need no change for this generalization. For each hole, we should drive counterclockwise rather than clockwise, to stay consistent with the orientation we have given the edges; the same argument shows that each hole produces a positive, rather than negative, total difference between the contributions to Pick’s formula and to the shoelace formula. Therefore, for a polygon with \(h\) holes, the total area becomes</p>
\[A=i+b/2+h-1.\]
<p>The basic idea for all this, by the way, comes from the paper “Pick’s theorem”, by Branko Grünbaum and G. C. Shephard (<a href="https://doi.org/10.2307/2323771"><em>Amer. Math. Monthly</em> 1993</a>). Rather than using the trapezoid version of the shoelace formula, Grünbaum and Shephard use a version of the formula that sums, over each edge of the polygon, the signed area of the triangle formed by each edge plus the origin. I think this makes the analysis a little messier, but it’s otherwise much the same as the analysis here. The formula for polygons with holes can be found in “On the compactness of subsets of digital pictures” by Sankar and Krishnamurthy (<a href="https://doi.org/10.1016/s0146-664x(78)80021-5"><em>CGIP</em> 1978</a>) but without the careful definition of polygons with holes that is necessary to avoid problems here.</p>
<p>Grünbaum and Shephard also generalize Pick’s formula in a slightly different direction: rather than allowing separate holes in their polygons, they define polygons to have only one boundary polygon, but they allow that polygon to cross itself. Their version of Pick’s theorem for this kind of generalized polygons sums a certain half-integral index of each lattice point (essentially, the winding number of the polygon around that point, averaged between open and closed versions of the polygon), with a correction term coming from the turning number of the whole polygon. It should be possible to combine both generalizations, and allow polygon boundaries that both cross and have multiple components. There’s a little ambiguity about which way the boundary is connected at vertices of degree higher than two, though, which would need to be resolved somehow if such vertices are to be allowed, because different choices are likely to cause pieces of the boundary to have different orientations leading to different areas.</p>
<p>(<a href="https://mathstodon.xyz/@11011110/106084926459254807">Discuss on Mastodon</a>)</p>David EppsteinTwo important methods for computing area of polygons in the plane are Pick’s theorem and the shoelace formula. For a simple lattice polygon (a polygon with a single non-crossing boundary cycle, all of whose vertex coordinates are integers) with \(i\) integer points in its interior and \(b\) on the boundary, Pick’s theorem computes the area asLinkage2021-04-15T22:15:00+00:002021-04-15T22:15:00+00:00https://11011110.github.io/blog/2021/04/15/linkage<ul>
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<p><a href="https://en.wikipedia.org/wiki/Keller%27s_conjecture">Keller’s conjecture</a> (<a href="https://mathstodon.xyz/@11011110/105994491983819823">\(\mathbb{M}\)</a>), another new Good Article on Wikipedia. The conjecture was falsified in 1992 with all remaining cases solved by 2019, but the name stuck. It’s about tilings of \(n\)-space by unit cubes, and pairs of cubes that share \((n-1)\)-faces. In 2d, all squares share an edge with a neighbor, but a 3d tiling derived from <a href="https://en.wikipedia.org/wiki/Tetrastix">tetrastix</a> has many cubes with no face-to-face neighbor. Up to 7d, some cubes must be face-to-face, but tilings in eight or more dimensions can have no face-to-face pair.</p>
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<p><a href="https://lucatrevisan.wordpress.com/2021/04/02/bocconi-hired-poorly-qualified-computer-scientist/">Italians and bibliometrics</a> (<a href="https://mathstodon.xyz/@11011110/105996551762053680">\(\mathbb{M}\)</a>): Luca Trevisan (a leading theorist with 7 SODA papers, 2 FOCS papers, a JACM paper and a SICOMP paper in the last four years) gets dinged for poor productivity as the Italian system only counts journal papers that do not match conference papers. The fact that these are all in top venues is irrelevant, and the conference papers count only negatively against matching journal papers. Comments discuss similar problems in other countries.</p>
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<p><a href="https://www.scottaaronson.com/blog/?p=5402">What is the computational complexity of dinosaur train tracks?</a> (<a href="https://mathstodon.xyz/@11011110/106009429519024889">\(\mathbb{M}\)</a>). Answer: not very high, because the only usable junction, a Y that remembers which way you came through it and sends you the same way if you come back through the other direction, is just not powerful enough to do much.</p>
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<p>Congratulations to Martín Farach-Colton, Shang-Hua Teng, and all of the other <a href="https://sinews.siam.org/Details-Page/siam-announces-class-of-2021-fellows">new SIAM Fellows</a> (<a href="https://mathstodon.xyz/@11011110/106016827432463028">\(\mathbb{M}\)</a>)!</p>
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<p><a href="https://writings.stephenwolfram.com/2021/03/a-little-closer-to-finding-what-became-of-moses-schonfinkel-inventor-of-combinators/">Stephen Wolfram tries to track down</a> what happened to logician <a href="https://en.wikipedia.org/wiki/Moses_Sch%C3%B6nfinkel">Moses Schönfinkel</a> (<a href="https://mathstodon.xyz/@11011110/106025064389253132">\(\mathbb{M}\)</a>, <a href="https://news.ycombinator.com/item?id=26694685">via</a>), who worked in Göttingen from 1914 to 1924, returned to Moscow, and then “basically vanished”. Wikipedia has more detail about what happened after (mental health issues, death around 1942), but Wolfram says the evidence for all that is weak. He doesn’t make direct progress on Schönfinkel himself but does find some relatives.</p>
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<p><a href="https://aperiodical.com/2021/04/my-robot-draws-tex/">How Christian Lawson-Perfect got a pen plotter to draw mathematical notation using TeX</a> (<a href="https://mathstodon.xyz/@christianp/106030474747952758">\(\mathbb{M}\)</a>).</p>
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<p><a href="https://www.improbable.com/2021/04/08/a-look-way-back-at-some-bearded-mathematicians/">When mathematicians wore geometric beards</a> (<a href="https://mathstodon.xyz/@11011110/106034106736789855">\(\mathbb{M}\)</a>).</p>
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<p><a href="https://www.scottaaronson.com/blog/?p=5437">Aaronson on politicization of research prizes</a> (<a href="https://mathstodon.xyz/@11011110/106039607813461033">\(\mathbb{M}\)</a>). Jeff Ullman won the Turing Award despite deplorable (some say racist) treatment of grad applicants for the crime of being Iranian, and Oded Goldreich was blocked from the Israel Prize for anti-settlement politics. Politicization is two-edged. I’d rather see Ullman awarded for his worthy contributions, and use the opportunity to decry his abhorrent actions and statements, than subject prizes to litmus tests from all sides.</p>
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<p><a href="https://thatsmaths.com/2021/04/08/circles-polygons-and-the-kepler-bouwkamp-constant/">Circles, polygons and the Kepler-Bouwkamp constant</a> (<a href="https://mathstodon.xyz/@11011110/106045568373359503">\(\mathbb{M}\)</a>). On the limiting behavior of infinitely-nested shapes alternating between circles and polygons with increasing numbers of sides.</p>
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<p><a href="https://www.technologyreview.com/2021/04/09/1022217/facebook-ad-algorithm-sex-discrimination">Continuing gender bias in who sees job-opening ads on Facebook</a> (<a href="https://mathstodon.xyz/@11011110/106051162687419819">\(\mathbb{M}\)</a>, <a href="https://news.ycombinator.com/item?id=26760790">via</a>): if an employer or industry has historically skewed male or female, Facebook replicates that bias, even for pairs of ads with identical qualifications. This is illegal, but Facebook appears unable to find a technical fix and unwilling to apply the obvious fix of not targeting its ads even when that targeting is illegal. As usual for Hacker News via links on topics related to social justice, don’t read the comments there.</p>
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<p>Amusing quote from McLarty’s 2003 “<a href="http://www.landsburg.com/grothendieck/mclarty1.pdf">Grothendieck on simplicity and generality</a>” (<a href="https://mathstodon.xyz/@11011110/106059214777848356">\(\mathbb{M}\)</a>, <a href="https://golem.ph.utexas.edu/category/2021/04/algebraic_closure.html">via</a>): “Serre created a series of concise elegant tools which Grothendieck and coworkers simplified into thousands of pages of category theory.” Nowadays I guess the people doing this sort of simplification are the ones formulating machine-verifiable proofs…</p>
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<p><a href="https://projects.cs.dal.ca/wads2021/wads-2021-accepted-papers/">WADS 2021 accepted papers</a> (<a href="https://mathstodon.xyz/@11011110/106062479142848045">\(\mathbb{M}\)</a>).
The biennial Algorithms and Data Structures Symposium is usually in Canada, and this time was supposed to be in Halifax, but is looking very likely to be completely online, this August. I have one paper on the list; I’ll write more about it later when I have a preprint version ready to share.</p>
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<p><a href="https://www.youtube.com/watch?v=S5fPwE7GQOA">A “confounding topological curiosity”</a> (<a href="https://mathstodon.xyz/@11011110/106068163014347342">\(\mathbb{M}\)</a>, <a href="https://boingboing.net/2021/04/13/heres-a-confounding-topological-curiosity.html">via</a>): a double torus with a line through one of its holes can be continuously transformed so that the line instead goes through both holes.</p>
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<p><a href="https://arxiv.org/abs/2102.11818">Strange inverses in the group rings of torsion-free groups</a> (<a href="https://mathstodon.xyz/@11011110/106072203556292078">\(\mathbb{M}\)</a>, <a href="https://www.quantamagazine.org/mathematician-disproves-group-algebra-unit-conjecture-20210412/">via</a>, <a href="https://www.uni-muenster.de/MathematicsMuenster/news/artikel/2021-03-04.shtml">see also</a>). This result of Giles Gardam disproves the strongest of the three <a href="https://en.wikipedia.org/wiki/Kaplansky%27s_conjectures">Kaplansky conjectures on group rings</a>. It’s just an isolated example at this point but it does show that group rings are less well-behaved than had been hoped.</p>
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</ul>David EppsteinKeller’s conjecture (\(\mathbb{M}\)), another new Good Article on Wikipedia. The conjecture was falsified in 1992 with all remaining cases solved by 2019, but the name stuck. It’s about tilings of \(n\)-space by unit cubes, and pairs of cubes that share \((n-1)\)-faces. In 2d, all squares share an edge with a neighbor, but a 3d tiling derived from tetrastix has many cubes with no face-to-face neighbor. Up to 7d, some cubes must be face-to-face, but tilings in eight or more dimensions can have no face-to-face pair.Islands2021-04-02T16:34:00+00:002021-04-02T16:34:00+00:00https://11011110.github.io/blog/2021/04/02/islands<p>In the neighborhood where I live, fire safety regulations require the streets to be super-wide (so wide that two fire trucks can pass even with cars parked along both sides of the street), and to have even wider turnarounds at the ends of the culs-de-sac. To break up the resulting vast expanses of pavement, we have occasional islands of green, public gardens too small to name as a park. They come in several different types: medians to separate the incoming and outgoing lanes at junctions with larger roads,</p>
<p style="text-align:center"><img src="https://www.ics.uci.edu/~eppstein/pix/islands/Frost-m.jpg" alt="Traffic island at Frost and Gabrielino, Irvine" style="border-style:solid;border-color:black;" /></p>
<p>barriers to separate small groups of houses from the main flow of the road,</p>
<p style="text-align:center"><img src="https://www.ics.uci.edu/~eppstein/pix/islands/LosTrancos-m.jpg" alt="Traffic island on Los Trancos Drive, Irvine" style="border-style:solid;border-color:black;" /></p>
<p>or giving some shape to the turnaround at the end of a cul-de-sac.</p>
<p style="text-align:center"><img src="https://www.ics.uci.edu/~eppstein/pix/islands/Harvey-m.jpg" alt="Traffic island on Harvey Court, Irvine" style="border-style:solid;border-color:black;" /></p>
<p>Most are ignored except by the community association’s gardeners and by passing cars, but I did find one set up as a neighborhood basketball court:</p>
<p style="text-align:center"><img src="https://www.ics.uci.edu/~eppstein/pix/islands/Perkins-m.jpg" alt="Traffic island on Perkins Court, Irvine" style="border-style:solid;border-color:black;" /></p>
<p><a href="https://www.ics.uci.edu/~eppstein/pix/islands/">The rest of the gallery</a>.</p>
<p>(<a href="https://mathstodon.xyz/@11011110/105998376302792384">Discuss on Mastodon</a>)</p>David EppsteinIn the neighborhood where I live, fire safety regulations require the streets to be super-wide (so wide that two fire trucks can pass even with cars parked along both sides of the street), and to have even wider turnarounds at the ends of the culs-de-sac. To break up the resulting vast expanses of pavement, we have occasional islands of green, public gardens too small to name as a park. They come in several different types: medians to separate the incoming and outgoing lanes at junctions with larger roads,Linkage2021-03-31T17:54:00+00:002021-03-31T17:54:00+00:00https://11011110.github.io/blog/2021/03/31/linkage<p>As is often the case, not all of these are links; they’re copies of posts that I made over on my Mastodon account because they were too short to make full posts here.</p>
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<p><a href="https://en.wikipedia.org/wiki/Borromean_rings">Borromean rings</a> (<a href="https://mathstodon.xyz/@11011110/105898319690639869">\(\mathbb{M}\)</a>), now a Good Article on Wikipedia. You can’t make the Borromean rings with geometric circles: as Tverberg observed, an inversion would make one of them a line. But then each of the other two circles would span an angle less than \(\pi\) as viewed from the line, leaving an unspanned direction along which the line could escape to infinity, contradicting the inseparability of the rings.</p>
<p>The same proof shows that <a href="https://en.wikipedia.org/wiki/Brunnian_link">Brunnian links</a> cannot be made from four geometric circles. But the original proof that Borromean rings are not circular, by <a href="https://doi.org/10.4310/jdg/1214440725">Freedman and Skora</a> using 4d hyperbolic geometry, applies more generally to any Brunnian link no matter how many components it has.</p>
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<p><a href="https://twitter.com/mathpuzzle/status/1366756737882218500">Squares can be divided into \(45^\circ\)–\(60^\circ\)–\(75^\circ\) triangles</a> (<a href="https://mathstodon.xyz/@11011110/105909672693909508">\(\mathbb{M}\)</a>), as Ed Pegg posts on twitter. This is an interesting variant on something I looked at a long time ago, <a href="https://www.ics.uci.edu/~eppstein/junkyard/acute-square/">triangulating the square to minimize the maximum angle</a>. But Pegg’s subdivision is not edge-to-edge. And the triangles on my page are not all similar. Do squares have edge-to-edge triangulations by similar acute triangles?</p>
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<p><a href="https://blogs.sciencemag.org/editors-blog/2021/02/18/a-new-name-change-policy/"><em>Science</em> joins other publishers in making it easy for authors to change their names on past publications</a> (<a href="https://mathstodon.xyz/@11011110/105915141700667095">\(\mathbb{M}\)</a>, <a href="https://www.metafilter.com/190813/Making-it-easier-for-published-scientists-to-change-their-names">via</a>). Transgender scientists made this change necessary but I think it benefits all of us. I’ve seen plenty of name changes in the literature for other reasons (marriage, divorce, escaping prejudice, …) and making it easier to find your old papers is generally a good thing.</p>
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<p>Cairo in Denmark? (<a href="https://mathstodon.xyz/@11011110/105920191343711124">\(\mathbb{M}\)</a>). This is a screenshot from Google Maps of Havnepromenade in Copenhagen, where it meets the end of Cort Adelers Gade. I’m sad to have missed this spot when I was in Copenhagen a couple years ago; it’s a nice example of a <a href="https://en.wikipedia.org/wiki/Cairo_pentagonal_tiling">Cairo pentagonal tiling</a>.</p>
<p style="text-align:center"><img src="/blog/assets/2021/Cairo-in-Copenhagen.jpg" alt="Pentagonal street pavers on Havnepromenade at Cort Adelers Gade in Copenhagen, taken as a screenshot from Google Maps" style="border-style:solid;border-color:black;" width="80%" /></p>
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<p>My latest illustration trick: make complex non-Boolean combinations of shapes by overlaying simple shapes, exploding the overlay into simple regions, and unioning the regions into the shapes I want (<a href="https://mathstodon.xyz/@11011110/105929236692195138">\(\mathbb{M}\)</a>). Exploding and unioning are single clicks in Illustrator’s pathfinder tool. I used two levels of this technique to make this diagram of a <a href="https://en.wikipedia.org/wiki/Brunnian_link">Brunnian link</a>: once to make the shape of a single component, and again to make the over-under relation between components.</p>
<p style="text-align:center"><img src="/blog/assets/2021/six-rubberband-link.svg" alt="Six-rubberband link" /></p>
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<p><a href="https://mattwthomas.com/blog/induction-on-reals/">Real induction</a> (<a href="https://mathstodon.xyz/@mwt/105862096547025004">\(\mathbb{M}\)</a>). An induction principle for proving statements about all real numbers in intervals.</p>
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<p><a href="https://www.patreon.com/posts/new-print-49071226">Recamán Sequence</a> (<a href="https://mastodon.social/@joshmillard/105935104495594723">\(\mathbb{M}\)</a>). Linocut art by Josh Millard in a style drawn from a <em>Numberphile</em> video.</p>
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<p><a href="https://www.quantamagazine.org/mathematicians-inch-closer-to-matrix-multiplication-goal-20210323/"><em>Quanta</em> surveys recent developments in fast matrix multiplication</a> (<a href="https://mathstodon.xyz/@11011110/105943717367375547">\(\mathbb{M}\)</a>), including <a href="https://arxiv.org/abs/2010.05846">a paper from last SODA by Josh Alman and Virginia Williams</a> improving the exponent from 2.3728639 to 2.3728596. Known barriers to the widely-expressed hope that the exponent can be reduced to 2 (see the SODA paper’s introduction) are, I think too specific to be convincing. On the other hand at this rate there’s still a long way from here to 2…</p>
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<p><a href="https://fredrikj.net/blog/2021/03/printing-algebraic-numbers/">Printing algebraic numbers</a> (<a href="https://mathstodon.xyz/@11011110/105946537048589616">\(\mathbb{M}\)</a>, <a href="https://news.ycombinator.com/item?id=26566090">via</a>). So your code can handle symbolic representations of exact algebraic numbers rather than just approximating everything as complex/float. How do you get those representations out to human users in a readable way? From Fredrik Johansson, long-time developer of open symbolic algebra and exact real arithmetic software including SymPy and Calcium.</p>
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<p><a href="https://en.wikipedia.org/wiki/Fermat%27s_right_triangle_theorem">Fermat’s right triangle theorem</a> (<a href="https://mathstodon.xyz/@11011110/105951398938627194">\(\mathbb{M}\)</a>), now a Good Article on Wikipedia. As Fibonacci observed and Fermat proved (as his only surviving proof), integer right triangles cannot have an area that is a perfect square. One corollary of this is the exponent-4 case of Fermat’s Last Theorem.</p>
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<p><a href="https://mastodon.social/@tomharris/105956972687307734">Tom Harris asks: What’s your favorite OEIS sequence?</a> Mine for this week at least is the <a href="https://en.wikipedia.org/wiki/Moser%E2%80%93De_Bruijn_sequence">Moser–De Bruijn sequence</a>, OEIS <a href="https://oeis.org/A000695">A000695</a>.</p>
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<p>The most recent image of me on my site was from 2009 and getting pretty out-of-date. So I took a screenshot of myself as I look these days on Zoom (<a href="https://mathstodon.xyz/@11011110/105964230654539136">\(\mathbb{M}\)</a>). This is far from the first time I’ve let my hair go at least this long, but I’m still looking forward to a time when I will feel more comfortable getting a haircut.
It occurs to me that there has been a reversal: now, instead of <a href="https://en.wikipedia.org/wiki/Almost_Cut_My_Hair">showing off rebelliousness</a>, I am visibly conforming to safety rules.</p>
<p style="text-align:center"><img src="/blog/assets/2021/zoom.jpg" alt="David Eppstein, self-portrait, Zoom screenshot" style="border-style:solid;border-color:black;" width="80%" /></p>
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<p><a href="https://www.eff.org/deeplinks/2021/03/free-climbing-rock-climbers-open-data-project-threatened-bogus-copyright-claims">Fight over open data in rock climbing</a> (<a href="https://mathstodon.xyz/@11011110/105971892719302236">\(\mathbb{M}\)</a>, <a href="https://news.ycombinator.com/item?id=26609945">via</a>).</p>
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<p><a href="https://mathoverflow.net/q/387543/440">Weird probability distributions on dyadic rationals from a simple averaging process</a> (<a href="https://mathstodon.xyz/@11011110/105977625108931414">\(\mathbb{M}\)</a>). Start with the two-element multiset \(\{0,1\}\), repeatedly draw two-element samples (without replacement) from the multiset and include the average into the multiset. The result tends to cluster around some value, not the same value every time but itself drawn from a bimodal distribution, with the clustering sort of looking Cauchy but not quite. Interesting new MathOverflow question.</p>
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<p><a href="https://blog.computationalcomplexity.org/2021/03/slicing-hypercube.html">How many hyperplanes does it take to slice all the edges of an -dimensional hypercube</a> (<a href="https://mathstodon.xyz/@11011110/105982764812531388">\(\mathbb{M}\)</a>)? Two if you can skim the ends of the edges or include entire edges in hyperplanes but more to really slice them. The obvious answer of axis-parallel cuts can be improved to \(5n/6\), and Lance Fortnow reports that the newest lower bound is \(\Omega(n^{0.57})\), by Gal Yehuda and Amir Yehudayoff, improving <a href="https://arxiv.org/abs/2102.05536">their own new preprint</a>.</p>
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<p><a href="https://rjlipton.wpcomstaging.com/2021/03/30/all-the-news-that-fits-we-print/">A list of current blogs on computer theory and related math</a> (<a href="https://mathstodon.xyz/@11011110/105986862218758838">\(\mathbb{M}\)</a>), from <em>Gödel’s Lost Letter and P=NP</em>.</p>
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</ul>David EppsteinAs is often the case, not all of these are links; they’re copies of posts that I made over on my Mastodon account because they were too short to make full posts here.Linkage for the Ides of March2021-03-15T18:08:00+00:002021-03-15T18:08:00+00:00https://11011110.github.io/blog/2021/03/15/linkage-ides-march<ul>
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<p>I am quite amused by the description of “<a href="https://golem.ph.utexas.edu/category/2020/12/the_algebraic_ktheory_of_the_i.html">some unimpressed Wikipedia editor</a>” (<a href="https://mathstodon.xyz/@11011110/105819260898827516">\(\mathbb{M}\)</a>) who didn’t think the evidence for or against the <a href="https://en.wikipedia.org/wiki/Kummer%E2%80%93Vandiver_conjecture">Kummer–Vandiver conjecture</a> in algebraic number theory is particularly strong, but I am unable to explain why it is so amusing. (No, it wasn’t me.)</p>
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<p>Yuansi Chen’s preprint, “<a href="https://arxiv.org/abs/2011.13661">An Almost Constant Lower Bound of the Isoperimetric Coefficient in the KLS Conjecture</a>” is described less technically by <a href="https://gilkalai.wordpress.com/2020/12/21/to-cheer-you-up-in-difficult-times-15-yuansi-chen-achieved-a-major-breakthrough-on-bourgains-slicing-problem-and-the-kannan-lovasz-and-simonovits-conjecture/">Gil Kalai</a> and now <a href="https://www.quantamagazine.org/statistics-postdoc-tames-decades-old-geometry-problem-20210301/"><em>Quanta</em></a> (<a href="https://mathstodon.xyz/@11011110/105821929361050909">\(\mathbb{M}\)</a>). It doesn’t quite solve the Bourgain slicing conjecture, that high-dimensional convex bodies of unit volume have cross-sections of constant volume, but it reduces the dependence on dimension from a power of \(d\) to something smaller.</p>
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<p><em>Inside Higher Ed</em> on <a href="https://www.insidehighered.com/news/2021/03/02/criminal-initiative-targeting-scholars-who-allegedly-hid-chinese-funding-and">arrests of US-based Chinese researchers for supposed instances of grant fraud</a> (<a href="https://mathstodon.xyz/@11011110/105830435111652902">\(\mathbb{M}\)</a>) that turn out to mean participating in official university-level collaborations with Chinese universities, taking notes at someone else’s talk, and serving as a peer reviewer: “Is it making major cases out of minor issues? Is it ethnic profiling?” The answers seem obvious.</p>
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<p><a href="https://www.thisiscolossal.com/2021/03/create-escape/">Oscar Wilde wrote <em>The Ballad of Reading Gaol</em> while staying with Bob Ross</a> (<a href="https://mathstodon.xyz/@11011110/105835524336926029">\(\mathbb{M}\)</a>, <a href="https://www.metafilter.com/190681/Create-Escape">also</a>, <a href="https://boingboing.net/2021/03/05/fantastic-new-banksy-video-narrated-by-bob-ross.html">also</a>), as I learned from this Banksy video.</p>
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<p><a href="https://en.wikipedia.org/wiki/De_quinque_corporibus_regularibus"><em>De quinque corporibus regularibus</em></a> (<a href="https://mathstodon.xyz/@11011110/105840200590322853">\(\mathbb{M}\)</a>). A 15th-century book on the mathematics of polyhedra by painter Piero della Francesca, lost for centuries except through a plagiarized translation by Luca Paciola, and rediscovered in the Vatican Library in the 19th century. Its contributions include the volume of bicylinders (independent of previous work by Archimedes and Zu Chongzhi) and a novel formula for the height of an irregular tetrahedron. Now a Good Article on Wikipedia.</p>
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<p><a href="https://twitter.com/wtgowers/status/1367401773355442180">British Universities are in negotiations with Elsevier</a> (<a href="https://mathstodon.xyz/@11011110/105847024238576398">\(\mathbb{M}\)</a>). Elsevier claim that because they have added many new open-access journals, there is more content to justify higher subscription prices. (Yes, that is blatant double-dipping.) The University of California system hasn’t been a subscriber to Elsevier since the end of 2018 and <a href="https://dailybruin.com/2020/02/27/ucs-termination-of-elsevier-contract-has-had-limited-negative-impact">we’ve been doing fine without</a>.</p>
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<p><a href="https://news.ycombinator.com/item?id=26325464">Building mechanical computers out of graphene nano-origami</a> (<a href="https://mathstodon.xyz/@11011110/105853362934281331">\(\mathbb{M}\)</a>, <a href="https://news.ycombinator.com/item?id=26325464">via</a>).</p>
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<p><a href="https://www.nytimes.com/2021/03/08/science/math-crumple-fragmentation-andrejevic.html">The latest wrinkle in crumple theory</a> (<a href="https://mathstodon.xyz/@11011110/105864188929091620">\(\mathbb{M}\)</a>, <a href="https://www.metafilter.com/190702/To-my-unfolding-lend-your-prosperous-ear">via</a>). Siobhan Roberts in the <em>NYT</em> on how paper crumples, based on <a href="https://www.nature.com/articles/s41467-021-21625-2">a new study in <em>Nature Communications</em> led by Jovana Andrejevic</a>. It builds on previous studies showing logarithmic growth of total crease length on repeated re-crumpling, but gains more insight by looking at the distribution of sizes of unfolded areas, and the ways those areas fragment into smaller areas, rather than just their total perimeter.</p>
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<p>Since I saw it <a href="/blog/2020/03/22/uci-ecological-preserve.html">a year ago</a> and again this week, I wondered: how can I see Los Angeles from my neighborhood, 65km away? (<a href="https://mathstodon.xyz/@11011110/105869692112016165">\(\mathbb{M}\)</a>) Doesn’t the earth’s curvature get in the way? Isn’t the horizon typically 5km away?</p>
<p>It isn’t refraction.</p>
<p><a href="https://en.wikipedia.org/wiki/University_Hills,_Irvine">I live on a hill</a>. Many LA skyscrapers are <a href="https://en.wikipedia.org/wiki/Bunker_Hill,_Los_Angeles">on another hill</a>. Both are around 100m high, and the land between is low and flat. That turns out to be almost exactly the right altitude for mutual visibility, even though a single hill would have to be 400m high to see as far.</p>
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<p><a href="https://news.ycombinator.com/item?id=26390040">ISO obstructs adoption of standards by paywalling them</a> (<a href="https://mathstodon.xyz/@11011110/105875625075011578">\(\mathbb{M}\)</a>); in particular <a href="https://news.ycombinator.com/item?id=26400089">the ISO 8601 date format standard is not public</a>. This makes these standards problematic when individuals, rather than deep-pocketed corporations, need to follow them. <a href="https://tools.ietf.org/html/rfc3339">RFC 3339</a> could be used in place of ISO 8601 in many uses, but is too restricted for some (can’t handle BC). Are there other good alternatives?</p>
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<p><a href="https://en.wikipedia.org/wiki/Wikipedia:Sockpuppet_investigations/Badtoothfairy">A large sockpuppet ring has been caught adding citations of Stephen Wolfram’s works to Wikipedia</a> (<a href="https://mathstodon.xyz/@11011110/105879489521944799">\(\mathbb{M}\)</a>). A typical addition: <a href="https://en.wikipedia.org/w/index.php?title=Riemann_hypothesis&type=revision&diff=1011413368&oldid=1009272577">equivalence of the Riemann hypothesis to a tag system</a>, a trivial corollary of Minsky’s proof of universality of tag systems. They’ve been going at it for years, so it will take significant effort to clean up. Technical information that might point to who did this (e.g. an overenthusiastic fan or corporate publicists) is private.</p>
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<p><a href="https://rjlipton.wordpress.com/2021/03/13/hilberts-tenth-again/">Hilbert’s Tenth again</a> (<a href="https://mathstodon.xyz/@11011110/105884148771833431">\(\mathbb{M}\)</a>). Matiyasevich famously proved undecidability of Diophantine equations over integers in the 1970s, but the same question over rationals remains open. A new paper by Prunescu makes progress by proving undecidability of exponential equations over rationals. As Richard Lipton explains in this blog post, a key step, parameterization by integers of rational solutions to \(x^y=y^x\), comes from a 30-year-old recreational-mathematics article by <a href="https://en.wikipedia.org/wiki/M%C3%A1rta_Sv%C3%A9d">Márta Svéd</a>.</p>
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<p><a href="https://scholar.archive.org/">Internet Archive Scholar</a> (<a href="https://idf.social/@arjen/105867224421425143">\(\mathbb{M}\)</a>), a new full-text database of “over 25 million research articles and other scholarly documents preserved in the Internet Archive”.</p>
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<p><a href="https://dianadavis.github.io/im/book.pdf">Sample pages from the gorgeous-looking new book <em>Illustrating Mathematics</em>, edited by Diana Davis</a> (<a href="https://mathstodon.xyz/@11011110/105896775391586479">\(\mathbb{M}\)</a>, <a href="https://www.maa.org/press/maa-reviews/illustrating-mathematics">via</a>).</p>
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</ul>David EppsteinI am quite amused by the description of “some unimpressed Wikipedia editor” (\(\mathbb{M}\)) who didn’t think the evidence for or against the Kummer–Vandiver conjecture in algebraic number theory is particularly strong, but I am unable to explain why it is so amusing. (No, it wasn’t me.)More mathematics books by women2021-03-08T18:28:00+00:002021-03-08T18:28:00+00:00https://11011110.github.io/blog/2021/03/08/more-mathematics-books<p>A year ago, for International Women’s Day, I made <a href="/blog/2020/03/08/mathematics-books-women.html">a list of mathematics books by women covered by then-new Wikipedia articles</a>. I thought it would be worthwhile to revisit the same topic and list several more mathematics books with at least one female author, at many different levels of audience, and again covered by new Wikipedia articles. They are (alphabetical by title):</p>
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<p><em><a href="https://en.wikipedia.org/wiki/Algorithmic_Combinatorics_on_Partial_Words">Algorithmic Combinatorics on Partial Words</a></em> (2008), Francine Blanchet-Sadri. Partial words are strings with “don’t care” symbols; Blanchet-Sadri looks at the combinatorics of repeated patterns within these strings.</p>
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<p><em><a href="https://en.wikipedia.org/wiki/Algorithmic_Geometry">Algorithmic Geometry</a></em> (1995), Jean-Daniel Boissonnat and Mariette Yvinec. One of several standard computational geometry textbooks; this is the French one, but it has also been published in translation into English.</p>
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<p><em><a href="https://en.wikipedia.org/wiki/Algorithmic_Puzzles">Algorithmic Puzzles</a></em> (2011), Anany and Maria Levitin. A nice collection of classic logic puzzles involving algorithmic thinking.</p>
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<p><em><a href="https://en.wikipedia.org/wiki/Braids,_Links,_and_Mapping_Class_Groups">Braids, Links, and Mapping Class Groups</a></em> (1975), Joan Birman. A classic research monograph on the topology of braid groups.</p>
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<p><em><a href="https://en.wikipedia.org/wiki/Code_of_the_Quipu">Code of the Quipu</a></em> (1981), Marcia and Robert Ascher. A general-audience book on how the Inca used knotted strings to record numbers and other information.</p>
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<p><em><a href="https://en.wikipedia.org/wiki/Combinatorics:_The_Rota_Way">Combinatorics: The Rota Way</a></em> (2009), Joseph P. S. Kung, Catherine Yan, and (posthumously) Gian-Carlo Rota. A graduate textbook on algebraic combinatorics.</p>
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<p><em><a href="https://en.wikipedia.org/wiki/Combinatorics_of_Experimental_Design">Combinatorics of Experimental Design</a></em> (1987), Anne Penfold Street and her daughter Deborah Street. A textbook on the design of experiments, an area that crosses between statistics and combinatorics.</p>
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<p><em><a href="https://en.wikipedia.org/wiki/Computability_in_Analysis_and_Physics">Computability in Analysis and Physics</a></em> (1989), Marian Pour-El and J. Ian Richards. A research monograph on problems involving differential equations including the wave equation whose initial conditions are continuous and computable, but that evolve to states whose values cannot be computed.</p>
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<p><em><a href="https://en.wikipedia.org/wiki/Diophantus_and_Diophantine_Equations">Diophantus and Diophantine Equations</a></em> (1972), Isabella Bashmakova. A somewhat idiosyncratic history based on the idea that Diophantus knew some very general techniques for finding rational-number solutions to equations, that can be inferred from the much more specific solutions to individual equations that have survived to us.</p>
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<p><em><a href="https://en.wikipedia.org/wiki/Elementary_Number_Theory,_Group_Theory_and_Ramanujan_Graphs">Elementary Number Theory, Group Theory, and Ramanujan Graphs</a></em> (2003), Giuliana Davidoff, Peter Sarnak, and Alain Valette. An attempt to make the construction of expander graphs accessible to undergraduate mathematics students.</p>
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<p><em><a href="https://en.wikipedia.org/wiki/Equivalents_of_the_Axiom_of_Choice">Equivalents of the Axiom of Choice</a></em> (1963, updated 1985), Herman and Jean Rubin. A large catalog of problems in mathematics whose solution is equivalent to the axiom of choice, from a time when the independence of choice from ZF set theory had not been proven.</p>
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<p><em><a href="https://en.wikipedia.org/wiki/Erd%C5%91s_on_Graphs">Erdős on Graphs: His Legacy of Unsolved Problems</a></em> (1998),
Fan Chung and Ronald Graham. The open problems in graph theory from this book have been further collected and updated on a web site, <a href="http://www.math.ucsd.edu/~erdosproblems/">Erdős’s Problems on Graphs</a>, maintained by Chung.</p>
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<p><em><a href="https://en.wikipedia.org/wiki/Extensions_of_First_Order_Logic">Extensions of First Order Logic</a></em> (1996), María Manzano. Attempts to unify second-order logic, modal logic, and dynamic logic, by translating them all into many-sorted logic.</p>
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<p><em><a href="https://en.wikipedia.org/wiki/Fat_Chance:_Probability_from_0_to_1">Fat Chance: Probability from 0 to 1</a></em> (2019), Benedict Gross, Joe Harris, and Emily Riehl. A general-audience undergraduate textbook on probability theory based on a metaphor of games of chance.</p>
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<p><em><a href="https://en.wikipedia.org/wiki/The_Fractal_Dimension_of_Architecture">The Fractal Dimension of Architecture</a></em> (2016), Michael J. Ostwald and Josephine Vaughan. Studies the fractal dimension of floor plans as a way to model the changing demands on the complexity of housing structures and to classify buildings by architect and style.</p>
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<p><em><a href="https://en.wikipedia.org/wiki/The_Geometry_of_Numbers">The Geometry of Numbers</a></em> (2000), Carl D. Olds, Anneli Cahn Lax, and Giuliana Davidoff. A textbook on connections between number theory and integer grids, rescued twice from the posthumous works of its first two coauthors.</p>
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<p><em><a href="https://en.wikipedia.org/wiki/The_History_of_Mathematical_Tables">The History of Mathematical Tables: from Sumer to Spreadsheets</a></em> (2003), Martin Campbell-Kelly, Mary Croarken, Raymond Flood, and Eleanor Robson. An edited volume with chapters on tables from many different periods in mathematical history.</p>
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<p><em><a href="https://en.wikipedia.org/wiki/Incidence_and_Symmetry_in_Design_and_Architecture">Incidence and Symmetry in Design and Architecture</a></em> (1983), Jenny Baglivo and Jack E. Graver. A textbook on graph theory and symmetry aimed at architecture students, also including interesting material on structural rigidity.</p>
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<p><em><a href="https://en.wikipedia.org/wiki/Introduction_to_the_Theory_of_Error-Correcting_Codes">Introduction to the Theory of Error-Correcting Codes</a></em> (1982, updated 1989 and 1998), Vera Pless. An advanced undergraduate textbook centered on algebraic constructions of linear block codes.</p>
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<p><em><a href="https://en.wikipedia.org/wiki/Introduction_to_3-Manifolds">Introduction to 3-Manifolds</a></em> (2014), Jennifer Schultens. An introductory graduate textbook on low-dimensional topology, leading up to the use of normal surfaces and Heegard splittings.</p>
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<p><em><a href="https://en.wikipedia.org/wiki/Journey_into_Geometries">Journey into Geometries</a></em> (1991), Márta Svéd. A conversational Alice-in-wonderland-inspired tour of non-Euclidean geometry.</p>
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<p><em><a href="https://en.wikipedia.org/wiki/Knots_Unravelled">Knots Unravelled: From String to Mathematics</a></em> (2011), Meike Akveld and Andrew Jobbings. Knot theory for schoolchildren, centered on knot invariants.</p>
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<p><em><a href="https://en.wikipedia.org/wiki/Lectures_in_Geometric_Combinatorics">Lectures in Geometric Combinatorics</a></em> (2006), Rekha R. Thomas. An advanced undergraduate or introductory graduate textbook on the combinatorics of convex polytopes and their connections to abstract algebra through secondary polytopes and toric varieties.</p>
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<p><em><a href="https://en.wikipedia.org/wiki/Making_Mathematics_with_Needlework">Making Mathematics with Needlework: Ten Papers and Ten Projects</a></em> (2008), sarah-marie belcastro and Carolyn Yackel. The projects come from eight different contributors and include photos, instructions, mathematical analyses, and teaching activities.</p>
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<p><em><a href="https://en.wikipedia.org/wiki/Mathematical_Excursions">Mathematical Excursions: Side Trips along Paths Not Generally Traveled in Elementary Courses in Mathematics</a></em> (1933), Helen Abbot Merrill. An early book on recreational mathematics, aimed at getting high school students interested in mathematics.</p>
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<p><em><a href="https://en.wikipedia.org/wiki/Mathematics_in_India">Mathematics in India: 500 BCE–1800 CE</a></em> (2009), Kim Plofker. Organized chronologically, this has become the standard overview of this large topic. It also includes material on the history of astronomy in India, which was often tied to the mathematics of its era.</p>
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<p><em><a href="https://en.wikipedia.org/wiki/The_Mathematics_of_Chip-Firing">The Mathematics of Chip-Firing</a></em> (2018), Caroline Klivans. A textbook on chip-firing games and abelian sandpile models.</p>
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<p><em><a href="https://en.wikipedia.org/wiki/Markov_Chains_and_Mixing_Times">Markov Chains and Mixing Times</a></em> (2009, 2017), David A. Levin and Yuval Peres, with contributions by Elizabeth Wilmer. A graduate-level text and research reference on how quickly random walks converge to their stable distributions.</p>
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<p><em><a href="https://en.wikipedia.org/wiki/Mirrors_and_Reflections">Mirrors and Reflections: The Geometry of Finite Reflection Groups</a></em> (2009), Alexandre V. and Anna Borovik. An undergraduate textbook on the classification of finite reflection groups and their associated root systems.</p>
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<p><em><a href="https://en.wikipedia.org/wiki/Pioneering_Women_in_American_Mathematics">Pioneering Women in American Mathematics: The Pre-1940 PhD’s</a></em> (2009), Judy Green and Jeanne LaDuke. Biographical profiles of over 200 women who earned doctorates in mathematics in the US before 1940, with some background material on what it was like for women to work in mathematics in those times.</p>
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<p><em><a href="https://en.wikipedia.org/wiki/Playing_with_Infinity">Playing with Infinity: Mathematical Explorations and Excursions</a></em> (1955, translated into English 1961), Rózsa Péter. An attempt to explain the nature of mathematics and of the infinite in mathematics to non-mathematicians, based on a series of letters from Péter to a literary friend.</p>
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<p><em><a href="https://en.wikipedia.org/wiki/Point_Processes">Point Processes</a></em> (1980), David Cox and Valerie Isham. A research reference on processes that randomly place points on the real line or other geometric spaces.</p>
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<p><em><a href="https://en.wikipedia.org/wiki/Power_in_Numbers:_The_Rebel_Women_of_Mathematics">Power in Numbers: The Rebel Women of Mathematics</a></em> (2018), Talithia Williams. A selection of profiles of famous women mathematicians, aimed at motivating young women to become mathematicians.</p>
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<p><em><a href="https://en.wikipedia.org/wiki/Primality_Testing_for_Beginners">Primality Testing for Beginners</a></em> (2009, translated into English 2014), Lasse Rempe-Gillen and Rebecca Waldecker. An undergraduate text on primality testing algorithms, based on a course from a summer research program for undergraduates.</p>
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<p><em><a href="https://en.wikipedia.org/wiki/Quantum_Computing:_A_Gentle_Introduction">Quantum Computing: A Gentle Introduction</a></em> (2011), Eleanor Rieffel and Wolfgang Polak. One of many texts on this fast-moving subject.</p>
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<p><em><a href="https://en.wikipedia.org/wiki/Robust_Regression_and_Outlier_Detection">Robust Regression and Outlier Detection</a></em> (1987), Peter Rousseeuw and Annick M. Leroy. A monograph on statistical methods that can tolerate the total corruption of a large fraction of the data points that they analyze, and still produce meaningful results.</p>
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<p><em><a href="https://en.wikipedia.org/wiki/Two-Sided_Matching">Two-Sided Matching: A Study in Game-Theoretic Modeling and Analysis</a></em> (1990), Alvin E. Roth and Marilda Sotomayor. A survey of methods related to stable matching, aimed at economics practitioners and focused on applications.</p>
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<p><em><a href="https://en.wikipedia.org/wiki/When_Topology_Meets_Chemistry">When Topology Meets Chemistry: A Topological Look At Molecular Chirality</a></em> (2000), Erica Flapan. Many biomolecules are different than their mirror images; classical examples include sugars, whose mirrored molecules may taste different and have different effects. This undergraduate-level text studies how to model this effect using a combination of graph theory and knot theory.</p>
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<p><em><a href="https://en.wikipedia.org/wiki/Women_in_Mathematics">Women in Mathematics</a></em> (1974), Lynn Osen. This is the one that based its coverage of Hypatia on an early-20th-century children’s book that gave her a made-up backstory and attributed made-up modern rationalist quotes to her. Not recommended, and included mainly as a warning not to use this as a reference.</p>
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</ul>
<p>To keep from ending on a sour note, I’ll add one more, that I found recently on Wikipedia (although the article there is very old) and I think is worthy of expansion: <em><a href="https://en.wikipedia.org/wiki/Logic_Made_Easy">Logic Made Easy: How to Know When Language Deceives You</a></em> (2004), Deborah J. Bennett, a popular-audience book on how to translate English phrases into logical formalisms and use that translation to understand more clearly what they mean.</p>
<p>(<a href="https://mathstodon.xyz/@11011110/105857580884627445">Discuss on Mastodon</a>)</p>David EppsteinA year ago, for International Women’s Day, I made a list of mathematics books by women covered by then-new Wikipedia articles. I thought it would be worthwhile to revisit the same topic and list several more mathematics books with at least one female author, at many different levels of audience, and again covered by new Wikipedia articles. They are (alphabetical by title):