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Polymath Professional is an impressive application through which users can apply effective numerical analysis methods during interactive problem solving on their systems.
#POLYMATH PROGRAM P5 16 FULL VERSION#
That leaves you with a shape that has four corners, a diameter of 1, and two edges that are curves of radius 0.5, and an area of about 0.64.Free download Polymath Professional 6.10.260 full version standalone offline installer for Windows PC, Polymath Professional Overview As an example, consider a unit-diameter circle, draw a square of side 1/Sqrt inside it, and delete the two slivers on opposite sides of the square. Yeah, it’s counterintuitive, but it’s true when one is fattening two edges at the same time. I would NEVER have guessed that the distance between, in your notation, (-c,0) and (h+g,3/4+a) would be more than 2 in the optimal solution. (Note that this is my notation except for your extra vertices, and that my “s” corresponds to both your d and your d_2.)Ĭongratulations on this remarkable tiling. So, I’ve hunted around the search space and the best I have is this: Undulation of the inter-arrowhead edges is needed (at least if any of the things that are >=4 are exactly 4), but does not introduce any additional constraints. This is all without curving any edges of diamonds or arrowheads, but you said that that makes only a small difference (6899 to 6906). In practice d,e,s are less than 0.1 and g,r,b,c are more than 0.8.Īlso we have a “Bock-esque” situation in the neighbourhood of ((b+g)/2,3/8), where two large tiles meet that need to be 1 away from tiles on the far side of relevant diamonds, giving: With all seven values positive, and with c,b,r,g in increasing order and less than 1. Then we have four distinct diameters of large tiles, which give the constraints: – the back of the arrowhead at y=1/4 is at (g,1/4) – the “trailing tip” of that arrowhead is at (r,s) – the tip of the arrowhead at y=0 is at (c,0) – the blunt corners of that diamond are at (+/-e,0) – the sharp corners of that diamond are at (0,+/-d) If we orient things following Pritikin and place the centre of a diamond at the origin, then here is my notation: I agree with you about Mathematica – I too am using FindMaximum and being careful about ranges. Thanks Jaan! But I’m still not reproducing this. Repost of my latest offering regarding Jaan’s 6-tiling: Graph minimization, focusing on the example of 5-chromatic unit-distance graphs in the plane, Geombinatorics, vol. A small 6-chromatic two-distance graph in the plane, Geombinatorics, vol. Heule, Trimming Graphs Using Clausal Proof Optimization, In International Conference on Principles and Practice of Constraint Programming, pp. Heule, Searching for a Unit-Distance Graph with Chromatic Number 6, SAT COMPETITION 2018: 66.
Heule, Computing a Smaller Unit-Distance Graph with Chromatic Number 5 via Proof Trimming, arXiv preprint arXiv:1907.00929 (2019). Pálvölgyi, Almost-monochromatic sets and the chromatic number of the plane, arXiv preprint arXiv:1912.02604 (2019). Ismailescu, The Hadwiger-Nelson problem with two forbidden distances, arXiv preprint arXiv:1805.06055 (2018). Ismailescu, The chromatic number of the plane is at least 5: A new proof, Discrete & Computational Geometry (2019): 1-11. Ismailescu, A 6-chromatic two-distance graph in the plane, arXiv preprint arXiv:1909.13177 (2019). Bock, Epsilon-colorings of strips, Acta Math. Ágoston, Probabilistic formulation of the Hadwiger–Nelson problem.į.
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(Feel free to any references I missed in the comments.) There has certainly been a bit of activity since Aubrey’s paper first hit the arXiv two years ago! For reference, I wanted to compile a list of related publications that have emerged since starting our project. The paper writing has found a second wind between Philip Gibbs, Aubrey de Grey, Jaan Parts and Tom Sirgedas. We will summarize progress on the Polymath wiki page. Discussion of the project of a non-research nature should continue in the Polymath proposal page. This project is a follow-up to Aubrey de Grey’s breakthrough result that the chromatic number of the plane is at least 5. This is the sixteenth “research” thread of the Polymath16 project to make progress on the Hadwiger–Nelson problem, continuing this post.
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