Jekyll2020-09-16T05:18:05+00:00https://11011110.github.io/blog/feed.xml11011110Geometry, graphs, algorithms, and moreDavid EppsteinLinkage2020-09-15T22:15:00+00:002020-09-15T22:15:00+00:00https://11011110.github.io/blog/2020/09/15/linkage<ul>
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<p><a href="https://www.quantamagazine.org/mathematicians-report-new-discovery-about-the-dodecahedron-20200831/">Closed quasigeodesics on the dodecahedron</a> (<a href="https://mathstodon.xyz/@11011110/104785420838924796">\(\mathbb{M}\)</a>), paths that start at a vertex and go straight across each edge until coming back to the same vertex from the other side. Original paper, <a href="https://arxiv.org/abs/1811.04131">arXiv:1811.04131</a>, <a href="https://doi.org/10.1080/10586458.2020.1712564">doi:10.1080/10586458.2020.1712564</a>. I saw this on Numberphile a few months back (video linked in article) but now it’s on <em>Quanta</em>.</p>
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<p><a href="https://blog.graphicine.com/lorenz-stoer-geometric-landscapes/">Lorenz Stöer’s geometric landscapes</a> (<a href="https://mathstodon.xyz/@11011110/104799765760054680">\(\mathbb{M}\)</a>). <a href="/blog/2014/09/30/linkage-for-end.html">In 2014 I linked a different page</a> with a few of Stöer’s 16th-century proto-surrealist combinations of landscape and geometry, but they were black and white. This one has more of them, in color.</p>
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<p><a href="https://en.wikipedia.org/wiki/Ideal_polyhedron">Ideal polyhedron</a>, a polyhedron in hyperbolic space with all vertices at infinity, and <a href="https://en.wikipedia.org/wiki/Sylvester%E2%80%93Gallai_theorem">Sylvester–Gallai theorem</a>, that every finite set of points in the Euclidean plane has a line that either passes through all of them or through exactly two of them. Both newly promoted to Good Article status on Wikipedia (<a href="https://mathstodon.xyz/@11011110/104803212564257211">\(\mathbb{M}\)</a>).</p>
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<p><a href="https://merveilles.town/@neauoire/104779168858836970">Escherian wiener-dog Cerberus fetches three impossible things</a>.</p>
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<p>Flamebait post of the day: <a href="http://nautil.us/issue/89/the-dark-side/why-mathematicians-should-stop-naming-things-after-each-other">Why mathematicians should stop naming things after each other</a> (<a href="https://mathstodon.xyz/@11011110/104813883920721252">\(\mathbb{M}\)</a>, <a href="https://news.ycombinator.com/item?id=24385389">via</a>). For once the via-link discussion is worth reading (main point: the alternative, using common English words to describe specialized technical concepts, can be even more confusing).</p>
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<p>Early Renaissance painter Piero della Francesca was also an accomplished mathematician, and his book on polyhedra, <em>De quinque corporibus regularibus</em> (subject of a <a href="https://en.wikipedia.org/wiki/De_quinque_corporibus_regularibus">new Wikipedia article</a>; <a href="https://mathstodon.xyz/@11011110/104820183749646319">\(\mathbb{M}\)</a>) has an interesting history that deserves to be better known. Rediscovery of the mathematics of Archimedes! “First full-blown case of plagiarism in the history of mathematics” (by Luca Pacioli, in Divina proportione)! Maybe owned by John Dee! Long lost and found centuries later in the Vatican Library!</p>
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<p><a href="https://cameroncounts.wordpress.com/2020/08/30/moonlighting/">Peter Cameron gives a nice roundup of two recent online conferences on group theory and combinatorics</a> (<a href="https://mathstodon.xyz/@11011110/104833470202099899">\(\mathbb{M}\)</a>) that he attended more-or-less simultaneously, something that would have been impossible for physical conferences. The parts on synchronizing automata and twin-width particularly caught my attention as stuff I should look up and find out more about.</p>
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<p><a href="https://twitter.com/RodBogart/status/455123609195802624">An hourglass that demonstrates Archimedes’ theorem that the volume of a cylinder is the sum of the volumes of its inscribed sphere and cone</a> (<a href="https://mathstodon.xyz/@mjd/104836143207567957">\(\mathbb{M}\)</a>), from Rod Bogart’s twitter feed.</p>
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<p>The <a href="/blog/2020/09/07/eberhards-theorem-bipartite.html">hexagon-minimizing simple bipartite polyhedra of my recent blog post</a> make nice shapes when converted to <a href="https://arxiv.org/abs/0912.0537">simple orthogonal polyhedra</a> (<a href="https://mathstodon.xyz/@11011110/104842837266010559">\(\mathbb{M}\)</a>): a squared-off amphitheater with L-shaped terraces of increasing length as they rise, or a diagonal staircase with congruent L-shaped steps. In each case the outer \(2n\)-gon is the underside of the polygon and the inner cycles are the horizontal faces.</p>
<p style="text-align:center"><img src="/blog/assets/2020/orthogonal-eberhard.svg" alt="Hexagon-minimizing simple bipartite polyhedra represented as simple orthogonal polyhedra" /></p>
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<p><a href="https://arxiv.org/abs/2008.11933">Open is not forever: a study of vanished open access journals</a> (<a href="https://mathstodon.xyz/@11011110/104850663521092974">\(\mathbb{M}\)</a>, <a href="https://news.ycombinator.com/item?id=24422593">via</a>, <a href="https://www.sciencemag.org/news/2020/09/dozens-scientific-journals-have-vanished-internet-and-no-one-preserved-them">via</a>). This study shows the need for systematic archiving and redundant copying of online open journals, but I suspect that the problem for small hand-run print-based journals without much library pickup might be much worse.</p>
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<p><a href="https://www.thisiscolossal.com/2018/10/a-prickly-structure-made-of-70000-reusable-hexapod-particles/">A prickly structure made of 70,000 reusable hexapod particles</a> (<a href="https://mathstodon.xyz/@11011110/104856358909259046">\(\mathbb{M}\)</a>). Sort of like those <a href="https://en.wikipedia.org/wiki/Tetrapod_(structure)">seawalls they build by jumbling together giant concrete caltrops</a>, only with pieces that are not quite so big and with usable spaces left void within it. Sometimes the article says “hexapod” and sometimes “decapod”; the pictures appear to show structures that mix two different kinds of particle.</p>
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<p><em><a href="http://math.sfsu.edu/beck/ct/board.php">Combinatorial Theory</a></em> (<a href="https://mathstodon.xyz/@bremner/104859257534118058">\(\mathbb{M}\)</a>, <a href="https://twitter.com/wtgowers/status/1305253478047068160">see also</a>), a new open-access combinatorics journal formed from the mass resignation of the Elsevier <em>JCTA</em> editorial board.</p>
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<p><a href="https://www.nytimes.com/2020/09/14/us/caputo-virus.html">Trump officials are now telling their supporters to buy guns and ammunition to use against scientists for being anti-Trump</a>. <a href="https://thehill.com/policy/healthcare/516319-top-hhs-official-accuses-scientists-of-plotting-against-trump-tells">No, seriously</a> (<a href="https://mathstodon.xyz/@11011110/104865069277930004">\(\mathbb{M}\)</a>).</p>
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<p><em><a href="https://www.cambridge.org/us/academic/subjects/mathematics/recreational-mathematics/origametry-mathematical-methods-paper-folding">Origametry: Mathematical Methods in Paper Folding</a></em> (<a href="https://mathstodon.xyz/@11011110/104870325812873444">\(\mathbb{M}\)</a>), new book coming out October 31 by Tom Hull. I haven’t seen anything more than the blurb linked here and the <a href="https://books.google.com/books?id=LdX7DwAAQBAJ">limited preview on Google Books</a>, but it looks interesting and worth waiting for.</p>
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</ul>David EppsteinClosed quasigeodesics on the dodecahedron (\(\mathbb{M}\)), paths that start at a vertex and go straight across each edge until coming back to the same vertex from the other side. Original paper, arXiv:1811.04131, doi:10.1080/10586458.2020.1712564. I saw this on Numberphile a few months back (video linked in article) but now it’s on Quanta.Eberhard’s theorem for bipartite polyhedra with one big face2020-09-07T22:28:00+00:002020-09-07T22:28:00+00:00https://11011110.github.io/blog/2020/09/07/eberhards-theorem-bipartite<p><a href="https://en.wikipedia.org/wiki/Eberhard%27s_theorem">Eberhard’s theorem</a> is a topic in the combinatorial theory of convex polyhedra that once saw a lot of research, but has faded from more recent interest. It’s named after <a href="https://en.wikipedia.org/wiki/Victor_Eberhard">Victor Eberhard</a>, a German mathematician from the late 19th and early 20th century who worked in geometry despite becoming blind at age 12 or 13. I find this hard to imagine, as my own research in geometry is based very heavily on visual thinking, but he was far from the only successful blind mathematician; <a href="https://en.wikipedia.org/wiki/Leonhard_Euler">Leonhard Euler</a>, <a href="https://en.wikipedia.org/wiki/Lev_Pontryagin">Lev Pontryagin</a>, and <a href="https://en.wikipedia.org/wiki/Bernard_Morin">Bernard Morin</a> also come to mind, and there are more.</p>
<p>Anyway, Eberhard’s theorem concerns the following question. Suppose I tell you that a polyhedron has a certain number of faces of certain types. For instance, after Archimedes’ work on polytopes was lost, all we knew about the <a href="https://en.wikipedia.org/wiki/Archimedean_solid">Archimedean solids</a> until their rediscovery in the Renaissance was a brief listing from <a href="https://en.wikipedia.org/wiki/Pappus_of_Alexandria">Pappus of Alexandria</a> giving this information: there is one with 8 triangles and 6 squares, etc. How can we tell that these counts of faces actually determine a polyhedron?</p>
<p>The given information for Eberhard’s theorem, then, is just a collection of counts of face types (triangles, quadrilaterals, etc.), without specifying the exact shapes of these faces. The goal is to use these faces to build a simple polyhedron, one for which three edges meet at every vertex (like a cube, unlike an octahedron). One necessary condition for this to be possible is that the polyhedron must obey Euler’s polyhedral formula \(v-e+f=2\). And it’s easy to calculate the numbers of vertices, edges, and faces appearing in this formula, from the face counts. Plugging these numbers into Euler’s formula leads to a linear equation that the face counts must obey. Crucially, this linear equation omits the count of hexagons: adding or removing hexagons will not change whether Euler’s formula holds. What Eberhard’s theorem states is that, as long as the face counts obey Euler’s formula in this way, there is always some number of hexagons that can be added or removed so that the remaining faces will form a polyhedron.</p>
<p>However, calculating the fewest number of hexagons needed, or even determining whether a given number of faces of all types (including hexagons) can be put together into a polyhedron, remains somewhat mysterious. So I thought I’d play with a case that would be both simple enough to solve and still interesting: the bipartite simple polyhedra (famous from <a href="https://en.wikipedia.org/wiki/Barnette%27s_conjecture">Barnette’s conjecture</a>), with one big face (a \(2n\)-gon for some \(n>3\)), many small faces (\(n+3\) quadrilaterals, the number needed to make Euler’s formula hold), and a mysterious number of hexagons. What is the smallest number of hexagons that will allow the construction of a simple polyhedron with these face counts? The answer turns out to be \(\lfloor (3n-6)/2\rfloor\), achieved with polyhedra (or polyhedral graphs) in which the outer \(2n\)-gon surrounds a <a href="https://en.wikipedia.org/wiki/Cactus_graph">cactus tree</a> of 6-vertex cycles (and possibly one 4-vertex cycle), connected to each other by bridge edges:</p>
<p style="text-align:center"><img src="/blog/assets/2020/eberhard.svg" alt="Three hexagon-minimal bipartite simple polyhedra" /></p>
<p>The central cactus tree can be rearranged, as long as no two bridge edges have adjacent endpoints. For instance, in the graph with the dodecagon outer face, at the bottom of the figure, it’s possible for the middle six-vertex loop to have three connections to the outside polygon on one side and only one connection on the other side, or to have the four-vertex loop in the middle. But I can prove that all optimal solutions have the same overall central cactus tree structure.</p>
<p>I find it easier to think about the following equivalent rephrasing of the optimization problem: instead of finding a minimum number of hexagons that will allow us to build a polyhedron with those face counts, let’s build a polyhedron with one \(2n\)-gon face and the rest quadrilaterals and hexagons, and concentrate on minimizing the number of vertices in this polyhedron. The number of quadrilaterals will automatically come out right, and the number of hexagons will be minimized if the number of vertices is minimized.</p>
<p>Now suppose that we have any simple polyhedron with one \(2n\)-gon face and the rest quadrilaterals and hexagons. Remove the outer \(2n\)-gon from the graph, leaving a conncted subgraph, and look at the biconnected components of this subgraph. For any one component, its outer face in its induced planar embedding must be a simple cycle, with some vertices having degree two in the component (the endpoints of edges connecting the component to the rest of the graph) and some having degree three. If the component is a 4-cycle or 6-cycle, then all of its vertices have degree two. But if not, then at most four consecutive vertices of its outer cycle can have degree two, because they and the two vertices connected to them on both sides form part of the boundary of a face interior to the component, which can have at most six vertices. And the degree-three vertices of the outer cycle must come in consecutive pairs, which cannot be adjacent to the endpoints of bridge edges connecting to other biconnected components, because a degree-three vertex next to a bridge edge or next to two other degree-three vertices would combine with part of the outer \(2n\)-gon to form a face with seven or more vertices, and a degree-three vertex by itself would form a pentagon, neither of which is allowed.</p>
<p>So in a component that is not a 4-cycle or 6-cycle, the degree-two and degree-three vertices alternate around the outer cycle of the component in consecutive sequences of at most four and exactly two vertices. This implies that the number of degree-two vertices is even (because the whole cycle is even by bipartiteness) and that the number of degree-three vertices in the component (even just counting the ones on its boundary) is at least half of the number of degree-two vertices on its boundary. For the cactus trees that we’ve been using, on the other hand, the number of degree-three vertices in each cactus tree is strictly less than half of the number of degree-two vertices. So if we replace a whole non-cycle component by a cactus tree, we can get a graph with the same number of exposed degree-2 vertices, but fewer total vertices. After repeated replacement of biconnected components, at each step reducing the number of vertices, we would reach a state where the subgraph inside the \(2n\)-gon is a cactus tree. It might not meet the requirement that its bridge edges have nonadjacent endpoints, but it could always be rearranged to do so. And it might not be a cactus with at most one 4-cycle, but if not we could replace two 4-cycles by one 6-cycle and make it even smaller. So the only graphs that cannot be made smaller are the ones we started with, the cactus trees of 6-cycles and at most one 4-cycle, surrounded by an outer \(2n\)-gon.</p>
<p>(<a href="https://mathstodon.xyz/@11011110/104827671950147352">Discuss on Mastodon</a>)</p>David EppsteinEberhard’s theorem is a topic in the combinatorial theory of convex polyhedra that once saw a lot of research, but has faded from more recent interest. It’s named after Victor Eberhard, a German mathematician from the late 19th and early 20th century who worked in geometry despite becoming blind at age 12 or 13. I find this hard to imagine, as my own research in geometry is based very heavily on visual thinking, but he was far from the only successful blind mathematician; Leonhard Euler, Lev Pontryagin, and Bernard Morin also come to mind, and there are more.Isosceles polyhedra2020-09-01T23:18:00+00:002020-09-01T23:18:00+00:00https://11011110.github.io/blog/2020/09/01/isosceles-polyhedra<p>My latest arXiv preprint is “On polyhedral realization with isosceles triangles”, <a href="http://arxiv.org/abs/2009.00116">arXiv:2009.00116</a>. As the title suggests, it studies polyhedra whose faces are all isosceles triangles. Despite several new results in it, there’s a lot I still don’t know. The paper finds a sort-of-new<sup id="fnref:1" role="doc-noteref"><a href="#fn:1" class="footnote">1</a></sup> infinite family of polyhedra with congruent isosceles faces, shown below, but I don’t know if there are any more such families.</p>
<p style="text-align:center"><img src="/blog/assets/2020/twisted.svg" alt="Twisted augmented bipyramid with isosceles-triangle faces" /></p>
<p>One of the other previously known families, shown below, has two integer parameters (the numbers of sides on the two half-bipyramids it combines), but I don’t know whether the same double parameterization is possible for the new family.</p>
<p style="text-align:center"><img src="/blog/assets/2020/biarc.svg" alt="Combination of two half-bipyramids with isosceles-triangle faces" width="80%" /></p>
<p>In 2001, Branko Grünbaum published an example of a polyhedron that could not be realized with congruent faces, even non-convexly,<sup id="fnref:2" role="doc-noteref"><a href="#fn:2" class="footnote">2</a></sup> but it can be realized as a convex polyhedron with all faces isosceles (or equilateral), as shown below. I don’t know whether there are polyhedra that cannot be realized with all faces isosceles, if one allows the realization to be non-convex (but non-self-crossing and combinatorially equivalent to a convex polyhedron) and the faces to be non-congruent.</p>
<p style="text-align:center"><img src="/blog/assets/2020/grunbaum.svg" alt="Grünbaum's example of a polyhedron that cannot be realized with congruent faces, realized convexly with isosceles and equilateral triangle faces" /></p>
<p>My new preprint proves that there exist polyhedra (iterated Kleetopes) that cannot be realized as convex polyhedra with isosceles faces. But the construction is a little non-explicit and I don’t know how complicated these polyhedra need to be. For instance, I don’t know whether there is a convex isosceles-face realization of the double Kleetope of the octahedron, shown below.</p>
<p style="text-align:center"><img src="/blog/assets/2020/kkoct.svg" alt="Double Kleetope of an octahedron" /></p>
<p>Grünbaum’s example can be realized convexly with only two edge lengths, and my non-isosceles-faced polyhedra require at least three edge lengths in any convex realization. I don’t know whether the number of required edge lengths can be unbounded, or whether non-convex realizations ever require three lengths (although certain stacked polyhedra require at least two).</p>
<div class="footnotes" role="doc-endnotes">
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<p>The family of polyhedra from the first image is only “sort-of-new” because the same combinatorial structure was previously described as a triangulation of the sphere by congruent spherical isosceles triangles: Dawson, Robert J. MacG. (2005), “<a href="https://archive.bridgesmathart.org/2005/bridges2005-489.html">Some new tilings of the sphere with congruent triangles</a>”, Renaissance Banff. In exchange for re-purposing Dawson’s triangulation, my paper describes another infinite family of spherical triangulations by congruent spherical isosceles triangles, not given by Dawson, based on applying a similar \(2\pi/3\) twist to an infinite family of non-convex bipyramids with congruent isosceles faces like the one below. Again, I don’t know whether there are other such families of spherical triangulations.</p>
<p style="text-align:center"><img src="/blog/assets/2020/nwb.svg" alt="Non-convex polyhedron with congruent isosceles-triangle faces" /> <a href="#fnref:1" class="reversefootnote" role="doc-backlink">↩</a></p>
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<p>Grünbaum, Branko (2001), “<a href="https://sites.math.washington.edu/~grunbaum/Nonequifacettablesphere.pdf">A convex polyhedron which is not equifacettable</a>”, <em>Geombinatorics</em> 10: 165–171. I don’t know how to access old papers on this journal in general, but fortunately Grünbaum made his one available on his web site. <a href="#fnref:2" class="reversefootnote" role="doc-backlink">↩</a></p>
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<p>(<a href="https://mathstodon.xyz/@11011110/104794000861684463">Discuss on Mastodon</a>)</p>David EppsteinMy latest arXiv preprint is “On polyhedral realization with isosceles triangles”, arXiv:2009.00116. As the title suggests, it studies polyhedra whose faces are all isosceles triangles. Despite several new results in it, there’s a lot I still don’t know. The paper finds a sort-of-new1 infinite family of polyhedra with congruent isosceles faces, shown below, but I don’t know if there are any more such families. The family of polyhedra from the first image is only “sort-of-new” because the same combinatorial structure was previously described as a triangulation of the sphere by congruent spherical isosceles triangles: Dawson, Robert J. MacG. (2005), “Some new tilings of the sphere with congruent triangles”, Renaissance Banff. In exchange for re-purposing Dawson’s triangulation, my paper describes another infinite family of spherical triangulations by congruent spherical isosceles triangles, not given by Dawson, based on applying a similar \(2\pi/3\) twist to an infinite family of non-convex bipyramids with congruent isosceles faces like the one below. Again, I don’t know whether there are other such families of spherical triangulations. ↩Linkage2020-08-30T17:47:00+00:002020-08-30T17:47:00+00:00https://11011110.github.io/blog/2020/08/30/linkage<ul>
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<p><a href="https://sinews.siam.org/Details-Page/the-evolution-of-mathematical-word-processing">The evolution of mathematical word processing</a> (<a href="https://mathstodon.xyz/@11011110/104701998974894412">\(\mathbb{M}\)</a>). “Developments in computing over the last 30 years have not done as much as one might have thought to make writing mathematics easier.”</p>
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<p><a href="https://prideout.net/blog/svg_wireframes/">3D Wireframes in SVG, via Python</a> (<a href="https://mathstodon.xyz/@11011110/104707509485635530">\(\mathbb{M}\)</a>, <a href="https://github.com/prideout/svg3d">code repository</a>). Once you generate the model, it does the rest. Its depth-ordering heuristics aren’t perfect (and will fail when the depth order is cyclic), but generally work pretty well. For example, here’s <a href="https://en.wikipedia.org/wiki/Jessen%27s_icosahedron">Jessen’s icosahedron</a> in a simple partially-transparent style without fancy shading (<a href="/blog/assets/2020/jessen.py">Python code</a>). To get it to look right I had to edit the resulting svg and manually reorder the faces.</p>
<p style="text-align:center"><img src="/blog/assets/2020/jessen.svg" alt="Jessen's icosahedron" /></p>
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<p><a href="https://www.insidehighered.com/news/2020/08/17/ip-grab-youngstown-state">Intellectual property grab at Youngstown State</a> (<a href="https://mathstodon.xyz/@11011110/104714502717608284">\(\mathbb{M}\)</a>). In negotiations with the faculty union, the university wants to replace the tradition of faculty holding copyright to research articles, textbooks, syllabi and lectures, etc by instead taking ownership of “all nonpatentable faculty work” as work-for-hire. They claim that they aren’t changing their policies and are merely doing this “to be consistent with the law” but this comes across as unlikely and disingenuous.</p>
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<p><a href="https://digital.lib.washington.edu/researchworks/bitstream/handle/1773/15700/Lost%20Mathematics.pdf?fterence=1">Lectures on lost mathematics</a> (<a href="https://mathstodon.xyz/@11011110/104718951109084030">\(\mathbb{M}\)</a>). In this 79-page pdf from 1975, Branko Grünbaum discusses mathematical questions studied by non-mathematicians but snubbed by pure mathematicians, including which polygons tile, the girth of infinitely-repeating cubic spatial graphs, the classification of vertex-transitive polyhedral manifolds in space, generalization of Kempe universality to surfaces, flexible polyhedra, and tensegrity.</p>
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<p><a href="https://xkcd.com/2348/">A recent xkcd on river crossing puzzles</a> particularly amused me (<a href="https://mathstodon.xyz/@11011110/104726172352360084">\(\mathbb{M}\)</a>).</p>
<p style="text-align:center"><img src="/blog/assets/2020/xkcd-2348.png" alt="xkcd comic "Boat Puzzle", https://xkcd.com/2348/" width="80%" /></p>
<p>It’s too bad xkcd uses an -NC clause in its CC license; if it didn’t, we could use it or its first frame to replace <a href="https://en.wikipedia.org/wiki/River_crossing_puzzle">the Wikipedia article’s illustration</a>, which is undergoing a long slow <a href="https://commons.wikimedia.org/wiki/Commons:Deletion_requests/File:Vovk_koza_kapusta.png">deletion discussion because copied from a 1954 Soviet book</a>. If you know something useful about the copyright status of 1954 Soviet books, please add your knowledge to that discussion.</p>
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<p><a href="https://www.flyingcoloursmaths.co.uk/eye-to-eye/">Eye to eye</a> (<a href="https://mathstodon.xyz/@11011110/104731872882549290">\(\mathbb{M}\)</a>). Let \(C\) and \(C'\) be circles with centers outside the other circle, and draw tangent rays from each center to the other circle. These rays cut their circles in chords of equal length. But I wonder: when only one circle has its center outside the other, its chord is still well defined, but the same length can’t be a chord of the other circle because it exceeds the diameter. But it should still have a geometric meaning with respect to the other circle: what is it?</p>
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<p><a href="https://petapixel.com/2020/08/20/lightroom-app-update-wipes-users-photos-and-presets-adobe-says-they-are-not-recoverable/">Lightroom App update irrecoverably loses users’ photos</a> (<a href="https://mathstodon.xyz/@11011110/104742916488291004">\(\mathbb{M}\)</a>, <a href="https://news.ycombinator.com/item?id=24229864">via</a>). My choice to continue using an oldish powerbook with dubiously-reliable keyboard, to avoid giving up my paid-for non-subscription Adobe apps which won’t run on newer Macs, has an unexpected benefit: my files haven’t been auto-deleted. Also, let this be a reminder to do your backups, and make sure that they include a local non-cloud backup of your files.</p>
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<p><a href="https://github.andrewt.net/mines/">Minesweeper where forced guesses are always safe, but unforced guesses always explode</a> (<a href="https://mastodon.technology/@andrewt/104701318997810776">\(\mathbb{M}\)</a>).</p>
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<p><a href="https://www.reddit.com/r/Scotland/comments/ig9jia/ive_discovered_that_almost_every_single_article/">“The Scots language version of Wikipedia is legendarily bad” — turns out because it was mostly written by an American teenager</a> (<a href="https://mathstodon.xyz/@11011110/104751072546716749">\(\mathbb{M}\)</a>, <a href="https://www.metafilter.com/188374/The-problem-is-that-this-person-cannot-speak-Scots">via</a>). See <a href="https://en.wikipedia.org/wiki/Wikipedia:Wikipedia_Signpost/2020-08-30/News_and_notes">the <em>Signpost</em> article</a> for an update.</p>
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<p>Today’s mini-episode of “not the Reuleaux triangle” (<a href="https://mathstodon.xyz/@11011110/104757581337870450">\(\mathbb{M}\)</a>): the logo of Polish football club Ruch Chorzów, which a supporter tried to add to the Wikipedia article. Unlike many non-Reuleaux round triangles, it appears to use circular arcs, but not centered at the corners and with non-equilateral corners. The arc across the bottom is longer than the two sides, and it is wider than it is tall. Image from <a href="http://kubamalicki.com/portfolio_page/ruch-chorzow-100-years-anniversary/">an article on a recent redesign of the official logo</a>.</p>
<p style="text-align:center"><img src="/blog/assets/2020/ruch-logo.png" alt="Geometric analysis of the Ruch Chorzów logo showing that it is not a Reuleaux triangle" width="80%" /></p>
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<p><a href="https://mathstodon.xyz/@jsiehler/104763100825420197">J. Siehler asks for unsolved problems in mathematics whose statements are understandable by elementary school students, other than in number theory </a>. Examples given so far include <a href="https://en.wikipedia.org/wiki/Inscribed_square_problem">square pegs</a>, <a href="https://en.wikipedia.org/wiki/Bellman%27s_lost_in_a_forest_problem">lost in a forest</a>, <a href="https://en.wikipedia.org/wiki/Moving_sofa_problem">sofa-moving</a>, <a href="https://en.wikipedia.org/wiki/Moser%27s_worm_problem">Moser’s worm</a>, <a href="https://en.wikipedia.org/wiki/Net_(polyhedron)#Existence_and_uniqueness">Dürer’s nets</a>, the <a href="https://en.wikipedia.org/wiki/Tur%C3%A1n%27s_brick_factory_problem">brick factory</a>, and <a href="https://en.wikipedia.org/wiki/Lonely_runner_conjecture">lonely runners</a>.</p>
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<p><a href="https://www.rayawolfsun.com/2015/02/06/the-romance-of-al-asturlabiya/">Concerning “Mariam” Al-Asturlabiya</a> (<a href="https://mathstodon.xyz/@11011110/104771298687510642">\(\mathbb{M}\)</a>). A warning about how romanticizing past figures (in this case <a href="https://en.wikipedia.org/wiki/Mariam_al-Asturlabi">the only woman astrolabist known from the medieval Islamic world</a>) can result in creating biographical details for them out of thin air.</p>
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<p><a href="https://prideout.net/knotgl/">Interactive 3d knot table</a> (<a href="https://mathstodon.xyz/@11011110/104779316951663485">\(\mathbb{M}\)</a>). One of many interesting visualizations on “<a href="https://prideout.net/">the little grasshopper</a>”, by Philip Rideout (author of the svg3d Python library linked above).</p>
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<p><a href="https://gd2020.cs.ubc.ca/program-no-links/">The Graph Drawing 2020 program is online</a> (<a href="https://mathstodon.xyz/@11011110/104781215682863021">\(\mathbb{M}\)</a>). It is September 16-18, from 8AM to noon Pacific daylight time (the time in Vancouver, where the conference was originally to be held). The format has talk videos available pre-conference, with sessions consisting of 1-minute reminders of each talk and 5 minutes of live questions per talk. Jeff Erickson and Sheelagh Carpendale will give live invited talks. It’s free but requires registration, with deadline September 10.</p>
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</ul>David EppsteinThe evolution of mathematical word processing (\(\mathbb{M}\)). “Developments in computing over the last 30 years have not done as much as one might have thought to make writing mathematics easier.”Bricard’s jumping octahedron2020-08-22T17:46:00+00:002020-08-22T17:46:00+00:00https://11011110.github.io/blog/2020/08/22/bricards-jumping-octahedron<p>The <a href="https://en.wikipedia.org/wiki/Sch%C3%B6nhardt_polyhedron">Schönhardt polyhedron</a> is a non-convex octahedron that can be formed from a convex regular octahedron by twisting two opposite faces, stretching and deforming the other faces as you twist. It’s well known for not having any interior diagonals, and for being impossible to subdivide into tetrahedra without introducing new vertices. But long before Erich Schönhardt described it in 1928 in connection with these properties, Raoul Bricard was investigating <a href="https://en.wikipedia.org/wiki/Bricard_octahedron">flexible octahedra</a>, in connection with <a href="https://en.wikipedia.org/wiki/Cauchy%27s_theorem_(geometry)">Cauchy’s theorem on the rigidity of polyhedra</a>. The Schönhardt polyhedron forms an interesting example of flexibility, as I learned from a 1975 collection of lecture notes by Branko Grünbaum on “<a href="https://digital.lib.washington.edu/researchworks/bitstream/handle/1773/15700/Lost%20Mathematics.pdf?fterence=1">Lost Mathematics</a>”). I’m not entirely sure that it was known to Bricard (it’s not clear from Bricard’s paper and Grünbaum doesn’t really say so) but it wouldn’t surprise me if it was.</p>
<p>Cauchy’s theorem states that the shape of every convex polyhedron is uniquely determined by the shapes and connectivity of its faces. There can be no other convex polyhedron that has faces of the same shape, connected in the same way. But in some cases (like the regular icosahedron) you can dent some of the faces in to make a different, non-convex polyhedron with the same face shapes and connectivity. So Cauchy’s theorem doesn’t immediately extend to non-convex polyhedra. In fact, certain non-convex “<a href="https://en.wikipedia.org/wiki/Flexible_polyhedron">flexible polyhedra</a>” can deform continuously into an infinite range of shapes, without changing the shape or connectivity of their faces. Bricard’s octahedra are self-crossing examples and later investigators found examples without self-crossings.</p>
<p>But Grünbaum describes a different, non-self-crossing non-convex octahedron, the “jumping octahedron”. Rather than having a continuous range of rigid shapes, it has exactly two shapes, both of which have faces of the same shapes and connectivity. Unlike the example of the regular icosahedron and dented icosahedron, the two shapes have dihedral angles that are convex and concave in the same places. If you make this polyhedron out of perfectly rigid faces, with hinged connections at their edges, it could only be in one or the other of its two shapes: you wouldn’t be able to get it to the other shape without taking it apart and rebuilding it. But if you make it out of a material that’s stiff enough to hold its shape but flexible enough to deform a little, you can make a model that jumps or snaps from one shape to the other when you twist it. If you deform it a little out of shape, it will snap back to the nearest of its two valid shapes. Here’s one I made very roughly from some light cardstock and transparent tape, in its two shapes, one with only slightly-concave long diagonals down its sides and the other much more twisted and folded up:</p>
<div><table style="margin-left:auto;margin-right:auto">
<tr style="text-align:center;vertical-align:middle">
<td style="padding:10px"><img src="http://www.ics.uci.edu/~eppstein/pix/jumping-octahedron/1-m.jpg" alt="Jumping octahedron" width="315" style="border-style:solid;border-color:black;" /></td>
<td style="padding:10px"><img src="http://www.ics.uci.edu/~eppstein/pix/jumping-octahedron/2-m.jpg" alt="Jumping octahedron" width="315" style="border-style:solid;border-color:black;" /></td>
</tr></table></div>
<p>My model makes an interesting squelchy sound when I twist it from the more upright shape to the more twisted one, because these two shapes have different volumes and the air has to get out through the cracks between the faces. If I had perfectly sealed all these cracks, the pressure change would prevent it from changing shape. This change in volume is a big contrast from the Bricard octahedra and other continuously-flexible polyhedra, which must maintain constant volume as they flex.</p>
<p>The net I folded it from looks like this:</p>
<p style="text-align:center"><img src="/blog/assets/2020/jumping-octahedron-net.svg" alt="Net for jumping octahedron" /></p>
<p>It’s in two parts in order to make the seams of my model be symmetric, but also that way it fits better onto a single sheet of paper or card. It consists of two equilateral-triangle faces (the faces at the top and bottom of the model shown above) and six identical obtuse triangles on the sides. In my net and model, these triangles are isosceles, but that’s not important. The important part is that, if one of these triangles is projected onto the line between it and the equilateral triangle, its projected length is slightly more than the length of the connecting edge (because it’s an obtuse triangle) but not too long: longer by a factor strictly between one and</p>
\[\frac{1}{2}+\frac{1}{\sqrt{3}}\approx 1.07735.\]
<p>If you make the projection too short (with a right or acute side triangle shape) then it will not have two different shapes that maintain the same faces and convex-concave relation at each dihedral. If you make the projection too long, then you won’t be able to put it together at all while keeping all the faces flat. An explanation for some of this behavior can be seen from the diagram below, which shows the bottom equilateral triangle and one of the obtuse side triangles of the polyhedron, flattened out into a single plane, from a top view.</p>
<p style="text-align:center"><img src="/blog/assets/2020/jumping-octahedron-overhead.svg" alt="Overhead view of two faces of the jumping octahedron" /></p>
<p>If you fold these two triangles on their connecting edge, keeping the bottom equilateral triangle fixed but lifting the obtuse triangle into space, then the outer vertex of the obtuse triangle will rotate through a semicircle, but the plane of this semicircle is perpendicular to the plane of view of the diagram, so in top view it just looks like a line, the red line in the diagram. The edge along which the two triangles are attached is the axis of rotation, so it’s perpendicular to the semicircle of rotation and to the projected red line. If you fold all three edges of the bottom equilateral triangle at equal angles, then by symmetry the tips of the three folded obtuse triangles will form another equilateral triangle, and the size of this equilateral triangle will depend on the fold angle. If the vertices of this second equilateral triangle project to points on the yellow circle (the circumcircle of the bottom equilateral triangle) then this triangle will have exactly the correct size to attach the top face. This is only possible when the vertex of the obtuse triangle in the drawing is folded to a point that projects to of the two crossings of the red line and the yellow circle. The two fold angles for which this happens give the two shapes of the jumping octahedron.</p>
<p>The constraint that the side triangles be obtuse is what is needed to make the two crossing points of the red line and the circle be in the same arc of the circle relative to the vertices of the bottom equilateral triangle. The constraint that their projected length should be only a little bit longer than the side length of the equilateral triangle is what is needed to make the red line cross the circle at all. So these two constraints are necessary to make the jumping octahedron work. There’s one more necessary constraint: the height of the obtuse triangle above the edge connecting it to the equilateral triangle has to be large enough to reach both of the crossing points of the red line. When I started to make the model I was worried about a different geometric constraint: maybe the twisted state of the model is so twisted that its inner folded parts cross each other near the center of the model? But that can’t happen. If it did happen, the projected view of the model would have the side triangles folded into a position where they cover the center of the yellow circle. But that would mean that the top triangle’s vertices are too far around the yellow circle, past the point where the farthest point of the red line can cross.</p>
<p>(<a href="https://mathstodon.xyz/@11011110/104737012685827990">Discuss on Mastodon</a>)</p>David EppsteinThe Schönhardt polyhedron is a non-convex octahedron that can be formed from a convex regular octahedron by twisting two opposite faces, stretching and deforming the other faces as you twist. It’s well known for not having any interior diagonals, and for being impossible to subdivide into tetrahedra without introducing new vertices. But long before Erich Schönhardt described it in 1928 in connection with these properties, Raoul Bricard was investigating flexible octahedra, in connection with Cauchy’s theorem on the rigidity of polyhedra. The Schönhardt polyhedron forms an interesting example of flexibility, as I learned from a 1975 collection of lecture notes by Branko Grünbaum on “Lost Mathematics”). I’m not entirely sure that it was known to Bricard (it’s not clear from Bricard’s paper and Grünbaum doesn’t really say so) but it wouldn’t surprise me if it was.Linkage2020-08-15T17:02:00+00:002020-08-15T17:02:00+00:00https://11011110.github.io/blog/2020/08/15/linkage<ul>
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<p>Two sites on toroidal polyhedra: <a href="https://www.spektrum.de/alias/raeumliche-geometrie/bonnie-stewarts-hohlkoerper/681891">Bonnie Stewarts Hohlkörper</a> and <a href="http://polyhedra.doskey.com/Stewart00.html">Alex Doskey’s virtual reality models of Stewart’s polyhedra</a> (<a href="https://mathstodon.xyz/@11011110/104618649607830730">\(\mathbb{M}\)</a>). Found while researching a new WP article on Stewart’s book <em><a href="https://en.wikipedia.org/wiki/Adventures_Among_the_Toroids">Adventures Among the Toroids</a></em>. The first link is in German but readable through Google translate and has lots of pretty pictures. The second needs VR software to be usable.</p>
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<p><a href="https://www.robertdickau.com/mapfolding.html">The map folding problem, illustrated by Robert Dickau</a> (<a href="https://mathstodon.xyz/@11011110/104630030819531499">\(\mathbb{M}\)</a>). See <a href="https://www.robertdickau.com/default.html#math">Dickau’s home page</a> for many more mathematical illustrations, mostly of combinatorial enumeration problems and fractals.</p>
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<p>For some reason I wanted the name of a surface of revolution of a circular arc less than \(\pi\) around its chord (<a href="https://mathstodon.xyz/@11011110/104635297508909080">\(\mathbb{M}\)</a>). <a href="https://en.wikipedia.org/wiki/Lemon_(geometry)">Wikipedia said “lemon”</a> but sourced to MathWorld so I thought maybe MathWorld had made it up. Not so. Better sources say the same. And the surface for the complementary arc is an “apple”. It looks like a North American football but <a href="http://modellsammlung.uni-goettingen.de/index.php?lang=en&r=5&sr=17&m=182">a “football” is a different surface of revolution, of constant positive Gaussian curvature</a>.</p>
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<p><a href="https://link.springer.com/journal/454/64/2">Special issue of <em>Discrete & Computational Geometry</em> in memory of Branko Grünbaum</a> (<a href="https://mathstodon.xyz/@11011110/104643955543481823">\(\mathbb{M}\)</a>). I think many of the research papers in it are interesting but I want to draw particular attention to <a href="https://link.springer.com/article/10.1007/s00454-020-00214-y">the preface by Gil Kalai, Bojan Mohar, and Isabella Novik</a>, which provides a nice brief survey both of Grünbaum’s many contributions to discrete geometry and of the lines of active research they have led to.</p>
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<p><a href="http://gallery.bridgesmathart.org/exhibitions/2020-Bridges-Conference">2020 Bridges Conference Mathematical Art Gallery</a> (<a href="https://mathstodon.xyz/@11011110/104646771923669610">\(\mathbb{M}\)</a>). Many are great but a couple of my favorites are <a href="http://gallery.bridgesmathart.org/exhibitions/2020-bridges-conference/conan-chadbourne">Conan Chadbourne’s grid partition enumeration</a> and <a href="http://gallery.bridgesmathart.org/exhibitions/2020-bridges-conference/mdlevin_publicmsncom">Martin Levin’s ten-tetrahedron tensegrity</a>. I didn’t participate but apparently the Bridges conference itself was held virtually a few days ago; see <a href="https://2020.bridgesmathart.org/">the conference site</a> for more including papers and videos.</p>
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<p><a href="https://felixboiii.github.io/paper-plotter/">Paper plotter</a> (<a href="https://mathstodon.xyz/@11011110/104655233453519187">\(\mathbb{M}\)</a>, <a href="https://news.ycombinator.com/item?id=24091297">via</a>): tool to make 3d paper cut-and-assemble models of the graphs of bivariate functions.</p>
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<p>Kowhaiwhai (<a href="https://mathstodon.xyz/@11011110/104663925881930344">\(\mathbb{M}\)</a>). are repeating decorative patterns used in New Zealand on Maori buildings. <a href="https://natlib.govt.nz/photos?text=kowhaiwhai&commit=Search">The National Library of NZ has a number of good examples</a>, including the <a href="https://natlib.govt.nz/records/23146518">sketches of patterns by Tamati Ngakoho (top) and of a traditional Arawa pattern (bottom)</a> shown below. There’s also <a href="http://www.maori.org.nz/whakairo/default.php?pid=sp55&parent=52">a brief guide to their interpretation online</a>. I can’t find much analysis of their structure, though, beyond pointing to frieze groups for their symmetries. The part that interests me more is their fractal-like swooping structure, reminiscent of (and in some cases directly modeled on) fern fronds.</p>
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<p style="text-align:center"><img src="/blog/assets/2020/Kowhaiwhai.jpg" alt="Godber, Albert Percy, 1875-1949. Godber, Albert Percy, 1876-1949. Drawings of Maori rafter patterns or kowhaiwhai. 16. 22W. MA22; 17. 21W. MA21; and, 18. 25W. MA25. Puhoro. [1939-1947]. Ref: E-302-q-1-016/018. Alexander Turnbull Library, Wellington, New Zealand. From https://natlib.govt.nz/records/23146518" /></p>
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<p><a href="https://www.wired.com/story/why-wikipedia-decided-to-stop-calling-fox-a-reliable-source/">Why Wikipedia decided to stop calling Fox a reliable source</a> (<a href="https://mathstodon.xyz/@11011110/104666160845673755">\(\mathbb{M}\)</a>). Note however that Fox has not actually been deemed unreliable, in general. <a href="https://en.wikipedia.org/wiki/Wikipedia:Reliable_sources/Noticeboard/Archive_303#RfC:_Fox_News">The discussion had a no-consensus close</a>.</p>
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<p><a href="https://sinews.siam.org/Details-Page/untangling-random-polygons-and-other-things">Untangling random polygons</a> (<a href="https://mathstodon.xyz/@11011110/104677553383067578">\(\mathbb{M}\)</a>): repeatedly rescaling midpoint polygons always leads to an ellipse.</p>
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<p><a href="https://www.atlasobscura.com/articles/kek-lapis-sarawak">The mesmerizing geometry of Malaysia’s most complex cakes:
Bold colors and designs set kek lapis Sarawak apart</a> (<a href="https://mathstodon.xyz/@11011110/104680812259935603">\(\mathbb{M}\)</a>, <a href="https://news.ycombinator.com/item?id=24116775">via</a>). As seen on The Great British Bake Off. These cakes have many parallel layers in bright colors, cut and rearranged to form complex designs. Mostly they involve 45 and 90-degree angles but at least one of the examples uses hexagonal symmetry instead.</p>
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<p>My Google Scholar profile has mildly broken down (<a href="https://mathstodon.xyz/@11011110/104683176033821672">\(\mathbb{M}\)</a>). When I go there, it offers me two new profiles to link as my coauthors: Man-Kwun Chiu and Matí Korman. They are indeed coauthors, from my new CCCG papers. But when I click to accept them as listed coauthors, it tells me I have too many coauthors, refuses to add them, and returns to offering me new profiles to link. I can see no way out of this other than to not accept my coauthors, which would be wrong. Google, fix this limitation!</p>
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<p>A use for old CDs: <a href="https://momath.org/home/math-monday-those-circles-are-great/">cut them up and glue the pieces together to make visualizations of great circle arrangements on the sphere</a> (<a href="https://mathstodon.xyz/@11011110/104690896919723051">\(\mathbb{M}\)</a>). The mathematical question posed by this is: for which numbers of great circles is it possible to make an arrangement in which all the arcs between pairs of neighbors have equal lengths?</p>
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<p><a href="https://thonyc.wordpress.com/">The Renaissance Mathematicus</a> (<a href="https://mathstodon.xyz/@pkra/104694183591138626">\(\mathbb{M}\)</a>), an interesting blogger on the history of science. See also the <a href="https://thonyc.wordpress.com/2020/08/15/keep-the-renaissance-mathematicus-online/">crowdfunding drive to replace their old creaky iMac</a>, from which I found this.</p>
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</ul>David EppsteinTwo sites on toroidal polyhedra: Bonnie Stewarts Hohlkörper and Alex Doskey’s virtual reality models of Stewart’s polyhedra (\(\mathbb{M}\)). Found while researching a new WP article on Stewart’s book Adventures Among the Toroids. The first link is in German but readable through Google translate and has lots of pretty pictures. The second needs VR software to be usable.Report from CCCG2020-08-07T17:46:00+00:002020-08-07T17:46:00+00:00https://11011110.github.io/blog/2020/08/07/report-from-cccg<p>I spent the last few days participating in the <a href="http://vga.usask.ca/cccg2020/">Canadian Conference in Computational Geometry</a>, originally planned for Saskatoon but organized virtually instead.</p>
<p>The way the conference was organized was that (after the usual submission reviewing process) the accepted authors provided both a proceedings paper and a 10-15 minute talk video to the conference organizers. Participants were required to register, but with no registration fee, and were provided with links to the papers and talks (which are all still live on the conference program). Then, during the conference itself, live online Zoom sessions ran for only 2-3 hours daily, scheduled for 10AM-1PM Saskatoon time: very convenient for anywhere in North America or Europe, not so much for the participants in Iran, India, China, Japan, and Korea (all of which did have participants). The sessions included a daily 1-hour live invited talk, and question-and-answer sessions for the contributed works, in which we were shown a one-minute teaser for each video and then invited to ask questions of authors, at least one of whom was required to be present.</p>
<p>I think it all worked very well; so well, in fact, that during the business meeting there were calls for having at least some of the content similarly online so that people could participate remotely again. The ability to ask and answer questions either by live video on Zoom or through Zoom chat was useful, and used. Attendance was far above previous levels: 162 registrants, and over 120 unique participants at the most heavily attended of the two daily parallel sessions, compared to roughly 60 attendees each at the last two physical conferences. Despite the free registration, the conference organization was not without cost: they spent roughly $1500 (Canadian) in video production costs and video conferencing fees, but this was more than made up for by institutional support for the conference, so they ended up running a surplus which may (if it can be kept) end up providing some float for future conference organizers.</p>
<p>There are two changes I would suggest for future events of this type:</p>
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<p>The contributed sessions were for a short enough time that holding them in parallel seemed unnecessary, and made it impossible to participate in all discussions. So this format may work better with less parallelism.</p>
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<p>The one-minute teaser videos were cut together by the conference organizers from the longer videos provided by the authors, but in some cases the pacing of the longer videos from which they were cut meant that these teaser videos could not clearly state the results of their papers. I think it would have been better to ask authors to provide these alongside the longer talk videos.</p>
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<p>Some of the highlights of the event:</p>
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<p>Wednesday’s invited talk by Erik Demaine was a moving tribute to Godfried Toussaint and a survey of some of both Toussaint’s research, and Erik’s research on problems started by Toussaint, including <a href="https://en.wikipedia.org/wiki/Erd%C5%91s%E2%80%93Nagy_theorem">convexifying polygons by flips</a>, sona curves (<a href="https://arxiv.org/abs/2007.15784">the subject of one of my own contributions</a>), the <a href="https://en.wikipedia.org/wiki/The_Geometry_of_Musical_Rhythm">geometry of musical rhythms</a>, and perhaps most importantly “supercollaboration”, the model of shared research and shared authorship developed by Toussaint at the Barbados research workshops and also used by Erik within many of his MIT classes. <a href="https://www.youtube.com/watch?v=exzxGODi2YU">Erik’s talk was recorded and is now on YouTube</a>; I hope the same will be true of the other two invited talks.</p>
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<p>In Wednesday’s contributed session on unfolding (in which I had two papers) I particularly liked Satyan Devadoss’s talk, “Nets of higher-dimensional cubes”. The main result is that if you unfold a hypercube, then no path of facets of the unfolded shape can contain a u-turn: if the path takes a step in one any coordinate direction, it cannot step in the opposite direction. This implies that all dual spanning trees unfold flat without self-intersection. The same property was known for all the Platonic solids, and Satyan can prove it for regular simplices in any dimension, but that still leaves several regular polytopes for which it is still open whether all unfoldings work: the cross-polytopes in all dimensions, and the three exceptional regular polytopes in four dimensions.</p>
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<p>Thursday’s invited talk was by Jeff Erickson, “Chasing puppies”. It was an entertaining presentation of an elegant topological proof of the following result: if you and a puppy can both move around a simple closed curve in the plane, with the puppy always moving along the curve to a local minimum of distance to you, then you can always find a path to follow that will bring you and the puppy to the same point.</p>
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<p>There wasn’t an official prize for best contributed presentation, but in the data structures session on Thursday, several comments nominated <a href="https://medium.com/photos-we-love/the-trinity-of-incongruity-or-why-i-still-love-this-tuxedo-sewing-machine-ups-truck-photograph-39712dd32fcb">Don Sheehy</a> as the unofficial winner, for a video artfully mixing live action with computer animations. His paper, “One-hop greedy permutations” concerned heuristics for improving the <a href="https://en.wikipedia.org/wiki/Farthest-first_traversal">farthest-first traversal</a> of a set of points by looking near each point as the sequence is constructed for a better point to use instead.</p>
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<p>There was an official best student paper award, and it went to <a href="https://www.cs.umd.edu/people/afloresv">Alejandro Flores Velazco</a> for his paper “Social distancing is good for points too!” It concerns the problem of reducing the size of a data set while preserving the quality of nearest-neighbor classification using the reduced set. It proves that FCNN, which it calls the most popular heuristic for this problem, can produce significantly less-reduced outputs than some other proven heuristics, and shows how to modify FCNN to get a heuristic with guaranteed output quality.</p>
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<p>The third of the invited sessions was by Yusu Wang, newly moved from Ohio State to UC San Diego. She gave a nice introduction to combinatorial methods for reconstructing road networks (or other networks embedded into higher-dimensional geometry) from noisy samples of points on the network by combining discrete Morse theory to find the ridge lines of sample density with persistent homology to clean some of the noise from the data.</p>
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<p>The final technical component of the conference was an open problem session, also recorded and presumably to be uploaded at some point. Satyan posed his question on regular polytope unfolding there. Mike Paterson asked whether one can construct “Plato’s torus”, an embedded torus with six equilateral-triangle faces meeting at each vertex; <a href="/blog/2009/02/03/flat-equilateral-tori.html">in a blog post I made on this problem in 2009 I traced its history to Nick Halloway in 1997</a> but Mike says he discussed it already with Christopher Zeeman in the 1970s. Another problem that caught my attention asked for an algorithmic version of the polyhedral <a href="https://en.wikipedia.org/wiki/Theorem_of_the_three_geodesics">theorem of the three geodesics</a>, the existence of a path across the surface of a convex polyhedron that stays straight across each edge or face of the polyhedron, and has at most \(\pi\) surface angle on each side of it when it passes through a vertex. Again there’s some history here: Joe O’Rourke says he once mentioned the problem to Gromov, who said it was easy but unfortunately didn’t elaborate.</p>
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<p>CCCG 2021 is planned for Halifax, colocated with WADS. One somewhat controversial issue is that the current plan is to have both conferences overlap for two days, with one overlap-free day for each conference at each end of the overlap period. But if both conferences are double-session, this means that participants can only choose one of four overlapping talks. At this point everyone is still hoping that events allow for a physical conference by then but that remains to be seen.</p>
<p>(<a href="https://mathstodon.xyz/@11011110/104651090140382005">Discuss on Mastodon</a>)</p>David EppsteinI spent the last few days participating in the Canadian Conference in Computational Geometry, originally planned for Saskatoon but organized virtually instead.Sona enumeration2020-08-02T23:48:00+00:002020-08-02T23:48:00+00:00https://11011110.github.io/blog/2020/08/02/sona-enumeration<p>The last of my CCCG 2020 papers is now on the arXiv: “New Results in Sona Drawing: Hardness and TSP Separation”, <a href="https://arxiv.org/abs/2007.15784">arXiv:2007.15784</a>, with Chiu, Demaine, Diomidov, Hearn, Hesterberg, Korman, Parada, and Rudoy. (As you might infer from the long list of coauthors, it’s a Barbados workshop paper.) The paper studies a mathematical formalization of the <a href="https://en.wikipedia.org/wiki/Lusona">lusona</a> drawings of southwest Africa; in this formalization, a sona curve for a given set of points is a curve that can be drawn in a single motion, intersecting itself only at simple crossings, and surrounding each given point in a separate region of the plane, with no empty regions. The paper proves that it’s hard to find the shortest one, hard even to find whether one exists when restricted to grid edges, and gives tighter bounds for the widest possible ratio between sona curve length and TSP tour length; see the preprint or <a href="/blog/2020/07/22/three-cccg-videos.html">the video I already posted</a> for more information.</p>
<p>To save this post from being content-free, here’s a research question that we didn’t even state in the paper, let alone make any progress on solving: just how many of these curves can a given set of points have? A sona curve can be described as a 4-regular plane multigraph (satisfying certain extra conditions) together with an assignment of the given points to its bounded faces, so there are finitely many of these things up to some sort of topological equivalence. And because this is topological it shouldn’t matter where the points are placed in the plane: the number of curves should be a function only of the number of points. I tried hand-enumerating the curves for up to three points but it was already messy and I’m not certain I got them all. (In an earlier version of this post I definitely didn’t get them all — I had to update the figure below after finding more.) Here are the ones I found:</p>
<p style="text-align:center"><img src="/blog/assets/2020/sona-enum.svg" alt="Sona curves for up to three points" /></p>
<p>If this hand enumeration is correct, then the numbers of sona curves for \(n\) labeled points form an integer sequence beginning \(1, 3, 24,\dots\) and the numbers for unlabeled points form a sequence beginning \(1, 2, 5,\dots\) but I don’t really know anything more than that for this problem.</p>
<p>Another research direction I don’t know much about yet: given a topological equivalence class of sona drawings, how can we find a good layout for it as an explicit drawing? There’s lots of research on drawing plane graphs nicely but it’s not clear how much of it carries over to making nice sona curves.</p>
<p>(<a href="https://mathstodon.xyz/@11011110/104624173377453724">Discuss on Mastodon</a>)</p>David EppsteinThe last of my CCCG 2020 papers is now on the arXiv: “New Results in Sona Drawing: Hardness and TSP Separation”, arXiv:2007.15784, with Chiu, Demaine, Diomidov, Hearn, Hesterberg, Korman, Parada, and Rudoy. (As you might infer from the long list of coauthors, it’s a Barbados workshop paper.) The paper studies a mathematical formalization of the lusona drawings of southwest Africa; in this formalization, a sona curve for a given set of points is a curve that can be drawn in a single motion, intersecting itself only at simple crossings, and surrounding each given point in a separate region of the plane, with no empty regions. The paper proves that it’s hard to find the shortest one, hard even to find whether one exists when restricted to grid edges, and gives tighter bounds for the widest possible ratio between sona curve length and TSP tour length; see the preprint or the video I already posted for more information.Linkage2020-07-31T21:03:00+00:002020-07-31T21:03:00+00:00https://11011110.github.io/blog/2020/07/31/linkage<ul>
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<p><a href="https://gilkalai.wordpress.com/2020/07/08/to-cheer-you-up-in-difficult-times-7-bloom-and-sisask-just-broke-the-logarithm-barrier-for-roths-theorem/">Gil Kalai on recent developments in Roth’s theorem</a> (<a href="https://mathstodon.xyz/@11011110/104533566344469712">\(\mathbb{M}\)</a>, <a href="https://blog.computationalcomplexity.org/2020/07/erdos-turan-for-k3-is-true.html">see also</a>). Salem and Spencer and later Behrend proved in the 1940s that <a href="https://en.wikipedia.org/wiki/Salem%E2%80%93Spencer_set">subsets of \([1,n]\) with no triple in arithmetic progression</a> can have nearly linear size, and Klaus Roth proved in 1953 that <a href="https://en.wikipedia.org/wiki/Roth%27s_theorem_on_arithmetic_progressions">they must be sublinear</a>. The upper bounds have slowly come down, to \(n/\log^{1+c} n\) in this new result, but they’re still far from Behrend’s \(n/e^{O(\sqrt{\log n})}\) lower bound.</p>
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<p><a href="https://cameroncounts.wordpress.com/2020/06/27/peter-sarnaks-hardy-lecture/">Peter Cameron describes Peter Sarnak’s Hardy Lecture</a> (<a href="https://mathstodon.xyz/@11011110/104539029304056696">\(\mathbb{M}\)</a>). It’s on the spectral theory of graphs. If you know about this you probably already know that regular graphs with a big gap between the largest eigenvalue (degree) and the second largest are very good expander graphs. It turns out that 3-regular graphs with gaps elsewhere in their spectrum are also important in the theories of waveguides and fullerenes, and some tight bounds on where those gaps can be are now known.</p>
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<p><a href="https://arxiv.org/abs/2007.07983">Optimal angle bounds for quadrilateral meshes</a> (<a href="https://mathstodon.xyz/@11011110/104544859698397049">\(\mathbb{M}\)</a>). Christopher J. Bishop meshes any simple polygon (why simple?) with max angle 120° and min angle max(60°, min of the polygon). Nice techniques involving conformal mapping, hyperbolic tessellation, and thick/thin decompositions of hyperbolic convex hulls of ideal sets. Also amusing to see him have to disambiguate my name from David B. A. Epstein’s within a single paragraph.</p>
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<p><a href="https://imgur.com/gallery/72lduu6">One-dimensional diagonal cellular automata generate Sierpinski carpets and intricate branching structures</a> (<a href="https://mathstodon.xyz/@11011110/104550317577491129">\(\mathbb{M}\)</a>, <a href="https://community.wolfram.com/groups/-/m/t/1890120">see also</a>). Via the June 27 update to <a href="http://www.mathpuzzle.com/">mathpuzzle.com</a> which also has plenty of other neat stuff involving tilings, drawings of symmetric graphs, graceful labeling, rectangle dissection into similar rectangles, etc.</p>
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<p><a href="https://mathoverflow.net/a/366118">Terry Tao on mathematical notation</a> (<a href="https://mathstodon.xyz/@JordiGH/104552943941946623">\(\mathbb{M}\)</a>), in response to a MathOverflow question about why there’s more than one way to write inner products.</p>
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<p><a href="http://matroidunion.org/?p=2693">Carmesin’s 3d version of Whitney’s planarity criterion</a> (<a href="https://mathstodon.xyz/@11011110/104567191288903767">\(\mathbb{M}\)</a>): a simply-connected 2-dimensional simplicial complex (meeting a technical condition, “locality”) can be topologically embedded into Euclidean space if and only if a certain ternary matroid on its faces has a graphic dual. The proof relies on Perelman’s proof of the Poincaré conjecture! Simply-connected complexes are pretty restrictive but they include e.g. the cone over a graph, which embeds if and only if the graph is planar.</p>
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<p><a href="https://cp4space.wordpress.com/2020/07/24/fast-growing-functions-revisited/">Fast-growing functions revisited</a> (<a href="https://mathstodon.xyz/@11011110/104573335749947907">\(\mathbb{M}\)</a>). News of recent developments relating the <a href="https://en.wikipedia.org/wiki/Busy_beaver">busy beaver function</a> with <a href="https://en.wikipedia.org/wiki/Graham%27s_number">Graham’s number</a>, and proofs of some older claims.</p>
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<p><a href="https://doi.org/10.1007/s12109-020-09750-0">Wikipedia, the free online medical encyclopedia anyone can plagiarize: Time to address wiki‑plagiarism</a> (<a href="https://mathstodon.xyz/@11011110/104576022559029845">\(\mathbb{M}\)</a>, <a href="https://retractionwatch.com/2020/07/25/weekend-reads-image-duplication-software-debuts-papers-that-plagiarize-wikipedia-time-to-get-serious-about-research-fraud/">via</a>). In this editorial in <em>Publishing Research Quarterly</em>, Michaël R. Laurent identifies five PubMed-indexed papers that copied content from Wikipedia without crediting it (noting that this is much more prevalent in predatory book and journal publishing), and argues that doing this should be treated as a form of academic misconduct.</p>
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<p><a href="https://andrewducker.dreamwidth.org/3861716.html">Facebook temporarily blocks posts of links to dreamwidth</a> (<a href="https://mathstodon.xyz/@11011110/104581191674329045">\(\mathbb{M}\)</a>, <a href="https://news.ycombinator.com/item?id=23956640">via</a>). Maybe it was just a mistake? And I guess the decentralization of Mastodon would make doing this to Mastodon posts somewhat harder. But this continued walling-off of the open web is not a good thing.</p>
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<p><a href="https://nebusresearch.wordpress.com/2020/07/23/my-all-2020-mathematics-a-to-z-fibonacci/">How much we don’t know about Fibonacci</a> (<a href="https://mathstodon.xyz/@nebusj/104581720945583863">\(\mathbb{M}\)</a>). Entry F in Joseph Nebus’s 2020 mathematics A-to-Z.</p>
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<p><a href="https://cp4space.wordpress.com/2020/07/25/rational-dodecahedron-inscribed-in-unit-sphere/">Rational dodecahedron inscribed in unit sphere</a> (<a href="https://mathstodon.xyz/@11011110/104595876537307480">\(\mathbb{M}\)</a>). It’s easy to inscribe a dodecahedron in the unit sphere: just use a regular one of the appropriate size. And it’s <a href="https://johncarlosbaez.wordpress.com/2011/09/12/fools-gold/">not hard to construct a dodecahedron combinatorially equivalent to the regular dodecahedron but with integer coordinates</a>. Now Adam Goucher shows how to do both at once, in answer to <a href="https://mathoverflow.net/q/234212/440">an old MathOverflow question</a>.</p>
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<p><em><a href="https://doi.org/10.1007/978-1-4612-5759-2">Descartes on Polyhedra</a></em> (<a href="https://mathstodon.xyz/@11011110/104606789909959594">\(\mathbb{M}\)</a>, <a href="https://en.wikipedia.org/wiki/Descartes_on_Polyhedra">see also</a>). This book is mainly on whether Descartes (circa 1630) knew Euler’s formula \(E-V+F=2\) (before Euler in 1752, but after Maurolico in 1537). It also covers Descartes’ invention of polyhedral figurate numbers beyond the cubes and pyramidal ones known to the Greeks. Descartes’ manuscript has an interesting history: found after his death in a desk, sunk in the Seine, copied by Leibniz, both copies lost, and Leibniz’s copy finally rediscovered in 1860.</p>
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<p><a href="https://www.youtube.com/watch?v=yY9GAyJtuJ0">Spherical geometry is stranger than hyperbolic (in how it looks from an in-universe viewpoint)</a> (<a href="https://mathstodon.xyz/@11011110/104611506183926064">\(\mathbb{M}\)</a>, <a href="https://news.ycombinator.com/item?id=24011727">via</a>).</p>
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</ul>David EppsteinGil Kalai on recent developments in Roth’s theorem (\(\mathbb{M}\), see also). Salem and Spencer and later Behrend proved in the 1940s that subsets of \([1,n]\) with no triple in arithmetic progression can have nearly linear size, and Klaus Roth proved in 1953 that they must be sublinear. The upper bounds have slowly come down, to \(n/\log^{1+c} n\) in this new result, but they’re still far from Behrend’s \(n/e^{O(\sqrt{\log n})}\) lower bound.Polyhedra with convex unfoldings2020-07-29T22:18:00+00:002020-07-29T22:18:00+00:00https://11011110.github.io/blog/2020/07/29/polyhedra-convex-unfoldings<p>My newest arXiv preprint is “Acutely triangulated, stacked, and very ununfoldable polyhedra” with Erik and Martin Demaine (<a href="https://arxiv.org/abs/2007.14525">arXiv:2007.14525</a>). It’s about polyhedra with acute-triangle faces that cannot be unfolded without cutting their surface into many separate polygons. I <a href="/blog/2020/07/22/three-cccg-videos.html">already posted a video for the paper</a> so see that for more information.</p>
<p>Instead, I thought I’d go into a little more detail about a throwaway remark in the video and the paper (one that I already got an email query about). It says that <a href="https://en.wikipedia.org/wiki/Ideal_polyhedron">ideal hyperbolic polyhedra</a> can always be unfolded (into the hyperbolic plane). These polyhedra are the hyperbolic convex hulls of finitely many limit points of the hyperbolic space; their faces are ideal polygons, glued together along entire hyperbolic lines. More strongly, if you cut an ideal polyhedron along any spanning tree of its vertices and edges, the result always unfolds into a convex ideal hyperbolic polygon. Here, for instance, is a net for an ideal cube:</p>
<p style="text-align:center"><img src="/blog/assets/2020/ideal-cube-net.svg" alt="Net for an ideal cube" /></p>
<p>I don’t know of a previous reference for this result, and the paper and video state it without proof (because it’s an introductory remark and not the topic of the paper), but it’s easy to prove a stronger statement by induction: any collection of ideal hyperbolic polygons (like the faces of an ideal polyhedron), when connected edge-to-edge in a complex with the connectivity of a tree (like the faces of any convex polyhedron when you cut it along a spanning tree), unfolds to an ideal convex polygon. As a base case, when you have one polygon in your collection, it unfolds to itself. When you have more than one, find a leaf polygon of the tree structure, remove it, and unfold the rest into a convex ideal polygon. Now add back the leaf. It needs to be connected to the rest of the complex along a hyperbolic line, which (by the induction hypothesis that the rest unfolds convexly) has the rest of the complex on one side and an empty hyperbolic halfplane on the other side. Any convex ideal polygon can be placed within this halfplane so that the side on which it should be glued matches up with the boundary line of the halfplane, with enough freedom to match up the points along this line that should be matched up.</p>
<p>This caused me to wonder: which Euclidean convex polyhedra have the same property, that cutting them along any spanning tree leads to a convex unfolding? The answer is: not very many. By <a href="https://en.wikipedia.org/wiki/Descartes%27_theorem_on_total_angular_defect">Descartes’ theorem on total angular defect</a>, the angular defects at the vertices of a convex polyhedron add up to \(4\pi\). If a polyhedron is to have all spanning trees produce a (weakly) convex unfolding, then each vertex has to have angular defect at least \(\pi\), because otherwise cutting along a spanning tree that has a leaf at that vertex will make an unfolding that is non-convex at that vertex. And this is the only thing that can go wrong, because if all angular defects are at least \(\pi\) then the unfolding will be convex at each of its vertices and cannot self-overlap.</p>
<p>So to answer the question about Euclidean polyhedra with all unfoldings convex, we need only look for ways to partition the total angular defect of \(4\pi\) among some set of vertices so that each one gets at least \(\pi\). If we know the defects of all the vertices and the distances between vertices, then by <a href="https://en.wikipedia.org/wiki/Alexandrov%27s_uniqueness_theorem">Alexandrov’s uniqueness theorem</a> the shape of the polyhedron will be determined. Since we’re using Alexandrov, we should also consider a <a href="https://en.wikipedia.org/wiki/Dihedron">dihedron</a> (two mirror-image convex faces glued at their edges) to be a special case of a polyhedron. This leaves, as the only cases:</p>
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<p>A triangular dihedron based on a right or acute triangle.</p>
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<p>A rectangular dihedron.</p>
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<p>A tetrahedron with angular defect exactly \(\pi\) at each vertex.</p>
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<p style="text-align:center"><img src="/blog/assets/2020/convex-unfoldings.svg" alt="Convex unfoldings of dihedra and a disphenoid" /></p>
<p>The unfoldings of the dihedra have two copies of their face, mirrored across a joining edge. The tetrahedra with all-convex unfoldings are exactly the <a href="https://en.wikipedia.org/wiki/Disphenoid">disphenoids</a>, the tetrahedra whose four faces are congruent. They unfold either to a copy of the same face shape,
expanded by a factor of two in each dimension and creased into four copies along its <a href="https://en.wikipedia.org/wiki/Medial_triangle">medial triangle</a>, or a parallelogram, creased to form a strip of four congruent triangles. Their unfoldings were discussed by Jin Akiyama in his paper “Tile-makers and semi-tile-makers” (<em>American Mathematical Monthly</em> 2007, <a href="https://doi.org/10.1080/00029890.2007.11920450">doi:10.1080/00029890.2007.11920450</a>, <a href="https://www.jstor.org/stable/27642275">jstor:27642275</a>), as part of a broader investigation of polyhedra whose every unfolding tiles the plane.</p>
<p>(<a href="https://mathstodon.xyz/@11011110/104601177683049272">Discuss on Mastodon</a>)</p>David EppsteinMy newest arXiv preprint is “Acutely triangulated, stacked, and very ununfoldable polyhedra” with Erik and Martin Demaine (arXiv:2007.14525). It’s about polyhedra with acute-triangle faces that cannot be unfolded without cutting their surface into many separate polygons. I already posted a video for the paper so see that for more information.