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  </channel><item rdf:about="https://arxiv.org/abs/2502.08396">
    <title>[2502.08396] Periodic double tilings of the plane</title>
    <dc:date>2026-04-20T15:57:35+00:00</dc:date>
    <link>https://arxiv.org/abs/2502.08396</link>
    <dc:creator>Vaguery</dc:creator><description><![CDATA[We study tilings of the plane composed of two repeating tiles of different assigned areas relative to an arbitrary periodic lattice. We classify isoperimetric configurations (i.e., configurations with minimal length of the interfaces) both in the case of a fixed lattice or for an arbitrary periodic lattice. We find three different configurations depending on the ratio between the assigned areas of the two tiles and compute the isoperimetric profile. The three different configurations are composed of tiles with a different number of circular edges, moreover, different configurations exhibit a different optimal lattice. Finally, we raise some open problems related to our investigation.
]]></description>
<dc:subject>tiling geometry plane-geometry rather-interesting constraint-satisfaction optimization to-understand consider:visualization consider:feature-discovery</dc:subject>
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<item rdf:about="https://arxiv.org/abs/2411.19864">
    <title>[2411.19864] An Elementary Proof of a Remarkable Relation Between the Squircle and Lemniscate</title>
    <dc:date>2025-11-03T17:54:39+00:00</dc:date>
    <link>https://arxiv.org/abs/2411.19864</link>
    <dc:creator>Vaguery</dc:creator><description><![CDATA[It is well known that there is a somewhat mysterious relation between the area of the quartic Fermat curve x4+y4=1, aka squircle, and the arc length of the lemniscate (x2+y2)2=x2−y2. The standardproof of this fact uses relations between elliptic integrals and the gamma function. In this article we generalize this result to relate areas of sectors of the squircle to arc lengths of segments of the lemniscate. We provide a geometric interpretation of this relation and an elementary proof of the relation, which only uses basic integral calculus. We also discuss an alternate version of this kind of relation, which is implicit in a calculation of Siegel.
]]></description>
<dc:subject>geometry mathematical-recreations mathematics polynomials plane-geometry rather-interesting amusing</dc:subject>
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<item rdf:about="https://arxiv.org/abs/math/9801088">
    <title>[math/9801088] Shapes of polyhedra and triangulations of the sphere</title>
    <dc:date>2025-04-16T19:09:10+00:00</dc:date>
    <link>https://arxiv.org/abs/math/9801088</link>
    <dc:creator>Vaguery</dc:creator><description><![CDATA[The space of shapes of a polyhedron with given total angles less than 2\pi at each of its n vertices has a Kaehler metric, locally isometric to complex hyperbolic space CH^{n-3}. The metric is not complete: collisions between vertices take place a finite distance from a nonsingular point. The metric completion is a complex hyperbolic cone-manifold. In some interesting special cases, the metric completion is an orbifold. The concrete description of these spaces of shapes gives information about the combinatorial classification of triangulations of the sphere with no more than 6 triangles at a vertex.
]]></description>
<dc:subject>geometry enumeration combinatorics classics amazing-papers to-write-about to-simulate consider:squares</dc:subject>
<dc:source>https://pinboard.in/</dc:source>
<dc:identifier>https://pinboard.in/u:Vaguery/b:ffcbd8d7924e/</dc:identifier>
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<item rdf:about="https://arxiv.org/abs/2402.18014">
    <title>[2402.18014] Set-valued Star-Shaped Risk Measures</title>
    <dc:date>2024-09-23T13:18:27+00:00</dc:date>
    <link>https://arxiv.org/abs/2402.18014</link>
    <dc:creator>Vaguery</dc:creator><description><![CDATA[In this paper, we introduce a new class of set-valued risk measures, named set-valued star-shaped risk measures. Motivated by the results of scalar monetary and star-shaped risk measures, this paper investigates the representation theorems in the set-valued framework. It is demonstrated that set-valued risk measures can be represented as the union of a family of set-valued convex risk measures, and set-valued normalized star-shaped risk measures can be represented as the union of a family of set-valued normalized convex risk measures. The link between set-valued risk measures and set-valued star-shaped risk measures is also established.
]]></description>
<dc:subject>approximation numerical-methods statistics risk rather-interesting portfolio-theory to-understand geometry error</dc:subject>
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<item rdf:about="https://link.springer.com/book/10.1007/978-3-642-03942-3">
    <title>Geometric Discrepancy: An Illustrated Guide | SpringerLink</title>
    <dc:date>2024-07-16T23:25:09+00:00</dc:date>
    <link>https://link.springer.com/book/10.1007/978-3-642-03942-3</link>
    <dc:creator>Vaguery</dc:creator><description><![CDATA[Discrepancy theory is also called the theory of irregularities of distribution. Here are some typical questions: What is the "most uniform" way of dis­ tributing n points in the unit square? How big is the "irregularity" necessarily present in any such distribution? For a precise formulation of these questions, we must quantify the irregularity of a given distribution, and discrepancy is a numerical parameter of a point set serving this purpose. Such questions were first tackled in the thirties, with a motivation com­ ing from number theory. A more or less satisfactory solution of the basic discrepancy problem in the plane was completed in the late sixties, and the analogous higher-dimensional problem is far from solved even today. In the meantime, discrepancy theory blossomed into a field of remarkable breadth and diversity. There are subfields closely connected to the original number­ theoretic roots of discrepancy theory, areas related to Ramsey theory and to hypergraphs, and also results supporting eminently practical methods and algorithms for numerical integration and similar tasks. The applications in­ clude financial calculations, computer graphics, and computational physics, just to name a few. This book is an introductory textbook on discrepancy theory. It should be accessible to early graduate students of mathematics or theoretical computer science. At the same time, about half of the book consists of material that up until now was only available in original research papers or in various surveys.
]]></description>
<dc:subject>to-read to-cite discrepancy-theory algorithmic-information-theory geometry statistics talking-about-stuff consider:tree-generation</dc:subject>
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<item rdf:about="https://arxiv.org/abs/2308.08834">
    <title>[2308.08834] Planar Doodles: Their Properties, Codes and Classification</title>
    <dc:date>2024-07-02T14:39:04+00:00</dc:date>
    <link>https://arxiv.org/abs/2308.08834</link>
    <dc:creator>Vaguery</dc:creator><description><![CDATA[We present those properties of planar doodles, especially when regarded as 4-valent graphs, that enable us to classify them into {\it prime} and {\it super prime} doodles by analogy to a knot sum. We describe a method for partially characterising a doodle diagram by a {\it doodle code} that describes the complementary regions of the diagram and use that code to enumerate all possible prime and super prime doodle diagrams via their dual graph. In addition we explore the relationship between planar doodles and twin groups, and note that a theorem of Tutte means that super prime doodles have a Hamiltonian circuit. We hope to expand upon this last point in a follow-up paper.
]]></description>
<dc:subject>topology geometry enumeration rather-interesting to-write-about to-simulate consider:representation consider:classification</dc:subject>
<dc:source>https://pinboard.in/</dc:source>
<dc:identifier>https://pinboard.in/u:Vaguery/b:cef8662227dd/</dc:identifier>
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<item rdf:about="https://arxiv.org/abs/1910.03379">
    <title>[1910.03379] The Triangle</title>
    <dc:date>2023-09-21T10:22:55+00:00</dc:date>
    <link>https://arxiv.org/abs/1910.03379</link>
    <dc:creator>Vaguery</dc:creator><description><![CDATA[If we label the vertices of a triangle with 1, 2 and 4, and the orthocentre with 7, then any of the four numbers 1, 2, 4, 7 is the nim-sum of the other three and is their orthocentre. Regard the triangle as an orthocentric quadrangle. Steiner's theorem states that the reflexions of a point on a circumcircle in each of the three edges of the corresponding triangle are collinear and collinear with its orthontre. This line intersects the circumcircles in new points to which the theorem may be applied. Iteration of this process with the triangle and the points rational leads to a "trisequence" whose properties merit study.
]]></description>
<dc:subject>mathematical-recreations plane-geometry compass-and-straightedge-constructions geometry miscellany rather-interesting</dc:subject>
<dc:source>https://pinboard.in/</dc:source>
<dc:identifier>https://pinboard.in/u:Vaguery/b:8cba8c0292ee/</dc:identifier>
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<item rdf:about="https://arxiv.org/abs/2002.06118">
    <title>[2002.06118] Covering of high-dimensional cubes and quantization</title>
    <dc:date>2023-08-19T22:16:45+00:00</dc:date>
    <link>https://arxiv.org/abs/2002.06118</link>
    <dc:creator>Vaguery</dc:creator><description><![CDATA[As the main problem, we consider covering of a d-dimensional cube by n balls with reasonably large d (10 or more) and reasonably small n, like n=100 or n=1000. We do not require the full coverage but only 90\% or 95\% coverage. We establish that efficient covering schemes have several important properties which are not seen in small dimensions and in asymptotical considerations, for very large n. One of these properties can be termed `do not try to cover the vertices' as the vertices of the cube and their close neighbourhoods are very hard to cover and for large d there are far too many of them. We clearly demonstrate that, contrary to a common belief, placing balls at points which form a low-discrepancy sequence in the cube, makes for a very inefficient covering scheme. For a family of random coverings, we are able to provide very accurate approximations to the coverage probability. We then extend our results to the problems of coverage of a cube by smaller cubes and quantization, the latter being also referred to as facility location. Along with theoretical considerations and derivation of approximations, we discuss results of a large-scale numerical investigation.
]]></description>
<dc:subject>computational-geometry low-discrepancy-numbers approximation looking-to-see geometry</dc:subject>
<dc:source>https://pinboard.in/</dc:source>
<dc:identifier>https://pinboard.in/u:Vaguery/b:0df86e75f223/</dc:identifier>
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<item rdf:about="https://arxiv.org/abs/2107.00518">
    <title>[2107.00518] Tiling of polyhedral sets</title>
    <dc:date>2022-04-12T12:42:17+00:00</dc:date>
    <link>https://arxiv.org/abs/2107.00518</link>
    <dc:creator>Vaguery</dc:creator><description><![CDATA[A self-affine tiling of a compact set G of positive Lebesgue measure is its partition to parallel shifts of a compact set which is affinely similar to G. We find all polyhedral sets (unions of finitely many convex polyhedra) that admit self-affine tilings. It is shown that in R^d there exist an infinite family of such polyhedral sets, not affinely equivalent to each other. A special attention is paid to an important particular case when the matrix of affine similarity and the translation vectors are integer. Applications to the approximation theory and to the functional analysis are discussed.
]]></description>
<dc:subject>tiling enumeration geometry to-understand consider:looking-to-see consider:visualization consider:2d-only consider:non-simple-polygons</dc:subject>
<dc:source>https://pinboard.in/</dc:source>
<dc:identifier>https://pinboard.in/u:Vaguery/b:ed209e66ce84/</dc:identifier>
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	<rdf:li rdf:resource="https://pinboard.in/u:Vaguery/t:to-understand"/>
	<rdf:li rdf:resource="https://pinboard.in/u:Vaguery/t:consider:looking-to-see"/>
	<rdf:li rdf:resource="https://pinboard.in/u:Vaguery/t:consider:visualization"/>
	<rdf:li rdf:resource="https://pinboard.in/u:Vaguery/t:consider:2d-only"/>
	<rdf:li rdf:resource="https://pinboard.in/u:Vaguery/t:consider:non-simple-polygons"/>
</rdf:Bag></taxo:topics>
</item>
<item rdf:about="https://arxiv.org/abs/2109.07817">
    <title>[2109.07817] Tiling of regular polygons with similar right triangles</title>
    <dc:date>2022-01-27T11:45:26+00:00</dc:date>
    <link>https://arxiv.org/abs/2109.07817</link>
    <dc:creator>Vaguery</dc:creator><description><![CDATA[We prove that for every N≠4 there is only one right triangle that tiles the regular N-gon.]]></description>
<dc:subject>geometry dissection-problems proof pictures-needed to-write-about to-visualize consider:equilateral-polygons</dc:subject>
<dc:source>https://pinboard.in/</dc:source>
<dc:identifier>https://pinboard.in/u:Vaguery/b:6205bbfcbc69/</dc:identifier>
<taxo:topics><rdf:Bag>	<rdf:li rdf:resource="https://pinboard.in/u:Vaguery/t:geometry"/>
	<rdf:li rdf:resource="https://pinboard.in/u:Vaguery/t:dissection-problems"/>
	<rdf:li rdf:resource="https://pinboard.in/u:Vaguery/t:proof"/>
	<rdf:li rdf:resource="https://pinboard.in/u:Vaguery/t:pictures-needed"/>
	<rdf:li rdf:resource="https://pinboard.in/u:Vaguery/t:to-write-about"/>
	<rdf:li rdf:resource="https://pinboard.in/u:Vaguery/t:to-visualize"/>
	<rdf:li rdf:resource="https://pinboard.in/u:Vaguery/t:consider:equilateral-polygons"/>
</rdf:Bag></taxo:topics>
</item>
<item rdf:about="https://arxiv.org/abs/2108.08680">
    <title>[2108.08680] Piecewise circular curves and positivity</title>
    <dc:date>2022-01-27T11:41:16+00:00</dc:date>
    <link>https://arxiv.org/abs/2108.08680</link>
    <dc:creator>Vaguery</dc:creator><description><![CDATA[We introduce the moduli space of generic piecewise circular n-gons in the Riemann sphere and relate it to a moduli space of Legendrian polygons. We prove that when n=2k, this moduli space contains a connected component homeomorphic to the Fock-Goncharov space of k-tuples of positive flags for 𝖯𝖲𝗉(4,ℝ) and hence is a topological ball. We characterize this component geometrically as the space of simple piecewise circular curves with decreasing curvature.
]]></description>
<dc:subject>geometry rather-interesting out-of-the-box non-Euclidean to-understand purdy-pitchers</dc:subject>
<dc:source>https://pinboard.in/</dc:source>
<dc:identifier>https://pinboard.in/u:Vaguery/b:38d439a75706/</dc:identifier>
<taxo:topics><rdf:Bag>	<rdf:li rdf:resource="https://pinboard.in/u:Vaguery/t:geometry"/>
	<rdf:li rdf:resource="https://pinboard.in/u:Vaguery/t:rather-interesting"/>
	<rdf:li rdf:resource="https://pinboard.in/u:Vaguery/t:out-of-the-box"/>
	<rdf:li rdf:resource="https://pinboard.in/u:Vaguery/t:non-Euclidean"/>
	<rdf:li rdf:resource="https://pinboard.in/u:Vaguery/t:to-understand"/>
	<rdf:li rdf:resource="https://pinboard.in/u:Vaguery/t:purdy-pitchers"/>
</rdf:Bag></taxo:topics>
</item>
<item rdf:about="https://arxiv.org/abs/2010.05052">
    <title>[2010.05052] Tiling of regular polygon with similar right triangles</title>
    <dc:date>2020-11-26T14:22:32+00:00</dc:date>
    <link>https://arxiv.org/abs/2010.05052</link>
    <dc:creator>Vaguery</dc:creator><description><![CDATA[A tiling is a decomposition of a polygon into finitely many non-overlapping triangles. We prove that if a regular n-gon, n≥5, n≠28, can be tiled with similar right triangles, then one of the angles of these triangles is in {πn,2πn,π6+2π3n}. Some related results were previously obtained by M.Laczkovich and B. Szegedy.]]></description>
<dc:subject>tiling dissection mathematical-recreations geometry rather-interesting proof to-write-about consider:looking-to-see</dc:subject>
<dc:source>https://pinboard.in/</dc:source>
<dc:identifier>https://pinboard.in/u:Vaguery/b:e3e5ba9ab577/</dc:identifier>
<taxo:topics><rdf:Bag>	<rdf:li rdf:resource="https://pinboard.in/u:Vaguery/t:tiling"/>
	<rdf:li rdf:resource="https://pinboard.in/u:Vaguery/t:dissection"/>
	<rdf:li rdf:resource="https://pinboard.in/u:Vaguery/t:mathematical-recreations"/>
	<rdf:li rdf:resource="https://pinboard.in/u:Vaguery/t:geometry"/>
	<rdf:li rdf:resource="https://pinboard.in/u:Vaguery/t:rather-interesting"/>
	<rdf:li rdf:resource="https://pinboard.in/u:Vaguery/t:proof"/>
	<rdf:li rdf:resource="https://pinboard.in/u:Vaguery/t:to-write-about"/>
	<rdf:li rdf:resource="https://pinboard.in/u:Vaguery/t:consider:looking-to-see"/>
</rdf:Bag></taxo:topics>
</item>
<item rdf:about="https://arxiv.org/abs/1908.07881">
    <title>[1908.07881] Apollonian Packing in Polydisperse Emulsions</title>
    <dc:date>2020-10-13T22:52:18+00:00</dc:date>
    <link>https://arxiv.org/abs/1908.07881</link>
    <dc:creator>Vaguery</dc:creator><description><![CDATA[We have discovered the existence of polydisperse High Internal-Phase-Ratio Emulsions (HIPE) in which the internal-phase droplets, present at 95% volume fraction, remain spherical and organize themselves in the available space according to Apollonian packing rules. These polydisperse HIPE are formed during emulsification of surfactant-poor compositions of oil-surfactant-water two-phase systems. Their droplet size-distributions evolve spontaneously towards power laws with the Apollonian exponent. Small-Angle X-Ray Scattering performed on aged HIPEs demonstrated that the droplet packing structure coincided with that of a numerically simulated Random Apollonian Packing. We argue that these peculiar, space-filling assemblies are a result of coalescence and fragmentation processes obeying simple geometrical rules of conserving total volume and minimizing surface area.
]]></description>
<dc:subject>materials-science indistinguishable-from-magic nanotechnology rather-interesting geometry physics! to-write-about to-simulate consider:3d packing</dc:subject>
<dc:source>https://pinboard.in/</dc:source>
<dc:identifier>https://pinboard.in/u:Vaguery/b:f7373d0d8635/</dc:identifier>
<taxo:topics><rdf:Bag>	<rdf:li rdf:resource="https://pinboard.in/u:Vaguery/t:materials-science"/>
	<rdf:li rdf:resource="https://pinboard.in/u:Vaguery/t:indistinguishable-from-magic"/>
	<rdf:li rdf:resource="https://pinboard.in/u:Vaguery/t:nanotechnology"/>
	<rdf:li rdf:resource="https://pinboard.in/u:Vaguery/t:rather-interesting"/>
	<rdf:li rdf:resource="https://pinboard.in/u:Vaguery/t:geometry"/>
	<rdf:li rdf:resource="https://pinboard.in/u:Vaguery/t:physics!"/>
	<rdf:li rdf:resource="https://pinboard.in/u:Vaguery/t:to-write-about"/>
	<rdf:li rdf:resource="https://pinboard.in/u:Vaguery/t:to-simulate"/>
	<rdf:li rdf:resource="https://pinboard.in/u:Vaguery/t:consider:3d"/>
	<rdf:li rdf:resource="https://pinboard.in/u:Vaguery/t:packing"/>
</rdf:Bag></taxo:topics>
</item>
<item rdf:about="https://en.wikipedia.org/wiki/Coxeter%E2%80%93Dynkin_diagram">
    <title>Coxeter–Dynkin diagram - Wikipedia</title>
    <dc:date>2020-05-23T12:12:47+00:00</dc:date>
    <link>https://en.wikipedia.org/wiki/Coxeter%E2%80%93Dynkin_diagram</link>
    <dc:creator>Vaguery</dc:creator><description><![CDATA[In geometry, a Coxeter–Dynkin diagram (or Coxeter diagram, Coxeter graph) is a graph with numerically labeled edges (called branches) representing the spatial relations between a collection of mirrors (or reflecting hyperplanes). It describes a kaleidoscopic construction: each graph "node" represents a mirror (domain facet) and the label attached to a branch encodes the dihedral angle order between two mirrors (on a domain ridge), that is, the amount by which the angle between the reflective planes can be multiplied by to get 180 degrees. An unlabeled branch implicitly represents order-3 (60 degrees).
]]></description>
<dc:subject>geometry polyhedra representation rather-interesting to-understand to-write-about to-simulate consider:visualization consider:animation</dc:subject>
<dc:source>https://pinboard.in/</dc:source>
<dc:identifier>https://pinboard.in/u:Vaguery/b:a728a64450bb/</dc:identifier>
<taxo:topics><rdf:Bag>	<rdf:li rdf:resource="https://pinboard.in/u:Vaguery/t:geometry"/>
	<rdf:li rdf:resource="https://pinboard.in/u:Vaguery/t:polyhedra"/>
	<rdf:li rdf:resource="https://pinboard.in/u:Vaguery/t:representation"/>
	<rdf:li rdf:resource="https://pinboard.in/u:Vaguery/t:rather-interesting"/>
	<rdf:li rdf:resource="https://pinboard.in/u:Vaguery/t:to-understand"/>
	<rdf:li rdf:resource="https://pinboard.in/u:Vaguery/t:to-write-about"/>
	<rdf:li rdf:resource="https://pinboard.in/u:Vaguery/t:to-simulate"/>
	<rdf:li rdf:resource="https://pinboard.in/u:Vaguery/t:consider:visualization"/>
	<rdf:li rdf:resource="https://pinboard.in/u:Vaguery/t:consider:animation"/>
</rdf:Bag></taxo:topics>
</item>
<item rdf:about="https://arxiv.org/abs/2001.09364">
    <title>[2001.09364] Classifying Regular Polyhedra and Polytopes using Wythoff's Construction</title>
    <dc:date>2020-05-23T12:03:20+00:00</dc:date>
    <link>https://arxiv.org/abs/2001.09364</link>
    <dc:creator>Vaguery</dc:creator><description><![CDATA[A polytope is the generalization of a polyhedron to any number of dimensions. The regular polyhedra are the Platonic solids: the tetrahedron, octahedron, cube, icosahedron, and dodecahedron. The hypercubes, hyperoctahedra, simplices, and regular polygons form four infinite fa milies of regular polytopes. Ludwig Schläfli proved that with the addition of five exceptional solids (the icosahedron and dodecahedron in 3 dimensions, and the 24-cell, 120-cell, and 600-cell in 4 dimensions) this list is complete. This paper provides an alternate proof to Schläfli's result, using Wythoff's construction and the theory of decorated Coxeter diagrams.
]]></description>
<dc:subject>polyhedra geometry generative-models classification to-understand algorithms representation isomorphic-play to-write-about to-simulate consider:representation consider:animation</dc:subject>
<dc:source>https://pinboard.in/</dc:source>
<dc:identifier>https://pinboard.in/u:Vaguery/b:4ebeab94a472/</dc:identifier>
<taxo:topics><rdf:Bag>	<rdf:li rdf:resource="https://pinboard.in/u:Vaguery/t:polyhedra"/>
	<rdf:li rdf:resource="https://pinboard.in/u:Vaguery/t:geometry"/>
	<rdf:li rdf:resource="https://pinboard.in/u:Vaguery/t:generative-models"/>
	<rdf:li rdf:resource="https://pinboard.in/u:Vaguery/t:classification"/>
	<rdf:li rdf:resource="https://pinboard.in/u:Vaguery/t:to-understand"/>
	<rdf:li rdf:resource="https://pinboard.in/u:Vaguery/t:algorithms"/>
	<rdf:li rdf:resource="https://pinboard.in/u:Vaguery/t:representation"/>
	<rdf:li rdf:resource="https://pinboard.in/u:Vaguery/t:isomorphic-play"/>
	<rdf:li rdf:resource="https://pinboard.in/u:Vaguery/t:to-write-about"/>
	<rdf:li rdf:resource="https://pinboard.in/u:Vaguery/t:to-simulate"/>
	<rdf:li rdf:resource="https://pinboard.in/u:Vaguery/t:consider:representation"/>
	<rdf:li rdf:resource="https://pinboard.in/u:Vaguery/t:consider:animation"/>
</rdf:Bag></taxo:topics>
</item>
<item rdf:about="https://arxiv.org/abs/1910.00418">
    <title>[1910.00418] Relationships Between Six Excircles</title>
    <dc:date>2020-05-18T21:47:06+00:00</dc:date>
    <link>https://arxiv.org/abs/1910.00418</link>
    <dc:creator>Vaguery</dc:creator><description><![CDATA[P is a point inside △ABC, then the cevians through P divide △ABC into smaller triangles of various sizes. We give theorems about the relationship between the radii of certain excircles of some of these triangles.]]></description>
<dc:subject>plane-geometry proof rather-interesting pattern-discovery geometry to-write-about to-simulate consider:ellipses</dc:subject>
<dc:source>https://pinboard.in/</dc:source>
<dc:identifier>https://pinboard.in/u:Vaguery/b:57700fc8a4a9/</dc:identifier>
<taxo:topics><rdf:Bag>	<rdf:li rdf:resource="https://pinboard.in/u:Vaguery/t:plane-geometry"/>
	<rdf:li rdf:resource="https://pinboard.in/u:Vaguery/t:proof"/>
	<rdf:li rdf:resource="https://pinboard.in/u:Vaguery/t:rather-interesting"/>
	<rdf:li rdf:resource="https://pinboard.in/u:Vaguery/t:pattern-discovery"/>
	<rdf:li rdf:resource="https://pinboard.in/u:Vaguery/t:geometry"/>
	<rdf:li rdf:resource="https://pinboard.in/u:Vaguery/t:to-write-about"/>
	<rdf:li rdf:resource="https://pinboard.in/u:Vaguery/t:to-simulate"/>
	<rdf:li rdf:resource="https://pinboard.in/u:Vaguery/t:consider:ellipses"/>
</rdf:Bag></taxo:topics>
</item>
<item rdf:about="https://arxiv.org/abs/1011.4034">
    <title>[1011.4034] Dense packing crystal structures of physical tetrahedra</title>
    <dc:date>2020-02-09T00:35:41+00:00</dc:date>
    <link>https://arxiv.org/abs/1011.4034</link>
    <dc:creator>Vaguery</dc:creator><description><![CDATA[We present a method for discovering dense packings of general convex hard particles and apply it to study the dense packing behavior of a one-parameter family of particles with tetrahedral symmetry representing a deformation of the ideal mathematical tetrahedron into a less ideal, physical, tetrahedron and all the way to the sphere. Thus, we also connect the two well studied problems of sphere packing and tetrahedron packing on a single axis. Our numerical results uncover a rich optimal-packing behavior, compared to that of other continuous families of particles previously studied. We present four structures as candidates for the optimal packing at different values of the parameter, providing an atlas of crystal structures which might be observed in systems of nano-particles with tetrahedral symmetry.
]]></description>
<dc:subject>packing granular-materials optimization geometry rather-interesting to-simulate consider:looking-to-see consider:particle-sims</dc:subject>
<dc:source>https://pinboard.in/</dc:source>
<dc:identifier>https://pinboard.in/u:Vaguery/b:5bf2339ca8ef/</dc:identifier>
<taxo:topics><rdf:Bag>	<rdf:li rdf:resource="https://pinboard.in/u:Vaguery/t:packing"/>
	<rdf:li rdf:resource="https://pinboard.in/u:Vaguery/t:granular-materials"/>
	<rdf:li rdf:resource="https://pinboard.in/u:Vaguery/t:optimization"/>
	<rdf:li rdf:resource="https://pinboard.in/u:Vaguery/t:geometry"/>
	<rdf:li rdf:resource="https://pinboard.in/u:Vaguery/t:rather-interesting"/>
	<rdf:li rdf:resource="https://pinboard.in/u:Vaguery/t:to-simulate"/>
	<rdf:li rdf:resource="https://pinboard.in/u:Vaguery/t:consider:looking-to-see"/>
	<rdf:li rdf:resource="https://pinboard.in/u:Vaguery/t:consider:particle-sims"/>
</rdf:Bag></taxo:topics>
</item>
<item rdf:about="https://arxiv.org/abs/1805.06512">
    <title>[1805.06512] The Broken Stick Project</title>
    <dc:date>2020-01-19T15:37:00+00:00</dc:date>
    <link>https://arxiv.org/abs/1805.06512</link>
    <dc:creator>Vaguery</dc:creator><description><![CDATA[The broken stick problem is the following classical question. 
You have a segment [0,1]. You choose two points on this segment at random. They divide the segment into three smaller segments. Show that the probability that the three segments form a triangle is 1/4. 
The MIT PRIMES program, together with Art of Problem Solving, organized a high school research project where participants worked on several variations of this problem. Participants were generally high school students who posted ideas and progress to the Art of Problem Solving forums over the course of an entire year, under the supervision of PRIMES mentors. This report summarizes the findings of this CrowdMath project.
]]></description>
<dc:subject>crowdsourcing see-author open-questions geometry probability-theory rather-interesting to-write-about to-simulate consider:genetic-programming consider:performance-measures</dc:subject>
<dc:source>https://pinboard.in/</dc:source>
<dc:identifier>https://pinboard.in/u:Vaguery/b:d7f139a0c81c/</dc:identifier>
<taxo:topics><rdf:Bag>	<rdf:li rdf:resource="https://pinboard.in/u:Vaguery/t:crowdsourcing"/>
	<rdf:li rdf:resource="https://pinboard.in/u:Vaguery/t:see-author"/>
	<rdf:li rdf:resource="https://pinboard.in/u:Vaguery/t:open-questions"/>
	<rdf:li rdf:resource="https://pinboard.in/u:Vaguery/t:geometry"/>
	<rdf:li rdf:resource="https://pinboard.in/u:Vaguery/t:probability-theory"/>
	<rdf:li rdf:resource="https://pinboard.in/u:Vaguery/t:rather-interesting"/>
	<rdf:li rdf:resource="https://pinboard.in/u:Vaguery/t:to-write-about"/>
	<rdf:li rdf:resource="https://pinboard.in/u:Vaguery/t:to-simulate"/>
	<rdf:li rdf:resource="https://pinboard.in/u:Vaguery/t:consider:genetic-programming"/>
	<rdf:li rdf:resource="https://pinboard.in/u:Vaguery/t:consider:performance-measures"/>
</rdf:Bag></taxo:topics>
</item>
<item rdf:about="https://divisbyzero.com/2017/03/30/a-geometric-proof-of-brookss-trisection/">
    <title>A Geometric Proof of Brooks’s Trisection? – David Richeson: Division by Zero</title>
    <dc:date>2019-12-26T13:14:39+00:00</dc:date>
    <link>https://divisbyzero.com/2017/03/30/a-geometric-proof-of-brookss-trisection/</link>
    <dc:creator>Vaguery</dc:creator><description><![CDATA[For instance, I made this applet showing how to use the “cycloid of Ceva” to trisect an angle. (It is based on Archimedes’s neusis [marked straightedge] construction.)
]]></description>
<dc:subject>geometry plane-geometry impossible-problems define-your-terms rather-interesting algorithms to-write-about consider:genetic-programming consider:rediscovery consider:inverse-problem consider:evolve-the-tool</dc:subject>
<dc:source>https://pinboard.in/</dc:source>
<dc:identifier>https://pinboard.in/u:Vaguery/b:097391bc02a8/</dc:identifier>
<taxo:topics><rdf:Bag>	<rdf:li rdf:resource="https://pinboard.in/u:Vaguery/t:geometry"/>
	<rdf:li rdf:resource="https://pinboard.in/u:Vaguery/t:plane-geometry"/>
	<rdf:li rdf:resource="https://pinboard.in/u:Vaguery/t:impossible-problems"/>
	<rdf:li rdf:resource="https://pinboard.in/u:Vaguery/t:define-your-terms"/>
	<rdf:li rdf:resource="https://pinboard.in/u:Vaguery/t:rather-interesting"/>
	<rdf:li rdf:resource="https://pinboard.in/u:Vaguery/t:algorithms"/>
	<rdf:li rdf:resource="https://pinboard.in/u:Vaguery/t:to-write-about"/>
	<rdf:li rdf:resource="https://pinboard.in/u:Vaguery/t:consider:genetic-programming"/>
	<rdf:li rdf:resource="https://pinboard.in/u:Vaguery/t:consider:rediscovery"/>
	<rdf:li rdf:resource="https://pinboard.in/u:Vaguery/t:consider:inverse-problem"/>
	<rdf:li rdf:resource="https://pinboard.in/u:Vaguery/t:consider:evolve-the-tool"/>
</rdf:Bag></taxo:topics>
</item>
<item rdf:about="https://link.springer.com/journal/volumesAndIssues/4">
    <title>Nexus Network Journal - All Volumes &amp; Issues - Springer</title>
    <dc:date>2019-10-14T12:07:05+00:00</dc:date>
    <link>https://link.springer.com/journal/volumesAndIssues/4</link>
    <dc:creator>Vaguery</dc:creator><description><![CDATA[Nexus Network Journal
Architecture and Mathematics]]></description>
<dc:subject>history-of-mathematics architecture geometry rather-interesting cannot-read to-browse</dc:subject>
<dc:source>https://pinboard.in/</dc:source>
<dc:identifier>https://pinboard.in/u:Vaguery/b:73261551888a/</dc:identifier>
<taxo:topics><rdf:Bag>	<rdf:li rdf:resource="https://pinboard.in/u:Vaguery/t:history-of-mathematics"/>
	<rdf:li rdf:resource="https://pinboard.in/u:Vaguery/t:architecture"/>
	<rdf:li rdf:resource="https://pinboard.in/u:Vaguery/t:geometry"/>
	<rdf:li rdf:resource="https://pinboard.in/u:Vaguery/t:rather-interesting"/>
	<rdf:li rdf:resource="https://pinboard.in/u:Vaguery/t:cannot-read"/>
	<rdf:li rdf:resource="https://pinboard.in/u:Vaguery/t:to-browse"/>
</rdf:Bag></taxo:topics>
</item>
<item rdf:about="https://markareynolds.com/about/">
    <title>Mark A Reynolds » About</title>
    <dc:date>2019-09-28T11:01:15+00:00</dc:date>
    <link>https://markareynolds.com/about/</link>
    <dc:creator>Vaguery</dc:creator><description><![CDATA[Mark A. Reynolds has worked with graphite, pen and ink, watercolor, pastel, and other drawing media for over thirty years, developing interpretations of traditional principles found in Euclidean and philosophical geometry. He has discovered systems and harmonic constructions with mixed ratios and new geometric relationships he calls “marriages of incommensurables” and “unions of opposites”. These geometric systems unite chaos and order in complex, intriguing, and beautiful geometric drawings. Mark’s work demonstrates his point-of-view regarding geometry as an art form and its interdisciplinary relationship with music, nature, mathematics, architecture, and personal discovery.

]]></description>
<dc:subject>art artist mathematical-recreations very-nice to-watch geometry</dc:subject>
<dc:source>https://pinboard.in/</dc:source>
<dc:identifier>https://pinboard.in/u:Vaguery/b:ae3d9d29b35e/</dc:identifier>
<taxo:topics><rdf:Bag>	<rdf:li rdf:resource="https://pinboard.in/u:Vaguery/t:art"/>
	<rdf:li rdf:resource="https://pinboard.in/u:Vaguery/t:artist"/>
	<rdf:li rdf:resource="https://pinboard.in/u:Vaguery/t:mathematical-recreations"/>
	<rdf:li rdf:resource="https://pinboard.in/u:Vaguery/t:very-nice"/>
	<rdf:li rdf:resource="https://pinboard.in/u:Vaguery/t:to-watch"/>
	<rdf:li rdf:resource="https://pinboard.in/u:Vaguery/t:geometry"/>
</rdf:Bag></taxo:topics>
</item>
<item rdf:about="https://arxiv.org/abs/1708.01559">
    <title>[1708.01559] Spherical Geometry and the Least Symmetric Triangle</title>
    <dc:date>2019-09-09T10:43:14+00:00</dc:date>
    <link>https://arxiv.org/abs/1708.01559</link>
    <dc:creator>Vaguery</dc:creator><description><![CDATA[We study the problem of determining the least symmetric triangle, which arises both from pure geometry and from the study of molecular chirality in chemistry. Using the correspondence between planar n-gons and points in the Grassmannian of 2-planes in real n-space introduced by Hausmann and Knutson, this corresponds to finding the point in the fundamental domain of the hyperoctahedral group action on the Grassmannian which is furthest from the boundary, which we compute exactly. We also determine the least symmetric obtuse and acute triangles. These calculations provide prototypes for computations on polygon and shape spaces.
]]></description>
<dc:subject>symmetry optimization rather-interesting define-your-terms performance-measure to-write-about to-emulate to-simulate consider:looking-to-see consider:robustness consider:stamp-collecting geometry plane-geometry</dc:subject>
<dc:source>https://pinboard.in/</dc:source>
<dc:identifier>https://pinboard.in/u:Vaguery/b:57d2fd877722/</dc:identifier>
<taxo:topics><rdf:Bag>	<rdf:li rdf:resource="https://pinboard.in/u:Vaguery/t:symmetry"/>
	<rdf:li rdf:resource="https://pinboard.in/u:Vaguery/t:optimization"/>
	<rdf:li rdf:resource="https://pinboard.in/u:Vaguery/t:rather-interesting"/>
	<rdf:li rdf:resource="https://pinboard.in/u:Vaguery/t:define-your-terms"/>
	<rdf:li rdf:resource="https://pinboard.in/u:Vaguery/t:performance-measure"/>
	<rdf:li rdf:resource="https://pinboard.in/u:Vaguery/t:to-write-about"/>
	<rdf:li rdf:resource="https://pinboard.in/u:Vaguery/t:to-emulate"/>
	<rdf:li rdf:resource="https://pinboard.in/u:Vaguery/t:to-simulate"/>
	<rdf:li rdf:resource="https://pinboard.in/u:Vaguery/t:consider:looking-to-see"/>
	<rdf:li rdf:resource="https://pinboard.in/u:Vaguery/t:consider:robustness"/>
	<rdf:li rdf:resource="https://pinboard.in/u:Vaguery/t:consider:stamp-collecting"/>
	<rdf:li rdf:resource="https://pinboard.in/u:Vaguery/t:geometry"/>
	<rdf:li rdf:resource="https://pinboard.in/u:Vaguery/t:plane-geometry"/>
</rdf:Bag></taxo:topics>
</item>
<item rdf:about="https://arxiv.org/abs/0910.5226">
    <title>[0910.5226] Dense periodic packings of tetrahedra with small repeating units</title>
    <dc:date>2019-09-07T12:12:24+00:00</dc:date>
    <link>https://arxiv.org/abs/0910.5226</link>
    <dc:creator>Vaguery</dc:creator><description><![CDATA[We present a one-parameter family of periodic packings of regular tetrahedra, with the packing fraction 100/117≈0.8547, that are simple in the sense that they are transitive and their repeating units involve only four tetrahedra. The construction of the packings was inspired from results of a numerical search that yielded a similar packing. We present an analytic construction of the packings and a description of their properties. We also present a transitive packing with a repeating unit of two tetrahedra and a packing fraction 139+4010√369≈0.7194.]]></description>
<dc:subject>geometry packing optimization rather-interesting to-understand to-simulate to-write-about consider:looking-to-see computational-geometry</dc:subject>
<dc:source>https://pinboard.in/</dc:source>
<dc:identifier>https://pinboard.in/u:Vaguery/b:c81bfd6924f6/</dc:identifier>
<taxo:topics><rdf:Bag>	<rdf:li rdf:resource="https://pinboard.in/u:Vaguery/t:geometry"/>
	<rdf:li rdf:resource="https://pinboard.in/u:Vaguery/t:packing"/>
	<rdf:li rdf:resource="https://pinboard.in/u:Vaguery/t:optimization"/>
	<rdf:li rdf:resource="https://pinboard.in/u:Vaguery/t:rather-interesting"/>
	<rdf:li rdf:resource="https://pinboard.in/u:Vaguery/t:to-understand"/>
	<rdf:li rdf:resource="https://pinboard.in/u:Vaguery/t:to-simulate"/>
	<rdf:li rdf:resource="https://pinboard.in/u:Vaguery/t:to-write-about"/>
	<rdf:li rdf:resource="https://pinboard.in/u:Vaguery/t:consider:looking-to-see"/>
	<rdf:li rdf:resource="https://pinboard.in/u:Vaguery/t:computational-geometry"/>
</rdf:Bag></taxo:topics>
</item>
<item rdf:about="https://arxiv.org/abs/1509.02241">
    <title>[1509.02241] The local optimality of the double lattice packing</title>
    <dc:date>2019-09-07T12:06:46+00:00</dc:date>
    <link>https://arxiv.org/abs/1509.02241</link>
    <dc:creator>Vaguery</dc:creator><description><![CDATA[This paper introduces a technique for proving the local optimality of packing configurations. Applying this technique to a general convex polygon, we prove that the construction of the optimal double lattice packing by Kuperberg and Kuperberg is also locally optimal in the full space of packings.
]]></description>
<dc:subject>geometry proof techniques algorithms packing plane-geometry optimization performance-measure to-understand to-write-about consider:looking-to-see consider:rediscovery consider:robustness</dc:subject>
<dc:source>https://pinboard.in/</dc:source>
<dc:identifier>https://pinboard.in/u:Vaguery/b:7caa3f823a4d/</dc:identifier>
<taxo:topics><rdf:Bag>	<rdf:li rdf:resource="https://pinboard.in/u:Vaguery/t:geometry"/>
	<rdf:li rdf:resource="https://pinboard.in/u:Vaguery/t:proof"/>
	<rdf:li rdf:resource="https://pinboard.in/u:Vaguery/t:techniques"/>
	<rdf:li rdf:resource="https://pinboard.in/u:Vaguery/t:algorithms"/>
	<rdf:li rdf:resource="https://pinboard.in/u:Vaguery/t:packing"/>
	<rdf:li rdf:resource="https://pinboard.in/u:Vaguery/t:plane-geometry"/>
	<rdf:li rdf:resource="https://pinboard.in/u:Vaguery/t:optimization"/>
	<rdf:li rdf:resource="https://pinboard.in/u:Vaguery/t:performance-measure"/>
	<rdf:li rdf:resource="https://pinboard.in/u:Vaguery/t:to-understand"/>
	<rdf:li rdf:resource="https://pinboard.in/u:Vaguery/t:to-write-about"/>
	<rdf:li rdf:resource="https://pinboard.in/u:Vaguery/t:consider:looking-to-see"/>
	<rdf:li rdf:resource="https://pinboard.in/u:Vaguery/t:consider:rediscovery"/>
	<rdf:li rdf:resource="https://pinboard.in/u:Vaguery/t:consider:robustness"/>
</rdf:Bag></taxo:topics>
</item>
<item rdf:about="https://arxiv.org/abs/1904.02043">
    <title>[1904.02043] Platonic Compounds of Cylinders</title>
    <dc:date>2019-09-07T11:59:16+00:00</dc:date>
    <link>https://arxiv.org/abs/1904.02043</link>
    <dc:creator>Vaguery</dc:creator><description><![CDATA[Generalizing the octahedral configuration of six congruent cylinders touching the unit sphere, we exhibit configurations of congruent cylinders associated to a pair of dual Platonic bodies.
]]></description>
<dc:subject>geometry combinatorics stamp-collecting rather-interesting optimization to-understand to-write-about consider:looking-to-see consider:representation counting</dc:subject>
<dc:source>https://pinboard.in/</dc:source>
<dc:identifier>https://pinboard.in/u:Vaguery/b:e04d95a00c99/</dc:identifier>
<taxo:topics><rdf:Bag>	<rdf:li rdf:resource="https://pinboard.in/u:Vaguery/t:geometry"/>
	<rdf:li rdf:resource="https://pinboard.in/u:Vaguery/t:combinatorics"/>
	<rdf:li rdf:resource="https://pinboard.in/u:Vaguery/t:stamp-collecting"/>
	<rdf:li rdf:resource="https://pinboard.in/u:Vaguery/t:rather-interesting"/>
	<rdf:li rdf:resource="https://pinboard.in/u:Vaguery/t:optimization"/>
	<rdf:li rdf:resource="https://pinboard.in/u:Vaguery/t:to-understand"/>
	<rdf:li rdf:resource="https://pinboard.in/u:Vaguery/t:to-write-about"/>
	<rdf:li rdf:resource="https://pinboard.in/u:Vaguery/t:consider:looking-to-see"/>
	<rdf:li rdf:resource="https://pinboard.in/u:Vaguery/t:consider:representation"/>
	<rdf:li rdf:resource="https://pinboard.in/u:Vaguery/t:counting"/>
</rdf:Bag></taxo:topics>
</item>
<item rdf:about="https://arxiv.org/abs/1611.10297">
    <title>[1611.10297] Configuration Spaces of Equal Spheres Touching a Given Sphere: The Twelve Spheres Problem</title>
    <dc:date>2019-09-07T11:35:37+00:00</dc:date>
    <link>https://arxiv.org/abs/1611.10297</link>
    <dc:creator>Vaguery</dc:creator><description><![CDATA[The problem of twelve spheres is to understand, as a function of r∈(0,rmax(12)], the configuration space of 12 non-overlapping equal spheres of radius r touching a central unit sphere. It considers to what extent, and in what fashion, touching spheres can be varied, subject to the constraint of always touching the central sphere. Such constrained motion problems are of interest in physics and materials science, and the problem involves topology and geometry. This paper reviews the history of work on this problem, presents some new results, and formulates some conjectures. It also presents general results on configuration spaces of N spheres of radius r touching a central unit sphere, with emphasis on 3≤N≤14. The problem of determining the maximal radius rmax(N) is a version of the Tammes problem, to which László Fejes Tóth made significant contributions.
]]></description>
<dc:subject>geometry constraint-satisfaction optimization open-questions rather-interesting classic-problems to-understand history-of-mathematics</dc:subject>
<dc:source>https://pinboard.in/</dc:source>
<dc:identifier>https://pinboard.in/u:Vaguery/b:985f7fe26d91/</dc:identifier>
<taxo:topics><rdf:Bag>	<rdf:li rdf:resource="https://pinboard.in/u:Vaguery/t:geometry"/>
	<rdf:li rdf:resource="https://pinboard.in/u:Vaguery/t:constraint-satisfaction"/>
	<rdf:li rdf:resource="https://pinboard.in/u:Vaguery/t:optimization"/>
	<rdf:li rdf:resource="https://pinboard.in/u:Vaguery/t:open-questions"/>
	<rdf:li rdf:resource="https://pinboard.in/u:Vaguery/t:rather-interesting"/>
	<rdf:li rdf:resource="https://pinboard.in/u:Vaguery/t:classic-problems"/>
	<rdf:li rdf:resource="https://pinboard.in/u:Vaguery/t:to-understand"/>
	<rdf:li rdf:resource="https://pinboard.in/u:Vaguery/t:history-of-mathematics"/>
</rdf:Bag></taxo:topics>
</item>
<item rdf:about="https://www.math3ma.com/blog/multiplying-non-numbers">
    <title>Multiplying Non-Numbers</title>
    <dc:date>2019-07-14T12:50:05+00:00</dc:date>
    <link>https://www.math3ma.com/blog/multiplying-non-numbers</link>
    <dc:creator>Vaguery</dc:creator><description><![CDATA[In last last week's episode of PBS Infinite Series, we talked about different flavors of multiplication (like associativity and commutativity) to think about when multiplying things that aren't numbers. My examples of multiplying non-numbers were vectors and matrices, which come from the land of algebra. Today I'd like to highlight another example:

]]></description>
<dc:subject>out-of-the-box geometry group-theory to-write-about have-written-about</dc:subject>
<dc:source>https://pinboard.in/</dc:source>
<dc:identifier>https://pinboard.in/u:Vaguery/b:8d0c9541444d/</dc:identifier>
<taxo:topics><rdf:Bag>	<rdf:li rdf:resource="https://pinboard.in/u:Vaguery/t:out-of-the-box"/>
	<rdf:li rdf:resource="https://pinboard.in/u:Vaguery/t:geometry"/>
	<rdf:li rdf:resource="https://pinboard.in/u:Vaguery/t:group-theory"/>
	<rdf:li rdf:resource="https://pinboard.in/u:Vaguery/t:to-write-about"/>
	<rdf:li rdf:resource="https://pinboard.in/u:Vaguery/t:have-written-about"/>
</rdf:Bag></taxo:topics>
</item>
<item rdf:about="https://arxiv.org/abs/1702.04199">
    <title>[1702.04199] The problem of camouflaging via mirror reflections</title>
    <dc:date>2019-05-06T11:46:40+00:00</dc:date>
    <link>https://arxiv.org/abs/1702.04199</link>
    <dc:creator>Vaguery</dc:creator><description><![CDATA[This work is related to billiards and their applications in geometric optics. It is known that perfectly invisible bodies with mirror surface do not exist. It is natural to search for bodies that are, in a sense, close to invisible. We introduce a {\it visibility index} of a body measuring the mean angle of deviation of incident light rays, and derive a lower estimate to this index. This estimate is a function of the body's volume and of the minimal radius of a ball containing the body. This result is far from being final and opens a possibility for further research.
]]></description>
<dc:subject>billiards optics optimization geometry inverse-problems rather-interesting approximation to-write-about nudge-targets consider:looking-to-see</dc:subject>
<dc:source>https://pinboard.in/</dc:source>
<dc:identifier>https://pinboard.in/u:Vaguery/b:e0c3966edb17/</dc:identifier>
<taxo:topics><rdf:Bag>	<rdf:li rdf:resource="https://pinboard.in/u:Vaguery/t:billiards"/>
	<rdf:li rdf:resource="https://pinboard.in/u:Vaguery/t:optics"/>
	<rdf:li rdf:resource="https://pinboard.in/u:Vaguery/t:optimization"/>
	<rdf:li rdf:resource="https://pinboard.in/u:Vaguery/t:geometry"/>
	<rdf:li rdf:resource="https://pinboard.in/u:Vaguery/t:inverse-problems"/>
	<rdf:li rdf:resource="https://pinboard.in/u:Vaguery/t:rather-interesting"/>
	<rdf:li rdf:resource="https://pinboard.in/u:Vaguery/t:approximation"/>
	<rdf:li rdf:resource="https://pinboard.in/u:Vaguery/t:to-write-about"/>
	<rdf:li rdf:resource="https://pinboard.in/u:Vaguery/t:nudge-targets"/>
	<rdf:li rdf:resource="https://pinboard.in/u:Vaguery/t:consider:looking-to-see"/>
</rdf:Bag></taxo:topics>
</item>
<item rdf:about="https://arxiv.org/abs/1305.4012">
    <title>[1305.4012] On the support of a body by a surface with random roughness</title>
    <dc:date>2019-05-06T10:40:52+00:00</dc:date>
    <link>https://arxiv.org/abs/1305.4012</link>
    <dc:creator>Vaguery</dc:creator><description><![CDATA[Suppose an interval is put on a horizontal line with random roughness. With probability one it is supported at two points, one from the left, and another from the right from its center. We compute probability distribution of support points provided the roughness is fine grained. We also solve an analogous problem where a circle is put on a rough plane. Some applications in static are given.
]]></description>
<dc:subject>extreme-values probability-theory geometry construction looking-to-see</dc:subject>
<dc:source>https://pinboard.in/</dc:source>
<dc:identifier>https://pinboard.in/u:Vaguery/b:24a56e4b0b95/</dc:identifier>
<taxo:topics><rdf:Bag>	<rdf:li rdf:resource="https://pinboard.in/u:Vaguery/t:extreme-values"/>
	<rdf:li rdf:resource="https://pinboard.in/u:Vaguery/t:probability-theory"/>
	<rdf:li rdf:resource="https://pinboard.in/u:Vaguery/t:geometry"/>
	<rdf:li rdf:resource="https://pinboard.in/u:Vaguery/t:construction"/>
	<rdf:li rdf:resource="https://pinboard.in/u:Vaguery/t:looking-to-see"/>
</rdf:Bag></taxo:topics>
</item>
<item rdf:about="https://arxiv.org/abs/1607.04758">
    <title>[1607.04758] Projective configuration theorems: old wine into new wineskins</title>
    <dc:date>2019-05-06T09:52:59+00:00</dc:date>
    <link>https://arxiv.org/abs/1607.04758</link>
    <dc:creator>Vaguery</dc:creator><description><![CDATA[This is a survey of select recent results by a number of authors, inspired by the classical configuration theorems of projective geometry.
]]></description>
<dc:subject>geometry review rather-interesting computational-geometry proof mathematical-recreations plane-geometry to-write-about to-simulate consider:rediscovery billiards</dc:subject>
<dc:source>https://pinboard.in/</dc:source>
<dc:identifier>https://pinboard.in/u:Vaguery/b:d7ad1f479e04/</dc:identifier>
<taxo:topics><rdf:Bag>	<rdf:li rdf:resource="https://pinboard.in/u:Vaguery/t:geometry"/>
	<rdf:li rdf:resource="https://pinboard.in/u:Vaguery/t:review"/>
	<rdf:li rdf:resource="https://pinboard.in/u:Vaguery/t:rather-interesting"/>
	<rdf:li rdf:resource="https://pinboard.in/u:Vaguery/t:computational-geometry"/>
	<rdf:li rdf:resource="https://pinboard.in/u:Vaguery/t:proof"/>
	<rdf:li rdf:resource="https://pinboard.in/u:Vaguery/t:mathematical-recreations"/>
	<rdf:li rdf:resource="https://pinboard.in/u:Vaguery/t:plane-geometry"/>
	<rdf:li rdf:resource="https://pinboard.in/u:Vaguery/t:to-write-about"/>
	<rdf:li rdf:resource="https://pinboard.in/u:Vaguery/t:to-simulate"/>
	<rdf:li rdf:resource="https://pinboard.in/u:Vaguery/t:consider:rediscovery"/>
	<rdf:li rdf:resource="https://pinboard.in/u:Vaguery/t:billiards"/>
</rdf:Bag></taxo:topics>
</item>
<item rdf:about="https://arxiv.org/abs/1304.5708">
    <title>[1304.5708] Pentagram Spirals</title>
    <dc:date>2019-05-06T09:21:32+00:00</dc:date>
    <link>https://arxiv.org/abs/1304.5708</link>
    <dc:creator>Vaguery</dc:creator><description><![CDATA[We introduce a geometric construction which relates to the pentagram map much in the way that a logarithmic spiral relates to a circle. After introducing the construction, we establish some basic geometric facts about it, and speculate on some of the deeper algebraic structure, such as the complete integrability of the associated dynamical system.
]]></description>
<dc:subject>dynamical-systems mathematical-recreations plane-geometry geometry construction rather-interesting looking-to-see</dc:subject>
<dc:source>https://pinboard.in/</dc:source>
<dc:identifier>https://pinboard.in/u:Vaguery/b:e6168d657aa1/</dc:identifier>
<taxo:topics><rdf:Bag>	<rdf:li rdf:resource="https://pinboard.in/u:Vaguery/t:dynamical-systems"/>
	<rdf:li rdf:resource="https://pinboard.in/u:Vaguery/t:mathematical-recreations"/>
	<rdf:li rdf:resource="https://pinboard.in/u:Vaguery/t:plane-geometry"/>
	<rdf:li rdf:resource="https://pinboard.in/u:Vaguery/t:geometry"/>
	<rdf:li rdf:resource="https://pinboard.in/u:Vaguery/t:construction"/>
	<rdf:li rdf:resource="https://pinboard.in/u:Vaguery/t:rather-interesting"/>
	<rdf:li rdf:resource="https://pinboard.in/u:Vaguery/t:looking-to-see"/>
</rdf:Bag></taxo:topics>
</item>
<item rdf:about="https://arxiv.org/abs/1804.06483">
    <title>[1804.06483] On Rigid Origami II: Quadrilateral Creased Papers</title>
    <dc:date>2019-05-04T13:50:49+00:00</dc:date>
    <link>https://arxiv.org/abs/1804.06483</link>
    <dc:creator>Vaguery</dc:creator><description><![CDATA[This paper describes several new variations of large rigid-foldable quadrilateral creased papers, which are generated by "stitching" together rigid-foldable Kokotsakis quadrilaterals. These creased papers are constructed with the following additional requirements: (a) There is at least one rigid folding motion for which no folding angle remains constant. (b) The quadrilateral creased paper is infinitely extendable in both longitudinal and transverse directions. (c) The sector angles, which define the crease directions, can be solved quadrilateral by quadrilateral. This work is based on a nearly complete classification of rigid-foldable Kokotsakis quadrilaterals from Ivan Izmestiev. All quadrilateral creased papers described in this paper have one degree of freedom in each branch of their rigid folding motion.
]]></description>
<dc:subject>origami engineering-design rather-interesting geometry materials-science kinematics planning to-write-about</dc:subject>
<dc:source>https://pinboard.in/</dc:source>
<dc:identifier>https://pinboard.in/u:Vaguery/b:e6ccf02ba6ef/</dc:identifier>
<taxo:topics><rdf:Bag>	<rdf:li rdf:resource="https://pinboard.in/u:Vaguery/t:origami"/>
	<rdf:li rdf:resource="https://pinboard.in/u:Vaguery/t:engineering-design"/>
	<rdf:li rdf:resource="https://pinboard.in/u:Vaguery/t:rather-interesting"/>
	<rdf:li rdf:resource="https://pinboard.in/u:Vaguery/t:geometry"/>
	<rdf:li rdf:resource="https://pinboard.in/u:Vaguery/t:materials-science"/>
	<rdf:li rdf:resource="https://pinboard.in/u:Vaguery/t:kinematics"/>
	<rdf:li rdf:resource="https://pinboard.in/u:Vaguery/t:planning"/>
	<rdf:li rdf:resource="https://pinboard.in/u:Vaguery/t:to-write-about"/>
</rdf:Bag></taxo:topics>
</item>
<item rdf:about="http://isohedral.ca/hexagonal-cross-stitch/">
    <title>Hexagonal Cross Stitch – Isohedral</title>
    <dc:date>2019-05-02T09:07:19+00:00</dc:date>
    <link>http://isohedral.ca/hexagonal-cross-stitch/</link>
    <dc:creator>Vaguery</dc:creator><description><![CDATA[If you are familiar with repeating two-dimensional patterns, you will know that there are precisely 17 distinct pattern types, known as wallpaper groups. This sampler shows only 12. But as Susan pointed out in her talk, Shephard’s piece is nevertheless a complete symmetry sampler. In cross-stitch, the stitches are applied to a woven fabric (called “aida”), which has holes for stitches arranged in a square lattice. Rectangular grids are fundamental to woven cloth, and necessarily constrain the symmetries that are achievable in cross-stitch. In particular, the square grid cannot support threefold or sixfold rotations (consider, for example, that no three points in a square lattice can form an equilateral triangle), which prohibits the five wallpaper groups that include them (for the record, they are p3, p31m, p3m1, p6, and p6m in the traditional crystallographic notation).

]]></description>
<dc:subject>making tiling geometry craft rather-interesting mathematical-recreations enumeration</dc:subject>
<dc:source>https://pinboard.in/</dc:source>
<dc:identifier>https://pinboard.in/u:Vaguery/b:761ce1a95699/</dc:identifier>
<taxo:topics><rdf:Bag>	<rdf:li rdf:resource="https://pinboard.in/u:Vaguery/t:making"/>
	<rdf:li rdf:resource="https://pinboard.in/u:Vaguery/t:tiling"/>
	<rdf:li rdf:resource="https://pinboard.in/u:Vaguery/t:geometry"/>
	<rdf:li rdf:resource="https://pinboard.in/u:Vaguery/t:craft"/>
	<rdf:li rdf:resource="https://pinboard.in/u:Vaguery/t:rather-interesting"/>
	<rdf:li rdf:resource="https://pinboard.in/u:Vaguery/t:mathematical-recreations"/>
	<rdf:li rdf:resource="https://pinboard.in/u:Vaguery/t:enumeration"/>
</rdf:Bag></taxo:topics>
</item>
<item rdf:about="https://arxiv.org/abs/1804.03770">
    <title>[1804.03770] Tilings of Sphere by Congruent Pentagons I: Edge Combinations $a^2b^2c$ and $a^3bc$</title>
    <dc:date>2019-04-24T14:17:41+00:00</dc:date>
    <link>https://arxiv.org/abs/1804.03770</link>
    <dc:creator>Vaguery</dc:creator><description><![CDATA[We develop the basic tools for classifying edge-to-edge tilings of sphere by congruent pentagons. Then we prove such tilings for edge combination a2b2c are three families of pentagonal subdivisions of the platonic solids, with 12, 24 and 60 tiles. We also prove that such tilings for edge combination a3bc are two unique double pentagonal subdivisions, with 48 and 120 tiles.
]]></description>
<dc:subject>tiling geometry topology rather-interesting enumeration representation to-write-about to-do consider:parameter-sweeps-and-animations</dc:subject>
<dc:source>https://pinboard.in/</dc:source>
<dc:identifier>https://pinboard.in/u:Vaguery/b:edccf47a30ab/</dc:identifier>
<taxo:topics><rdf:Bag>	<rdf:li rdf:resource="https://pinboard.in/u:Vaguery/t:tiling"/>
	<rdf:li rdf:resource="https://pinboard.in/u:Vaguery/t:geometry"/>
	<rdf:li rdf:resource="https://pinboard.in/u:Vaguery/t:topology"/>
	<rdf:li rdf:resource="https://pinboard.in/u:Vaguery/t:rather-interesting"/>
	<rdf:li rdf:resource="https://pinboard.in/u:Vaguery/t:enumeration"/>
	<rdf:li rdf:resource="https://pinboard.in/u:Vaguery/t:representation"/>
	<rdf:li rdf:resource="https://pinboard.in/u:Vaguery/t:to-write-about"/>
	<rdf:li rdf:resource="https://pinboard.in/u:Vaguery/t:to-do"/>
	<rdf:li rdf:resource="https://pinboard.in/u:Vaguery/t:consider:parameter-sweeps-and-animations"/>
</rdf:Bag></taxo:topics>
</item>
<item rdf:about="https://arxiv.org/abs/1903.02712">
    <title>[1903.02712] Tilings of Sphere by Congruent Pentagons II: Edge Combination $a^3b^2$</title>
    <dc:date>2019-04-24T14:16:37+00:00</dc:date>
    <link>https://arxiv.org/abs/1903.02712</link>
    <dc:creator>Vaguery</dc:creator><description><![CDATA[There are fifteen edge-to-edge tilings of sphere by congruent pentagons with edge combination a3b2: five one-parameter families of pentagonal subdivisions, two tilings by 24 copies of a unique pentagon, four tilings by 60 copies of second unique pentagon, four tilings by 60 copies of third unique pentagon.
]]></description>
<dc:subject>tiling geometry rather-odd to-simulate consider:genetic-programming consider:robustness consider:scanning-parameter-space</dc:subject>
<dc:source>https://pinboard.in/</dc:source>
<dc:identifier>https://pinboard.in/u:Vaguery/b:102c91044eca/</dc:identifier>
<taxo:topics><rdf:Bag>	<rdf:li rdf:resource="https://pinboard.in/u:Vaguery/t:tiling"/>
	<rdf:li rdf:resource="https://pinboard.in/u:Vaguery/t:geometry"/>
	<rdf:li rdf:resource="https://pinboard.in/u:Vaguery/t:rather-odd"/>
	<rdf:li rdf:resource="https://pinboard.in/u:Vaguery/t:to-simulate"/>
	<rdf:li rdf:resource="https://pinboard.in/u:Vaguery/t:consider:genetic-programming"/>
	<rdf:li rdf:resource="https://pinboard.in/u:Vaguery/t:consider:robustness"/>
	<rdf:li rdf:resource="https://pinboard.in/u:Vaguery/t:consider:scanning-parameter-space"/>
</rdf:Bag></taxo:topics>
</item>
<item rdf:about="https://arxiv.org/abs/math/0604585">
    <title>[math/0604585] A Strong Law for the Largest Nearest-Neighbor Link on Normally Distributed Points</title>
    <dc:date>2019-04-20T22:03:26+00:00</dc:date>
    <link>https://arxiv.org/abs/math/0604585</link>
    <dc:creator>Vaguery</dc:creator><description><![CDATA[Let n points be placed independently in d−dimensional space according to the standard d−dimensional normal distribution. Let dn be the longest edge length for the nearest neighbor graph on these points. We show that
\lim_{n
\rar \infty} \frac{\sqrt{\log n} d_n}{\log \log n} = \frac{d}{\sqrt{2}}, \qquad
d \geq 2, {a.s.}]]></description>
<dc:subject>probability-theory geometry consider:looking-to-see consider:genetic-programming consider:rediscovery</dc:subject>
<dc:source>https://pinboard.in/</dc:source>
<dc:identifier>https://pinboard.in/u:Vaguery/b:497d723f8d70/</dc:identifier>
<taxo:topics><rdf:Bag>	<rdf:li rdf:resource="https://pinboard.in/u:Vaguery/t:probability-theory"/>
	<rdf:li rdf:resource="https://pinboard.in/u:Vaguery/t:geometry"/>
	<rdf:li rdf:resource="https://pinboard.in/u:Vaguery/t:consider:looking-to-see"/>
	<rdf:li rdf:resource="https://pinboard.in/u:Vaguery/t:consider:genetic-programming"/>
	<rdf:li rdf:resource="https://pinboard.in/u:Vaguery/t:consider:rediscovery"/>
</rdf:Bag></taxo:topics>
</item>
<item rdf:about="https://arxiv.org/abs/math/0702073">
    <title>[math/0702073] Unbounded Orbits for Outer Billiards</title>
    <dc:date>2019-04-20T21:18:07+00:00</dc:date>
    <link>https://arxiv.org/abs/math/0702073</link>
    <dc:creator>Vaguery</dc:creator><description><![CDATA[Outer billiards is a basic dynamical system, defined relative to a planar convex shape. This system was introduced in the 1950's by B.H. Neumann and later popularized in the 1970's by J. Moser. All along, one of the central questions has been: is there an outer billiards system with an unbounded orbit. We answer this question by proving that outer billiards defined relative to the Penrose Kite has an unbounded orbit. The Penrose kite is the quadrilateral that appears in the famous Penrose tiling. We also analyze some of the finer orbit structure of outer billiards on the penrose kite. This analysis shows that there is an uncountable set of unbounded orbits. Our method of proof relates the problem to self-similar tilings, polygon exchange maps, and arithmetic dynamics.
]]></description>
<dc:subject>billiards mathematical-recreations generative-models geometry dynamical-systems rather-interesting chaos to-simulate to-write-about</dc:subject>
<dc:source>https://pinboard.in/</dc:source>
<dc:identifier>https://pinboard.in/u:Vaguery/b:85199ae66c7d/</dc:identifier>
<taxo:topics><rdf:Bag>	<rdf:li rdf:resource="https://pinboard.in/u:Vaguery/t:billiards"/>
	<rdf:li rdf:resource="https://pinboard.in/u:Vaguery/t:mathematical-recreations"/>
	<rdf:li rdf:resource="https://pinboard.in/u:Vaguery/t:generative-models"/>
	<rdf:li rdf:resource="https://pinboard.in/u:Vaguery/t:geometry"/>
	<rdf:li rdf:resource="https://pinboard.in/u:Vaguery/t:dynamical-systems"/>
	<rdf:li rdf:resource="https://pinboard.in/u:Vaguery/t:rather-interesting"/>
	<rdf:li rdf:resource="https://pinboard.in/u:Vaguery/t:chaos"/>
	<rdf:li rdf:resource="https://pinboard.in/u:Vaguery/t:to-simulate"/>
	<rdf:li rdf:resource="https://pinboard.in/u:Vaguery/t:to-write-about"/>
</rdf:Bag></taxo:topics>
</item>
<item rdf:about="https://arxiv.org/abs/1612.09295">
    <title>[1612.09295] First Families of Regular Polygons and their Mutations</title>
    <dc:date>2019-04-19T21:02:39+00:00</dc:date>
    <link>https://arxiv.org/abs/1612.09295</link>
    <dc:creator>Vaguery</dc:creator><description><![CDATA[Every regular N-gon generates a canonical family of regular polygons which are conforming to the bounds of the 'star polygons' determined by N. These star polygons are formed from truncated extended edges of the N-gon and the intersection points determine a scaling which defines the parameters of the family. In 'First Families of Regular Polygons' (arXiv:1503.05536) we showed that this scaling forms a basis for the maximal real subfield of the cyclotomic field of N. The traditional generator for this subfield is 2cos(2Pi/N) so it has order Phi(N)/2 where Phi is the Euler totient function. This order is known as the 'algebraic complexity' of N. The family of regular polygons shares this same scaling and complexity, so members of this family are an intrinsic part of any regular polygon - and we call them the First Family of N. 
Here we start from first principles and give an algebraic derivation of the First Families showing how each star[k] point of N defines a scale[k] and also an S[k] 'tile' of the family. Under a piecewise isometry such as the outer-billiards map the 'singularity set' W can be formed by iterating the extended edges of N and we show that W can be reduced to a 'shear and rotation' which preserves the S[k], so the 'web' W can be regarded as the disjoint union (coproduct) of the local webs of the S[k]. These local webs are still very complex but we show in Lemma 4.1 that the center of each S[k] has a constant step-k orbit around N. These 'resonant' orbits set bounds on global orbits and establish a connection between geometry and dynamics. For the first time it is possible to make predictions about the small-scale geometry of the S[1] and S[2] tiles on the edges of N and give a plausible explanation for the long-standing '4k+1'conjecture of arXiv:1311.6763 about extended families of tiles when N = 8k+2. Each 8k+j family has unique edge geometry.]]></description>
<dc:subject>geometry plane-geometry dynamical-systems rather-interesting to-understand billiards generative-models</dc:subject>
<dc:source>https://pinboard.in/</dc:source>
<dc:identifier>https://pinboard.in/u:Vaguery/b:d73f43c0d985/</dc:identifier>
<taxo:topics><rdf:Bag>	<rdf:li rdf:resource="https://pinboard.in/u:Vaguery/t:geometry"/>
	<rdf:li rdf:resource="https://pinboard.in/u:Vaguery/t:plane-geometry"/>
	<rdf:li rdf:resource="https://pinboard.in/u:Vaguery/t:dynamical-systems"/>
	<rdf:li rdf:resource="https://pinboard.in/u:Vaguery/t:rather-interesting"/>
	<rdf:li rdf:resource="https://pinboard.in/u:Vaguery/t:to-understand"/>
	<rdf:li rdf:resource="https://pinboard.in/u:Vaguery/t:billiards"/>
	<rdf:li rdf:resource="https://pinboard.in/u:Vaguery/t:generative-models"/>
</rdf:Bag></taxo:topics>
</item>
<item rdf:about="http://blog.eikeland.se/2019/04/13/oloid/">
    <title>Coding a rolling Oloid</title>
    <dc:date>2019-04-15T10:40:27+00:00</dc:date>
    <link>http://blog.eikeland.se/2019/04/13/oloid/</link>
    <dc:creator>Vaguery</dc:creator><description><![CDATA[Did you know there's a whole class rolling objects that's not spherical or cylindrical and still able to roll in a given direction? I didn't!

Some of them have them even have interesting properties like the fact that every point on their surface touch the ground as they're rolling. The Sphericon and the Oloid are two such objects.

As a fun little experiment I wanted to code and print one.

]]></description>
<dc:subject>Clojure IDE rather-interesting geometry mathematical-recreations 3d 3d-printing looking-to-see</dc:subject>
<dc:source>https://pinboard.in/</dc:source>
<dc:identifier>https://pinboard.in/u:Vaguery/b:67bb65487ff6/</dc:identifier>
<taxo:topics><rdf:Bag>	<rdf:li rdf:resource="https://pinboard.in/u:Vaguery/t:Clojure"/>
	<rdf:li rdf:resource="https://pinboard.in/u:Vaguery/t:IDE"/>
	<rdf:li rdf:resource="https://pinboard.in/u:Vaguery/t:rather-interesting"/>
	<rdf:li rdf:resource="https://pinboard.in/u:Vaguery/t:geometry"/>
	<rdf:li rdf:resource="https://pinboard.in/u:Vaguery/t:mathematical-recreations"/>
	<rdf:li rdf:resource="https://pinboard.in/u:Vaguery/t:3d"/>
	<rdf:li rdf:resource="https://pinboard.in/u:Vaguery/t:3d-printing"/>
	<rdf:li rdf:resource="https://pinboard.in/u:Vaguery/t:looking-to-see"/>
</rdf:Bag></taxo:topics>
</item>
<item rdf:about="https://arxiv.org/abs/1805.02222">
    <title>[1805.02222] A Computer Approach to Determine the Densest Translative Tetrahedron Packings</title>
    <dc:date>2019-03-09T13:34:56+00:00</dc:date>
    <link>https://arxiv.org/abs/1805.02222</link>
    <dc:creator>Vaguery</dc:creator><description><![CDATA[In 1900, as a part of his 18th problem, Hilbert proposed the question to determine the densest congruent (or translative) packings of a given solid, such as the unit ball or the regular tetrahedron of unit edges. Up to now, our knowledge about this problem is still very limited, excepting the ball case. It is conjectured that, for some particular solids such as tetrahedra, cuboctahedra and octahedra, their maximal translative packing densities and their maximal lattice packing densities are identical. To attack this conjecture, this paper suggests a computer approach to determine the maximal local translative packing density of a given polytope, by studying associated color graphs and applying optimization. In particular, all the tetrahedral case, the cuboctahedral case and the octahedral case of the conjecture have been reduced into finite numbers of manageable optimization problems.
]]></description>
<dc:subject>packing open-questions geometry looking-to-see representation feature-construction to-write-about</dc:subject>
<dc:source>https://pinboard.in/</dc:source>
<dc:identifier>https://pinboard.in/u:Vaguery/b:2283ff677902/</dc:identifier>
<taxo:topics><rdf:Bag>	<rdf:li rdf:resource="https://pinboard.in/u:Vaguery/t:packing"/>
	<rdf:li rdf:resource="https://pinboard.in/u:Vaguery/t:open-questions"/>
	<rdf:li rdf:resource="https://pinboard.in/u:Vaguery/t:geometry"/>
	<rdf:li rdf:resource="https://pinboard.in/u:Vaguery/t:looking-to-see"/>
	<rdf:li rdf:resource="https://pinboard.in/u:Vaguery/t:representation"/>
	<rdf:li rdf:resource="https://pinboard.in/u:Vaguery/t:feature-construction"/>
	<rdf:li rdf:resource="https://pinboard.in/u:Vaguery/t:to-write-about"/>
</rdf:Bag></taxo:topics>
</item>
<item rdf:about="http://marctenbosch.com/quaternions/">
    <title>Let’s remove Quaternions from every 3D Engine (An Interactive Introduction to Rotors from Geometric Algebra)</title>
    <dc:date>2019-03-03T15:57:18+00:00</dc:date>
    <link>http://marctenbosch.com/quaternions/</link>
    <dc:creator>Vaguery</dc:creator><description><![CDATA[(An Interactive Introduction to Rotors from Geometric Algebra) 
]]></description>
<dc:subject>3d algebra geometry very-good vector-algebra representation to-write-about to-include-in-languages wedge-product oriented-linear-subspaces</dc:subject>
<dc:source>https://pinboard.in/</dc:source>
<dc:identifier>https://pinboard.in/u:Vaguery/b:407e4d090e97/</dc:identifier>
<taxo:topics><rdf:Bag>	<rdf:li rdf:resource="https://pinboard.in/u:Vaguery/t:3d"/>
	<rdf:li rdf:resource="https://pinboard.in/u:Vaguery/t:algebra"/>
	<rdf:li rdf:resource="https://pinboard.in/u:Vaguery/t:geometry"/>
	<rdf:li rdf:resource="https://pinboard.in/u:Vaguery/t:very-good"/>
	<rdf:li rdf:resource="https://pinboard.in/u:Vaguery/t:vector-algebra"/>
	<rdf:li rdf:resource="https://pinboard.in/u:Vaguery/t:representation"/>
	<rdf:li rdf:resource="https://pinboard.in/u:Vaguery/t:to-write-about"/>
	<rdf:li rdf:resource="https://pinboard.in/u:Vaguery/t:to-include-in-languages"/>
	<rdf:li rdf:resource="https://pinboard.in/u:Vaguery/t:wedge-product"/>
	<rdf:li rdf:resource="https://pinboard.in/u:Vaguery/t:oriented-linear-subspaces"/>
</rdf:Bag></taxo:topics>
</item>
<item rdf:about="https://publications.mfo.de/handle/mfo/1378">
    <title>Tropical geometry</title>
    <dc:date>2019-02-24T11:37:15+00:00</dc:date>
    <link>https://publications.mfo.de/handle/mfo/1378</link>
    <dc:creator>Vaguery</dc:creator><description><![CDATA[What kind of strange spaces hide behind the enigmatic
 name of tropical geometry? In the tropics, just
 as in other geometries, one of the simplest objects is
 a line. Therefore, we begin our exploration by considering
 tropical lines. Afterwards, we take a look at
 tropical arithmetic and algebra, and describe how to
 define tropical curves using tropical polynomials.
]]></description>
<dc:subject>tropical-arithmetic geometry rather-interesting representation category-theory algebra review</dc:subject>
<dc:source>https://pinboard.in/</dc:source>
<dc:identifier>https://pinboard.in/u:Vaguery/b:08d9d8279fe2/</dc:identifier>
<taxo:topics><rdf:Bag>	<rdf:li rdf:resource="https://pinboard.in/u:Vaguery/t:tropical-arithmetic"/>
	<rdf:li rdf:resource="https://pinboard.in/u:Vaguery/t:geometry"/>
	<rdf:li rdf:resource="https://pinboard.in/u:Vaguery/t:rather-interesting"/>
	<rdf:li rdf:resource="https://pinboard.in/u:Vaguery/t:representation"/>
	<rdf:li rdf:resource="https://pinboard.in/u:Vaguery/t:category-theory"/>
	<rdf:li rdf:resource="https://pinboard.in/u:Vaguery/t:algebra"/>
	<rdf:li rdf:resource="https://pinboard.in/u:Vaguery/t:review"/>
</rdf:Bag></taxo:topics>
</item>
<item rdf:about="https://arxiv.org/abs/1812.05410">
    <title>[1812.05410] Peeling Digital Potatoes</title>
    <dc:date>2019-02-06T00:12:28+00:00</dc:date>
    <link>https://arxiv.org/abs/1812.05410</link>
    <dc:creator>Vaguery</dc:creator><description><![CDATA[The potato-peeling problem (also known as convex skull) is a fundamental computational geometry problem and the fastest algorithm to date runs in O(n8) time for a polygon with n vertices that may have holes. In this paper, we consider a digital version of the problem. A set K⊂ℤ2 is digital convex if conv(K)∩ℤ2=K, where conv(K) denotes the convex hull of K. Given a set S of n lattice points, we present polynomial time algorithms to the problems of finding the largest digital convex subset K of S (digital potato-peeling problem) and the largest union of two digital convex subsets of S. The two algorithms take roughly O(n3) and O(n9) time, respectively. We also show that those algorithms provide an approximation to the continuous versions.]]></description>
<dc:subject>computational-complexity computational-geometry algorithms geometry to-write-about nudge-targets</dc:subject>
<dc:source>https://pinboard.in/</dc:source>
<dc:identifier>https://pinboard.in/u:Vaguery/b:efa18ecfb1f2/</dc:identifier>
<taxo:topics><rdf:Bag>	<rdf:li rdf:resource="https://pinboard.in/u:Vaguery/t:computational-complexity"/>
	<rdf:li rdf:resource="https://pinboard.in/u:Vaguery/t:computational-geometry"/>
	<rdf:li rdf:resource="https://pinboard.in/u:Vaguery/t:algorithms"/>
	<rdf:li rdf:resource="https://pinboard.in/u:Vaguery/t:geometry"/>
	<rdf:li rdf:resource="https://pinboard.in/u:Vaguery/t:to-write-about"/>
	<rdf:li rdf:resource="https://pinboard.in/u:Vaguery/t:nudge-targets"/>
</rdf:Bag></taxo:topics>
</item>
<item rdf:about="https://samjshah.com/2018/03/06/a-nice-proof-for-the-law-of-cosines/">
    <title>A nice proof for the Law of Cosines | Continuous Everywhere but Differentiable Nowhere</title>
    <dc:date>2019-02-05T11:58:26+00:00</dc:date>
    <link>https://samjshah.com/2018/03/06/a-nice-proof-for-the-law-of-cosines/</link>
    <dc:creator>Vaguery</dc:creator><description><![CDATA[And I kinda told them what to do… Meh. I was jumping way ahead to get to the formula. We weren’t savoring the thinking to get to the formula. Now we are.

That being said, I ran across something quite beautiful. A stunning proof of the Law of Cosines (at least for acute triangles) on the site trigonography.]]></description>
<dc:subject>geometry pedagogy visualization rather-interesting visual-proof feature-construction explanation consider:the-mangle</dc:subject>
<dc:source>https://pinboard.in/</dc:source>
<dc:identifier>https://pinboard.in/u:Vaguery/b:f6409052060b/</dc:identifier>
<taxo:topics><rdf:Bag>	<rdf:li rdf:resource="https://pinboard.in/u:Vaguery/t:geometry"/>
	<rdf:li rdf:resource="https://pinboard.in/u:Vaguery/t:pedagogy"/>
	<rdf:li rdf:resource="https://pinboard.in/u:Vaguery/t:visualization"/>
	<rdf:li rdf:resource="https://pinboard.in/u:Vaguery/t:rather-interesting"/>
	<rdf:li rdf:resource="https://pinboard.in/u:Vaguery/t:visual-proof"/>
	<rdf:li rdf:resource="https://pinboard.in/u:Vaguery/t:feature-construction"/>
	<rdf:li rdf:resource="https://pinboard.in/u:Vaguery/t:explanation"/>
	<rdf:li rdf:resource="https://pinboard.in/u:Vaguery/t:consider:the-mangle"/>
</rdf:Bag></taxo:topics>
</item>
<item rdf:about="https://arxiv.org/abs/1809.04243">
    <title>[1809.04243] Self-foldability of monohedral quadrilateral origami tessellations</title>
    <dc:date>2019-02-05T11:03:54+00:00</dc:date>
    <link>https://arxiv.org/abs/1809.04243</link>
    <dc:creator>Vaguery</dc:creator><description><![CDATA[Using a mathematical model for self-foldability of rigid origami, we determine which monohedral quadrilateral tilings of the plane are uniquely self-foldable. In particular, the Miura-ori and Chicken Wire patterns are not self-foldable under our definition, but such tilings that are rotationally-symmetric about the midpoints of the tile are uniquely self-foldable.
]]></description>
<dc:subject>origami constraint-satisfaction geometry classification rather-interesting to-write-about consider:algorithms</dc:subject>
<dc:source>https://pinboard.in/</dc:source>
<dc:identifier>https://pinboard.in/u:Vaguery/b:873805f9e770/</dc:identifier>
<taxo:topics><rdf:Bag>	<rdf:li rdf:resource="https://pinboard.in/u:Vaguery/t:origami"/>
	<rdf:li rdf:resource="https://pinboard.in/u:Vaguery/t:constraint-satisfaction"/>
	<rdf:li rdf:resource="https://pinboard.in/u:Vaguery/t:geometry"/>
	<rdf:li rdf:resource="https://pinboard.in/u:Vaguery/t:classification"/>
	<rdf:li rdf:resource="https://pinboard.in/u:Vaguery/t:rather-interesting"/>
	<rdf:li rdf:resource="https://pinboard.in/u:Vaguery/t:to-write-about"/>
	<rdf:li rdf:resource="https://pinboard.in/u:Vaguery/t:consider:algorithms"/>
</rdf:Bag></taxo:topics>
</item>
<item rdf:about="https://arxiv.org/abs/1802.04148">
    <title>[1802.04148] On Dendrites Generated By Symmetric Polygonal Systems: The Case of Regular Polygons</title>
    <dc:date>2019-01-08T11:21:25+00:00</dc:date>
    <link>https://arxiv.org/abs/1802.04148</link>
    <dc:creator>Vaguery</dc:creator><description><![CDATA[We define G-symmetric polygonal systems of similarities and study the properties of symmetric dendrites, which appear as their attractors. This allows us to find the conditions under which the attractor of a zipper becomes a dendrite.]]></description>
<dc:subject>fractals geometry nonlinear-dynamics rather-interesting purdy-pitchers</dc:subject>
<dc:source>https://pinboard.in/</dc:source>
<dc:identifier>https://pinboard.in/u:Vaguery/b:bf823012bd72/</dc:identifier>
<taxo:topics><rdf:Bag>	<rdf:li rdf:resource="https://pinboard.in/u:Vaguery/t:fractals"/>
	<rdf:li rdf:resource="https://pinboard.in/u:Vaguery/t:geometry"/>
	<rdf:li rdf:resource="https://pinboard.in/u:Vaguery/t:nonlinear-dynamics"/>
	<rdf:li rdf:resource="https://pinboard.in/u:Vaguery/t:rather-interesting"/>
	<rdf:li rdf:resource="https://pinboard.in/u:Vaguery/t:purdy-pitchers"/>
</rdf:Bag></taxo:topics>
</item>
<item rdf:about="https://arxiv.org/abs/1806.04129">
    <title>[1806.04129] Angels' staircases, Sturmian sequences, and trajectories on homothety surfaces</title>
    <dc:date>2018-12-25T13:05:57+00:00</dc:date>
    <link>https://arxiv.org/abs/1806.04129</link>
    <dc:creator>Vaguery</dc:creator><description><![CDATA[A homothety surface can be assembled from polygons by identifying their edges in pairs via homotheties, which are compositions of translation and scaling. We consider linear trajectories on a 1-parameter family of genus-2 homothety surfaces. The closure of a trajectory on each of these surfaces always has Hausdorff dimension 1, and contains either a closed loop or a lamination with Cantor cross-section. Trajectories have cutting sequences that are either eventually periodic or eventually Sturmian. Although no two of these surfaces are affinely equivalent, their linear trajectories can be related directly to those on the square torus, and thence to each other, by means of explicit functions. We also briefly examine two related families of surfaces and show that the above behaviors can be mixed; for instance, the closure of a linear trajectory can contain both a closed loop and a lamination.]]></description>
<dc:subject>plane-geometry geometry algebra continued-fractions fractals to-understand topology several-overlaps</dc:subject>
<dc:source>https://pinboard.in/</dc:source>
<dc:identifier>https://pinboard.in/u:Vaguery/b:a445a4a828fb/</dc:identifier>
<taxo:topics><rdf:Bag>	<rdf:li rdf:resource="https://pinboard.in/u:Vaguery/t:plane-geometry"/>
	<rdf:li rdf:resource="https://pinboard.in/u:Vaguery/t:geometry"/>
	<rdf:li rdf:resource="https://pinboard.in/u:Vaguery/t:algebra"/>
	<rdf:li rdf:resource="https://pinboard.in/u:Vaguery/t:continued-fractions"/>
	<rdf:li rdf:resource="https://pinboard.in/u:Vaguery/t:fractals"/>
	<rdf:li rdf:resource="https://pinboard.in/u:Vaguery/t:to-understand"/>
	<rdf:li rdf:resource="https://pinboard.in/u:Vaguery/t:topology"/>
	<rdf:li rdf:resource="https://pinboard.in/u:Vaguery/t:several-overlaps"/>
</rdf:Bag></taxo:topics>
</item>
<item rdf:about="https://arxiv.org/abs/1611.03052">
    <title>[1611.03052] Cluster algebraic interpretation of infinite friezes</title>
    <dc:date>2018-12-20T12:24:37+00:00</dc:date>
    <link>https://arxiv.org/abs/1611.03052</link>
    <dc:creator>Vaguery</dc:creator><description><![CDATA[Originally studied by Conway and Coxeter, friezes appeared in various recreational mathematics publications in the 1970s. More recently, in 2015, Baur, Parsons, and Tschabold constructed periodic infinite friezes and related them to matching numbers in the once-punctured disk and annulus. In this paper, we study such infinite friezes with an eye towards cluster algebras of type D and affine A, respectively. By examining infinite friezes with Laurent polynomial entries, we discover new symmetries and formulas relating the entries of this frieze to one another. Lastly, we also present a correspondence between Broline, Crowe and Isaacs's classical matching tuples and combinatorial interpretations of elements of cluster algebras from surfaces.]]></description>
<dc:subject>combinatorics algebra enumeration representation to-understand geometry</dc:subject>
<dc:source>https://pinboard.in/</dc:source>
<dc:identifier>https://pinboard.in/u:Vaguery/b:51d868803982/</dc:identifier>
<taxo:topics><rdf:Bag>	<rdf:li rdf:resource="https://pinboard.in/u:Vaguery/t:combinatorics"/>
	<rdf:li rdf:resource="https://pinboard.in/u:Vaguery/t:algebra"/>
	<rdf:li rdf:resource="https://pinboard.in/u:Vaguery/t:enumeration"/>
	<rdf:li rdf:resource="https://pinboard.in/u:Vaguery/t:representation"/>
	<rdf:li rdf:resource="https://pinboard.in/u:Vaguery/t:to-understand"/>
	<rdf:li rdf:resource="https://pinboard.in/u:Vaguery/t:geometry"/>
</rdf:Bag></taxo:topics>
</item>
<item rdf:about="https://arxiv.org/abs/1801.08003">
    <title>[1801.08003] Threadable Curves</title>
    <dc:date>2018-10-07T18:24:35+00:00</dc:date>
    <link>https://arxiv.org/abs/1801.08003</link>
    <dc:creator>Vaguery</dc:creator><description><![CDATA[We define a plane curve to be threadable if it can rigidly pass through a point-hole in a line L without otherwise touching L. Threadable curves are in a sense generalizations of monotone curves. We have two main results. The first is a linear-time algorithm for deciding whether a polygonal curve is threadable---O(n) for a curve of n vertices---and if threadable, finding a sequence of rigid motions to thread it through a hole. We also sketch an argument that shows that the threadability of algebraic curves can be decided in time polynomial in the degree of the curve. The second main result is an O(n polylog n)-time algorithm for deciding whether a 3D polygonal curve can thread through hole in a plane in R^3, and if so, providing a description of the rigid motions that achieve the threading.]]></description>
<dc:subject>computational-geometry geometry rather-interesting definition nudge-targets consider:feature-discovery to-write-about consider:algorithms</dc:subject>
<dc:source>https://pinboard.in/</dc:source>
<dc:identifier>https://pinboard.in/u:Vaguery/b:fb6b2344d0f0/</dc:identifier>
<taxo:topics><rdf:Bag>	<rdf:li rdf:resource="https://pinboard.in/u:Vaguery/t:computational-geometry"/>
	<rdf:li rdf:resource="https://pinboard.in/u:Vaguery/t:geometry"/>
	<rdf:li rdf:resource="https://pinboard.in/u:Vaguery/t:rather-interesting"/>
	<rdf:li rdf:resource="https://pinboard.in/u:Vaguery/t:definition"/>
	<rdf:li rdf:resource="https://pinboard.in/u:Vaguery/t:nudge-targets"/>
	<rdf:li rdf:resource="https://pinboard.in/u:Vaguery/t:consider:feature-discovery"/>
	<rdf:li rdf:resource="https://pinboard.in/u:Vaguery/t:to-write-about"/>
	<rdf:li rdf:resource="https://pinboard.in/u:Vaguery/t:consider:algorithms"/>
</rdf:Bag></taxo:topics>
</item>
<item rdf:about="https://arxiv.org/abs/1810.00173">
    <title>[1810.00173] On solids whose (entire) surface can be unfolded onto a plane</title>
    <dc:date>2018-10-07T16:00:25+00:00</dc:date>
    <link>https://arxiv.org/abs/1810.00173</link>
    <dc:creator>Vaguery</dc:creator><description><![CDATA[This is the English translation of Leonhard Euler's Latin paper "De solidis quorum superficiem in planum explicare licet". Euler explains several methods to obtain equations for developable surfaces. Therefore, this paper might be interesting for anyone studying the history of Differential Geometry.
]]></description>
<dc:subject>geometry history mathematics translation rather-interesting</dc:subject>
<dc:source>https://pinboard.in/</dc:source>
<dc:identifier>https://pinboard.in/u:Vaguery/b:053eb3b6178d/</dc:identifier>
<taxo:topics><rdf:Bag>	<rdf:li rdf:resource="https://pinboard.in/u:Vaguery/t:geometry"/>
	<rdf:li rdf:resource="https://pinboard.in/u:Vaguery/t:history"/>
	<rdf:li rdf:resource="https://pinboard.in/u:Vaguery/t:mathematics"/>
	<rdf:li rdf:resource="https://pinboard.in/u:Vaguery/t:translation"/>
	<rdf:li rdf:resource="https://pinboard.in/u:Vaguery/t:rather-interesting"/>
</rdf:Bag></taxo:topics>
</item>
<item rdf:about="https://mathlesstraveled.com/2018/07/30/smt-solutions/">
    <title>SMT solutions | The Math Less Traveled</title>
    <dc:date>2018-08-20T12:23:45+00:00</dc:date>
    <link>https://mathlesstraveled.com/2018/07/30/smt-solutions/</link>
    <dc:creator>Vaguery</dc:creator><description><![CDATA[In my last post I described the general approach I used to draw orthogons using an SMT solver, but left some of the details as exercises. In this post I’ll explain the solutions I came up with.

]]></description>
<dc:subject>geometry programming rather-interesting representation testing to-write-about mathematical-recreations nudge-targets consider:looking-to-see</dc:subject>
<dc:source>https://pinboard.in/</dc:source>
<dc:identifier>https://pinboard.in/u:Vaguery/b:d8ff96f5c1be/</dc:identifier>
<taxo:topics><rdf:Bag>	<rdf:li rdf:resource="https://pinboard.in/u:Vaguery/t:geometry"/>
	<rdf:li rdf:resource="https://pinboard.in/u:Vaguery/t:programming"/>
	<rdf:li rdf:resource="https://pinboard.in/u:Vaguery/t:rather-interesting"/>
	<rdf:li rdf:resource="https://pinboard.in/u:Vaguery/t:representation"/>
	<rdf:li rdf:resource="https://pinboard.in/u:Vaguery/t:testing"/>
	<rdf:li rdf:resource="https://pinboard.in/u:Vaguery/t:to-write-about"/>
	<rdf:li rdf:resource="https://pinboard.in/u:Vaguery/t:mathematical-recreations"/>
	<rdf:li rdf:resource="https://pinboard.in/u:Vaguery/t:nudge-targets"/>
	<rdf:li rdf:resource="https://pinboard.in/u:Vaguery/t:consider:looking-to-see"/>
</rdf:Bag></taxo:topics>
</item>
<item rdf:about="https://arxiv.org/abs/1806.00521">
    <title>[1806.00521] The lemniscate tree of a random polynomial</title>
    <dc:date>2018-06-30T11:29:07+00:00</dc:date>
    <link>https://arxiv.org/abs/1806.00521</link>
    <dc:creator>Vaguery</dc:creator><description><![CDATA[To each generic complex polynomial p(z) there is associated a labeled binary tree (here referred to as a "lemniscate tree") that encodes the topological type of the graph of |p(z)|. The branching structure of the lemniscate tree is determined by the configuration (i.e., arrangement in the plane) of the singular components of those level sets |p(z)|=t passing through a critical point. 
In this paper, we address the question "How many branches appear in a typical lemniscate tree?" We answer this question first for a lemniscate tree sampled uniformly from the combinatorial class and second for the lemniscate tree arising from a random polynomial generated by i.i.d. zeros. From a more general perspective, these results take a first step toward a probabilistic treatment (within a specialized setting) of Arnold's program of enumerating algebraic Morse functions.
]]></description>
<dc:subject>geometry topology rather-interesting probability-theory data-structures feature-construction to-understand to-write-about</dc:subject>
<dc:source>https://pinboard.in/</dc:source>
<dc:identifier>https://pinboard.in/u:Vaguery/b:82d560335b43/</dc:identifier>
<taxo:topics><rdf:Bag>	<rdf:li rdf:resource="https://pinboard.in/u:Vaguery/t:geometry"/>
	<rdf:li rdf:resource="https://pinboard.in/u:Vaguery/t:topology"/>
	<rdf:li rdf:resource="https://pinboard.in/u:Vaguery/t:rather-interesting"/>
	<rdf:li rdf:resource="https://pinboard.in/u:Vaguery/t:probability-theory"/>
	<rdf:li rdf:resource="https://pinboard.in/u:Vaguery/t:data-structures"/>
	<rdf:li rdf:resource="https://pinboard.in/u:Vaguery/t:feature-construction"/>
	<rdf:li rdf:resource="https://pinboard.in/u:Vaguery/t:to-understand"/>
	<rdf:li rdf:resource="https://pinboard.in/u:Vaguery/t:to-write-about"/>
</rdf:Bag></taxo:topics>
</item>
<item rdf:about="https://seekecho.blogspot.co.uk/2018/03/exploring-golden-ratio.html">
    <title>Exploring the golden ratio Φ</title>
    <dc:date>2018-05-28T12:09:03+00:00</dc:date>
    <link>https://seekecho.blogspot.co.uk/2018/03/exploring-golden-ratio.html</link>
    <dc:creator>Vaguery</dc:creator><description><![CDATA[This post is really just things I've learned from other people, but things that surprised me]]></description>
<dc:subject>mathematical-recreations phi geometry to-write-about</dc:subject>
<dc:source>https://pinboard.in/</dc:source>
<dc:identifier>https://pinboard.in/u:Vaguery/b:1c67c858a49a/</dc:identifier>
<taxo:topics><rdf:Bag>	<rdf:li rdf:resource="https://pinboard.in/u:Vaguery/t:mathematical-recreations"/>
	<rdf:li rdf:resource="https://pinboard.in/u:Vaguery/t:phi"/>
	<rdf:li rdf:resource="https://pinboard.in/u:Vaguery/t:geometry"/>
	<rdf:li rdf:resource="https://pinboard.in/u:Vaguery/t:to-write-about"/>
</rdf:Bag></taxo:topics>
</item>
<item rdf:about="https://mathlesstraveled.com/2018/01/31/orthogonal-polygons/">
    <title>Orthogonal polygons | The Math Less Traveled</title>
    <dc:date>2018-05-03T11:06:37+00:00</dc:date>
    <link>https://mathlesstraveled.com/2018/01/31/orthogonal-polygons/</link>
    <dc:creator>Vaguery</dc:creator><description><![CDATA[Quite a few commenters figured out what was going on, and mentioned several nice (equivalent) ways to think about it. Primarily, the idea is to draw all possible orthogonal polygons, that is, polygons with only right angles, organized by the total number of vertices. (So, for example, the picture above shows all orthogonal polygons with exactly ten vertices.) However, we have to be careful what we mean by the phrase “all possible”: there would be an infinite number of such polygons if we think about things like the precise lengths of edges. So we have to say when two polygons are considered the same, and when they are distinct. My rules are as follows:

]]></description>
<dc:subject>mathematical-recreations looking-to-see geometry polyominoes to-write-about</dc:subject>
<dc:source>https://pinboard.in/</dc:source>
<dc:identifier>https://pinboard.in/u:Vaguery/b:09a3f870657e/</dc:identifier>
<taxo:topics><rdf:Bag>	<rdf:li rdf:resource="https://pinboard.in/u:Vaguery/t:mathematical-recreations"/>
	<rdf:li rdf:resource="https://pinboard.in/u:Vaguery/t:looking-to-see"/>
	<rdf:li rdf:resource="https://pinboard.in/u:Vaguery/t:geometry"/>
	<rdf:li rdf:resource="https://pinboard.in/u:Vaguery/t:polyominoes"/>
	<rdf:li rdf:resource="https://pinboard.in/u:Vaguery/t:to-write-about"/>
</rdf:Bag></taxo:topics>
</item>
<item rdf:about="https://blogs.ams.org/visualinsight/2015/09/15/mcgee-graph/">
    <title>McGee Graph | Visual Insight</title>
    <dc:date>2018-04-27T00:34:09+00:00</dc:date>
    <link>https://blogs.ams.org/visualinsight/2015/09/15/mcgee-graph/</link>
    <dc:creator>Vaguery</dc:creator><description><![CDATA[The Heawood and Tutte–Coxeter graphs are Levi graphs. In other words, they arise from a finite configuration of points and lines, with each vertex corresponding to either a point or line, and an edge connecting two vertices whenever we have a point lying on a line. The McGee graph cannot be a Levi graph, because it is not bipartite. In other words, it doesn’t admit a 2-coloring of its vertices, where the edges only connect vertices of one color to vertices of the other color. The picture above shows a 3-coloring of the McGee graph.

]]></description>
<dc:subject>graph-theory geometry rather-interesting to-write-about nudge-targets</dc:subject>
<dc:source>https://pinboard.in/</dc:source>
<dc:identifier>https://pinboard.in/u:Vaguery/b:26ef92e0c171/</dc:identifier>
<taxo:topics><rdf:Bag>	<rdf:li rdf:resource="https://pinboard.in/u:Vaguery/t:graph-theory"/>
	<rdf:li rdf:resource="https://pinboard.in/u:Vaguery/t:geometry"/>
	<rdf:li rdf:resource="https://pinboard.in/u:Vaguery/t:rather-interesting"/>
	<rdf:li rdf:resource="https://pinboard.in/u:Vaguery/t:to-write-about"/>
	<rdf:li rdf:resource="https://pinboard.in/u:Vaguery/t:nudge-targets"/>
</rdf:Bag></taxo:topics>
</item>
<item rdf:about="http://dmccooey.com/polyhedra/index.html">
    <title>Visual Polyhedra</title>
    <dc:date>2017-12-23T10:16:19+00:00</dc:date>
    <link>http://dmccooey.com/polyhedra/index.html</link>
    <dc:creator>Vaguery</dc:creator><description><![CDATA[These pages present interactive graphical polyhedra organized in several categories. Each polyhedron's page contains a 3-dimensional virtual model of the polyhedron, followed by a summary of the polyhedron's vital statistics. The model provides an opaque visual mode, a translucent visual mode, and a metrics mode. In the visual modes, the polyhedron can be rotated and scaled, the perspective can be changed, and individual faces can be examined in isolation. The metrics mode computes the polyhedron's vital statistics empirically.

]]></description>
<dc:subject>geometry stamp-collecting polyhedra rather-interesting 3d to-write-about</dc:subject>
<dc:source>https://pinboard.in/</dc:source>
<dc:identifier>https://pinboard.in/u:Vaguery/b:661ea969ff47/</dc:identifier>
<taxo:topics><rdf:Bag>	<rdf:li rdf:resource="https://pinboard.in/u:Vaguery/t:geometry"/>
	<rdf:li rdf:resource="https://pinboard.in/u:Vaguery/t:stamp-collecting"/>
	<rdf:li rdf:resource="https://pinboard.in/u:Vaguery/t:polyhedra"/>
	<rdf:li rdf:resource="https://pinboard.in/u:Vaguery/t:rather-interesting"/>
	<rdf:li rdf:resource="https://pinboard.in/u:Vaguery/t:3d"/>
	<rdf:li rdf:resource="https://pinboard.in/u:Vaguery/t:to-write-about"/>
</rdf:Bag></taxo:topics>
</item>
<item rdf:about="https://arxiv.org/abs/1702.01027">
    <title>[1702.01027] Random Triangles and Polygons in the Plane</title>
    <dc:date>2017-10-20T13:19:47+00:00</dc:date>
    <link>https://arxiv.org/abs/1702.01027</link>
    <dc:creator>Vaguery</dc:creator><description><![CDATA[We consider the problem of finding the probability that a random triangle is obtuse, which was first raised by Lewis Caroll. Our investigation leads us to a natural correspondence between plane polygons and the Grassmann manifold of 2-planes in real n-space proposed by Allen Knutson and Jean-Claude Hausmann. This correspondence defines a natural probability measure on plane polygons. In these terms, we answer Caroll's question. We then explore the Grassmannian geometry of planar quadrilaterals, providing an answer to Sylvester's four-point problem, and describing explicitly the moduli space of unordered quadrilaterals. All of this provides a concrete introduction to a family of metrics used in shape classification and computer vision.]]></description>
<dc:subject>probability-theory open-problems mathematical-recreations geometry rather-interesting to-write-about</dc:subject>
<dc:source>https://pinboard.in/</dc:source>
<dc:identifier>https://pinboard.in/u:Vaguery/b:82808c2b27ca/</dc:identifier>
<taxo:topics><rdf:Bag>	<rdf:li rdf:resource="https://pinboard.in/u:Vaguery/t:probability-theory"/>
	<rdf:li rdf:resource="https://pinboard.in/u:Vaguery/t:open-problems"/>
	<rdf:li rdf:resource="https://pinboard.in/u:Vaguery/t:mathematical-recreations"/>
	<rdf:li rdf:resource="https://pinboard.in/u:Vaguery/t:geometry"/>
	<rdf:li rdf:resource="https://pinboard.in/u:Vaguery/t:rather-interesting"/>
	<rdf:li rdf:resource="https://pinboard.in/u:Vaguery/t:to-write-about"/>
</rdf:Bag></taxo:topics>
</item>
<item rdf:about="https://arxiv.org/abs/math/0103224">
    <title>[math/0103224] On the Minimum Ropelength of Knots and Links</title>
    <dc:date>2017-10-20T11:47:18+00:00</dc:date>
    <link>https://arxiv.org/abs/math/0103224</link>
    <dc:creator>Vaguery</dc:creator><description><![CDATA[The ropelength of a knot is the quotient of its length and its thickness, the radius of the largest embedded normal tube around the knot. We prove existence and regularity for ropelength minimizers in any knot or link type; these are C1,1 curves, but need not be smoother. We improve the lower bound for the ropelength of a nontrivial knot, and establish new ropelength bounds for small knots and links, including some which are sharp.]]></description>
<dc:subject>knot-theory optimization rather-interesting feature-construction geometry engineering-design nudge-targets consider:approximation</dc:subject>
<dc:source>https://pinboard.in/</dc:source>
<dc:identifier>https://pinboard.in/u:Vaguery/b:5b65f547564c/</dc:identifier>
<taxo:topics><rdf:Bag>	<rdf:li rdf:resource="https://pinboard.in/u:Vaguery/t:knot-theory"/>
	<rdf:li rdf:resource="https://pinboard.in/u:Vaguery/t:optimization"/>
	<rdf:li rdf:resource="https://pinboard.in/u:Vaguery/t:rather-interesting"/>
	<rdf:li rdf:resource="https://pinboard.in/u:Vaguery/t:feature-construction"/>
	<rdf:li rdf:resource="https://pinboard.in/u:Vaguery/t:geometry"/>
	<rdf:li rdf:resource="https://pinboard.in/u:Vaguery/t:engineering-design"/>
	<rdf:li rdf:resource="https://pinboard.in/u:Vaguery/t:nudge-targets"/>
	<rdf:li rdf:resource="https://pinboard.in/u:Vaguery/t:consider:approximation"/>
</rdf:Bag></taxo:topics>
</item>
<item rdf:about="https://arxiv.org/abs/math/0204106">
    <title>[math/0204106] The Second Hull of a Knotted Curve</title>
    <dc:date>2017-10-20T11:39:34+00:00</dc:date>
    <link>https://arxiv.org/abs/math/0204106</link>
    <dc:creator>Vaguery</dc:creator><description><![CDATA[The convex hull of a set K in space consists of points which are, in a certain sense, "surrounded" by K. When K is a closed curve, we define its higher hulls, consisting of points which are "multiply surrounded" by the curve. Our main theorem shows that if a curve is knotted then it has a nonempty second hull. This provides a new proof of the Fary/Milnor theorem that every knotted curve has total curvature at least 4pi.]]></description>
<dc:subject>knot-theory geometry topology rather-interesting feature-construction proof nudge-targets consider:representation consider:looking-to-see</dc:subject>
<dc:source>https://pinboard.in/</dc:source>
<dc:identifier>https://pinboard.in/u:Vaguery/b:a5799c956ddc/</dc:identifier>
<taxo:topics><rdf:Bag>	<rdf:li rdf:resource="https://pinboard.in/u:Vaguery/t:knot-theory"/>
	<rdf:li rdf:resource="https://pinboard.in/u:Vaguery/t:geometry"/>
	<rdf:li rdf:resource="https://pinboard.in/u:Vaguery/t:topology"/>
	<rdf:li rdf:resource="https://pinboard.in/u:Vaguery/t:rather-interesting"/>
	<rdf:li rdf:resource="https://pinboard.in/u:Vaguery/t:feature-construction"/>
	<rdf:li rdf:resource="https://pinboard.in/u:Vaguery/t:proof"/>
	<rdf:li rdf:resource="https://pinboard.in/u:Vaguery/t:nudge-targets"/>
	<rdf:li rdf:resource="https://pinboard.in/u:Vaguery/t:consider:representation"/>
	<rdf:li rdf:resource="https://pinboard.in/u:Vaguery/t:consider:looking-to-see"/>
</rdf:Bag></taxo:topics>
</item>
<item rdf:about="https://www.quantamagazine.org/new-shapes-solve-infinite-pool-table-problem-20170808/">
    <title>New Shapes Solve Infinite Pool-Table Problem | Quanta Magazine</title>
    <dc:date>2017-10-11T00:31:55+00:00</dc:date>
    <link>https://www.quantamagazine.org/new-shapes-solve-infinite-pool-table-problem-20170808/</link>
    <dc:creator>Vaguery</dc:creator><description><![CDATA[Two “rare jewels” have illuminated a mysterious multidimensional object that connects a huge variety of mathematical work.
]]></description>
<dc:subject>dynamical-systems geometry rather-interesting mathematical-recreations nudge-targets consider:looking-to-see consider:feature-discovery</dc:subject>
<dc:source>https://pinboard.in/</dc:source>
<dc:identifier>https://pinboard.in/u:Vaguery/b:7249118a18f7/</dc:identifier>
<taxo:topics><rdf:Bag>	<rdf:li rdf:resource="https://pinboard.in/u:Vaguery/t:dynamical-systems"/>
	<rdf:li rdf:resource="https://pinboard.in/u:Vaguery/t:geometry"/>
	<rdf:li rdf:resource="https://pinboard.in/u:Vaguery/t:rather-interesting"/>
	<rdf:li rdf:resource="https://pinboard.in/u:Vaguery/t:mathematical-recreations"/>
	<rdf:li rdf:resource="https://pinboard.in/u:Vaguery/t:nudge-targets"/>
	<rdf:li rdf:resource="https://pinboard.in/u:Vaguery/t:consider:looking-to-see"/>
	<rdf:li rdf:resource="https://pinboard.in/u:Vaguery/t:consider:feature-discovery"/>
</rdf:Bag></taxo:topics>
</item>
<item rdf:about="http://aperiodical.com/2017/09/hlf-blogs-math-art-what-is-a-rotogon/">
    <title>HLF Blogs – Math ⇔ Art: what is a rotogon? | The Aperiodical</title>
    <dc:date>2017-10-10T20:56:01+00:00</dc:date>
    <link>http://aperiodical.com/2017/09/hlf-blogs-math-art-what-is-a-rotogon/</link>
    <dc:creator>Vaguery</dc:creator><description><![CDATA[This animation (Eli Spizzichino, 2017) shows a cube being simultaneously rotated around all its edges. You can see the complex shapes created inside the object, before they’re finally covered by the outside layers of the shape as the cube completes its rotation.

One of the artworks in the exhibit shows a cross-sectional slice through one such shape – a rotogon based on a heptagon. Cross-sections like this reveal the internal structure of the shape and create beautiful images. Spizzichino points out that these internal slices are equivalent to looking at the pieces of the shape part-way through the rotating process.]]></description>
<dc:subject>mathematical-recreations geometry rather-interesting art mathematics</dc:subject>
<dc:source>https://pinboard.in/</dc:source>
<dc:identifier>https://pinboard.in/u:Vaguery/b:ba546e92fc9b/</dc:identifier>
<taxo:topics><rdf:Bag>	<rdf:li rdf:resource="https://pinboard.in/u:Vaguery/t:mathematical-recreations"/>
	<rdf:li rdf:resource="https://pinboard.in/u:Vaguery/t:geometry"/>
	<rdf:li rdf:resource="https://pinboard.in/u:Vaguery/t:rather-interesting"/>
	<rdf:li rdf:resource="https://pinboard.in/u:Vaguery/t:art"/>
	<rdf:li rdf:resource="https://pinboard.in/u:Vaguery/t:mathematics"/>
</rdf:Bag></taxo:topics>
</item>
<item rdf:about="https://arxiv.org/abs/1708.02891v1">
    <title>[1708.02891v1] Area difference bounds for dissections of a square into an odd number of triangles</title>
    <dc:date>2017-09-30T23:57:27+00:00</dc:date>
    <link>https://arxiv.org/abs/1708.02891v1</link>
    <dc:creator>Vaguery</dc:creator><description><![CDATA[Monsky's theorem from 1970 states that a square cannot be dissected into an odd number of triangles of the same area, but it does not give a lower bound for the area differences that must occur. 
We extend Monsky's theorem to "constrained framed maps"; based on this we can apply a gap theorem from semi-algebraic geometry to a polynomial area difference measure and thus get a lower bound for the area differences that decreases doubly-exponentially with the number of triangles. On the other hand, we obtain the first superpolynomial upper bounds for this problem, derived from an explicit construction that uses the Thue-Morse sequence.]]></description>
<dc:subject>computational-geometry limits geometry constraint-satisfaction nudge-targets consider:looking-to-see consider:representation to-write-about</dc:subject>
<dc:source>https://pinboard.in/</dc:source>
<dc:identifier>https://pinboard.in/u:Vaguery/b:0e54fd4021c2/</dc:identifier>
<taxo:topics><rdf:Bag>	<rdf:li rdf:resource="https://pinboard.in/u:Vaguery/t:computational-geometry"/>
	<rdf:li rdf:resource="https://pinboard.in/u:Vaguery/t:limits"/>
	<rdf:li rdf:resource="https://pinboard.in/u:Vaguery/t:geometry"/>
	<rdf:li rdf:resource="https://pinboard.in/u:Vaguery/t:constraint-satisfaction"/>
	<rdf:li rdf:resource="https://pinboard.in/u:Vaguery/t:nudge-targets"/>
	<rdf:li rdf:resource="https://pinboard.in/u:Vaguery/t:consider:looking-to-see"/>
	<rdf:li rdf:resource="https://pinboard.in/u:Vaguery/t:consider:representation"/>
	<rdf:li rdf:resource="https://pinboard.in/u:Vaguery/t:to-write-about"/>
</rdf:Bag></taxo:topics>
</item>
<item rdf:about="https://arxiv.org/abs/math/9410209">
    <title>[math/9410209] Simulation of simplicity: a technique to cope with degenerate cases in geometric algorithms</title>
    <dc:date>2017-09-27T12:09:08+00:00</dc:date>
    <link>https://arxiv.org/abs/math/9410209</link>
    <dc:creator>Vaguery</dc:creator><description><![CDATA[This paper describes a general-purpose programming technique, called the Simulation of Simplicity, which can be used to cope with degenerate input data for geometric algorithms. It relieves the programmer from the task to provide a consistent treatment for every single special case that can occur. The programs that use the technique tend to be considerably smaller and more robust than those that do not use it. We believe that this technique will become a standard tool in writing geometric software.
]]></description>
<dc:subject>geometry algorithms computational-geometry rather-interesting to-write-about consider:kata</dc:subject>
<dc:source>https://pinboard.in/</dc:source>
<dc:identifier>https://pinboard.in/u:Vaguery/b:3d275179313b/</dc:identifier>
<taxo:topics><rdf:Bag>	<rdf:li rdf:resource="https://pinboard.in/u:Vaguery/t:geometry"/>
	<rdf:li rdf:resource="https://pinboard.in/u:Vaguery/t:algorithms"/>
	<rdf:li rdf:resource="https://pinboard.in/u:Vaguery/t:computational-geometry"/>
	<rdf:li rdf:resource="https://pinboard.in/u:Vaguery/t:rather-interesting"/>
	<rdf:li rdf:resource="https://pinboard.in/u:Vaguery/t:to-write-about"/>
	<rdf:li rdf:resource="https://pinboard.in/u:Vaguery/t:consider:kata"/>
</rdf:Bag></taxo:topics>
</item>
<item rdf:about="https://publications.mfo.de/handle/mfo/447">
    <title>Friezes and tilings</title>
    <dc:date>2017-09-27T11:58:44+00:00</dc:date>
    <link>https://publications.mfo.de/handle/mfo/447</link>
    <dc:creator>Vaguery</dc:creator><description><![CDATA[Friezes have occured as architectural ornaments for many centuries. In this snapshot, we consider the mathematical analogue of friezes as introduced in the 1970s by Conway and Coxeter. Recently, infinite versions of such friezes have appeared in current research. We are going to describe them and explain how they can be classified using some nice geometric pictures.
]]></description>
<dc:subject>tiling mathematics essay classification geometry to-write-about</dc:subject>
<dc:source>https://pinboard.in/</dc:source>
<dc:identifier>https://pinboard.in/u:Vaguery/b:13672d6cf3ff/</dc:identifier>
<taxo:topics><rdf:Bag>	<rdf:li rdf:resource="https://pinboard.in/u:Vaguery/t:tiling"/>
	<rdf:li rdf:resource="https://pinboard.in/u:Vaguery/t:mathematics"/>
	<rdf:li rdf:resource="https://pinboard.in/u:Vaguery/t:essay"/>
	<rdf:li rdf:resource="https://pinboard.in/u:Vaguery/t:classification"/>
	<rdf:li rdf:resource="https://pinboard.in/u:Vaguery/t:geometry"/>
	<rdf:li rdf:resource="https://pinboard.in/u:Vaguery/t:to-write-about"/>
</rdf:Bag></taxo:topics>
</item>
<item rdf:about="https://theinnerframe.wordpress.com/2017/09/18/safely-footed-spiderwebs/">
    <title>Safely Footed Spiderwebs – The Inner Frame</title>
    <dc:date>2017-09-23T11:22:40+00:00</dc:date>
    <link>https://theinnerframe.wordpress.com/2017/09/18/safely-footed-spiderwebs/</link>
    <dc:creator>Vaguery</dc:creator><description><![CDATA[The first three are equiangular still, making angles of 10, 90 and 240 degrees at the corners, respectively. The spiderwebs are conformal images of polar coordinates on the disk, thus illustrating the Schwarz-Christoffel formula for circular polygons. The bat down below is a neat optical illusion, too: Would you think that the vertices are at the corners of an equilateral triangle?]]></description>
<dc:subject>mathematical-recreations visualization to-write-about geometry algebra</dc:subject>
<dc:source>https://pinboard.in/</dc:source>
<dc:identifier>https://pinboard.in/u:Vaguery/b:d09527a165f7/</dc:identifier>
<taxo:topics><rdf:Bag>	<rdf:li rdf:resource="https://pinboard.in/u:Vaguery/t:mathematical-recreations"/>
	<rdf:li rdf:resource="https://pinboard.in/u:Vaguery/t:visualization"/>
	<rdf:li rdf:resource="https://pinboard.in/u:Vaguery/t:to-write-about"/>
	<rdf:li rdf:resource="https://pinboard.in/u:Vaguery/t:geometry"/>
	<rdf:li rdf:resource="https://pinboard.in/u:Vaguery/t:algebra"/>
</rdf:Bag></taxo:topics>
</item>
<item rdf:about="https://theinnerframe.wordpress.com/2017/09/11/the-desargues-configuration-a-quick-tour/">
    <title>The Desargues Configuration – A Quick Tour – The Inner Frame</title>
    <dc:date>2017-09-19T11:53:08+00:00</dc:date>
    <link>https://theinnerframe.wordpress.com/2017/09/11/the-desargues-configuration-a-quick-tour/</link>
    <dc:creator>Vaguery</dc:creator><description><![CDATA[This is illustrating Desargue’s Theorem: Take two triangles p1, p2, p3 and q1, q2, q3 which are in perspective, i.e. have the three lines lines p(i)q(i) pass through a common point p. Now consider the intersection r12 of the two lines p1q1 and p2q2, and likewise the intersections r13 and r23. Desargue tells us that these three points lie on a common line. This sounds complicated, but given that we are only using the most elementary notion from geometry, namely incidence, it is surprising enough that there are non-trivial theorems at all. Why is this true? A miracle? Let’s move back into space and pretend for a second that the two triangles lie in different planes.

]]></description>
<dc:subject>mathematical-recreations geometry rather-interesting to-write-about nudge-targets consider:looking-to-see</dc:subject>
<dc:source>https://pinboard.in/</dc:source>
<dc:identifier>https://pinboard.in/u:Vaguery/b:65547fd68d74/</dc:identifier>
<taxo:topics><rdf:Bag>	<rdf:li rdf:resource="https://pinboard.in/u:Vaguery/t:mathematical-recreations"/>
	<rdf:li rdf:resource="https://pinboard.in/u:Vaguery/t:geometry"/>
	<rdf:li rdf:resource="https://pinboard.in/u:Vaguery/t:rather-interesting"/>
	<rdf:li rdf:resource="https://pinboard.in/u:Vaguery/t:to-write-about"/>
	<rdf:li rdf:resource="https://pinboard.in/u:Vaguery/t:nudge-targets"/>
	<rdf:li rdf:resource="https://pinboard.in/u:Vaguery/t:consider:looking-to-see"/>
</rdf:Bag></taxo:topics>
</item>
<item rdf:about="https://arxiv.org/abs/1612.08473">
    <title>[1612.08473] Doodles on surfaces I: An introduction to their basic properties</title>
    <dc:date>2017-05-09T17:06:04+00:00</dc:date>
    <link>https://arxiv.org/abs/1612.08473</link>
    <dc:creator>Vaguery</dc:creator><description><![CDATA[Doodles were introduced in [R. Fenn and P. Taylor, Introducing doodles, Topology of low-dimensional manifolds, pp. 37--43, Lecture Notes in Math., 722, Springer, Berlin, 1979] but were restricted to embedded circles in the 2-sphere. Khovanov, [M. Khovanov, Doodle groups, Trans. Amer. Math. Soc. 349 (1997), 2297--2315], extended the idea to immersed circles in the 2-sphere. In this paper we further extend the range of doodles to any closed orientable surfaces.
]]></description>
<dc:subject>topology geometry to-understand</dc:subject>
<dc:source>https://pinboard.in/</dc:source>
<dc:identifier>https://pinboard.in/u:Vaguery/b:8e87b08251d5/</dc:identifier>
<taxo:topics><rdf:Bag>	<rdf:li rdf:resource="https://pinboard.in/u:Vaguery/t:topology"/>
	<rdf:li rdf:resource="https://pinboard.in/u:Vaguery/t:geometry"/>
	<rdf:li rdf:resource="https://pinboard.in/u:Vaguery/t:to-understand"/>
</rdf:Bag></taxo:topics>
</item>
<item rdf:about="https://arxiv.org/abs/1704.08156">
    <title>[1704.08156] Locally optimal 2-periodic sphere packings</title>
    <dc:date>2017-05-03T09:46:43+00:00</dc:date>
    <link>https://arxiv.org/abs/1704.08156</link>
    <dc:creator>Vaguery</dc:creator><description><![CDATA[The sphere packing problem is an old puzzle. We consider packings with m spheres in the unit cell (m-periodic packings). For the case m = 1 (lattice packings), Voronoi presented an algorithm to enumerate all local optima in a finite computation, which has been implemented in up to d = 8 dimensions. We generalize Voronoi's algorithm to m > 1 and use this new algorithm to enumerate all locally optimal 2-periodic sphere packings in d = 3, 4, and 5. In particular, we show that no 2-periodic packing surpasses the density of the optimal lattice in these dimensions. A partial enumeration is performed in d = 6.
]]></description>
<dc:subject>packing generalization geometry optimization nudge-targets consider:rediscovery consider:representation</dc:subject>
<dc:source>https://pinboard.in/</dc:source>
<dc:identifier>https://pinboard.in/u:Vaguery/b:d2ace8d8d57b/</dc:identifier>
<taxo:topics><rdf:Bag>	<rdf:li rdf:resource="https://pinboard.in/u:Vaguery/t:packing"/>
	<rdf:li rdf:resource="https://pinboard.in/u:Vaguery/t:generalization"/>
	<rdf:li rdf:resource="https://pinboard.in/u:Vaguery/t:geometry"/>
	<rdf:li rdf:resource="https://pinboard.in/u:Vaguery/t:optimization"/>
	<rdf:li rdf:resource="https://pinboard.in/u:Vaguery/t:nudge-targets"/>
	<rdf:li rdf:resource="https://pinboard.in/u:Vaguery/t:consider:rediscovery"/>
	<rdf:li rdf:resource="https://pinboard.in/u:Vaguery/t:consider:representation"/>
</rdf:Bag></taxo:topics>
</item>
<item rdf:about="https://arxiv.org/abs/1306.2741">
    <title>[1306.2741] Convex Equipartitions: The Spicy Chicken Theorem</title>
    <dc:date>2017-04-23T02:27:52+00:00</dc:date>
    <link>https://arxiv.org/abs/1306.2741</link>
    <dc:creator>Vaguery</dc:creator><description><![CDATA[We show that, for any prime power n and any convex body K (i.e., a compact convex set with interior) in Rd, there exists a partition of K into n convex sets with equal volumes and equal surface areas. Similar results regarding equipartitions with respect to continuous functionals and absolutely continuous measures on convex bodies are also proven. These include a generalization of the ham-sandwich theorem to arbitrary number of convex pieces confirming a conjecture of Kaneko and Kano, a similar generalization of perfect partitions of a cake and its icing, and a generalization of the Gromov-Borsuk-Ulam theorem for convex sets in the model spaces of constant curvature. 
Most of the results in this paper appear in arxiv:1011.4762 and in arxiv:1010.4611. Since the main results and techniques there are essentially the same, we have merged the papers for journal publication. In this version we also provide a technical alternative to a part of the proof of the main topological result that avoids the use of compactly supported homology.]]></description>
<dc:subject>geometry multiobjective-optimization rather-interesting proof consider:looking-to-see nudge-targets</dc:subject>
<dc:source>https://pinboard.in/</dc:source>
<dc:identifier>https://pinboard.in/u:Vaguery/b:dd4e4e64f693/</dc:identifier>
<taxo:topics><rdf:Bag>	<rdf:li rdf:resource="https://pinboard.in/u:Vaguery/t:geometry"/>
	<rdf:li rdf:resource="https://pinboard.in/u:Vaguery/t:multiobjective-optimization"/>
	<rdf:li rdf:resource="https://pinboard.in/u:Vaguery/t:rather-interesting"/>
	<rdf:li rdf:resource="https://pinboard.in/u:Vaguery/t:proof"/>
	<rdf:li rdf:resource="https://pinboard.in/u:Vaguery/t:consider:looking-to-see"/>
	<rdf:li rdf:resource="https://pinboard.in/u:Vaguery/t:nudge-targets"/>
</rdf:Bag></taxo:topics>
</item>
<item rdf:about="https://arxiv.org/abs/1412.0371">
    <title>[1412.0371] Realization spaces of arrangements of convex bodies</title>
    <dc:date>2017-04-23T02:20:13+00:00</dc:date>
    <link>https://arxiv.org/abs/1412.0371</link>
    <dc:creator>Vaguery</dc:creator><description><![CDATA[We introduce combinatorial types of arrangements of convex bodies, extending order types of point sets to arrangements of convex bodies, and study their realization spaces. Our main results witness a trade-off between the combinatorial complexity of the bodies and the topological complexity of their realization space. First, we show that every combinatorial type is realizable and its realization space is contractible under mild assumptions. Second, we prove a universality theorem that says the restriction of the realization space to arrangements polygons with a bounded number of vertices can have the homotopy type of any primary semialgebraic set.
]]></description>
<dc:subject>geometry rather-interesting mathematical-recreations combinatorics nudge-targets consider:magic-squares</dc:subject>
<dc:source>https://pinboard.in/</dc:source>
<dc:identifier>https://pinboard.in/u:Vaguery/b:8dc415c33e13/</dc:identifier>
<taxo:topics><rdf:Bag>	<rdf:li rdf:resource="https://pinboard.in/u:Vaguery/t:geometry"/>
	<rdf:li rdf:resource="https://pinboard.in/u:Vaguery/t:rather-interesting"/>
	<rdf:li rdf:resource="https://pinboard.in/u:Vaguery/t:mathematical-recreations"/>
	<rdf:li rdf:resource="https://pinboard.in/u:Vaguery/t:combinatorics"/>
	<rdf:li rdf:resource="https://pinboard.in/u:Vaguery/t:nudge-targets"/>
	<rdf:li rdf:resource="https://pinboard.in/u:Vaguery/t:consider:magic-squares"/>
</rdf:Bag></taxo:topics>
</item>
</rdf:RDF>