wp-plugin-hostgator domain was triggered too early. This is usually an indicator for some code in the plugin or theme running too early. Translations should be loaded at the init action or later. Please see Debugging in WordPress for more information. (This message was added in version 6.7.0.) in /home4/scienrds/scienceandnerds/wp-includes/functions.php on line 6114ol-scrapes domain was triggered too early. This is usually an indicator for some code in the plugin or theme running too early. Translations should be loaded at the init action or later. Please see Debugging in WordPress for more information. (This message was added in version 6.7.0.) in /home4/scienrds/scienceandnerds/wp-includes/functions.php on line 6114Source:https:\/\/www.quantamagazine.org\/the-biggest-smallest-triangle-just-got-smaller-20230908\/#comments<\/a><\/br> Consider a square with a bunch of points inside. Take three of those points, and you can make a triangle. Four points define four different triangles. Ten points define 120 triangles. The numbers grow quickly from there \u2014 100 points define 161,700 different triangles. Each of those triangles, of course, has a particular area.<\/p>\n Hans Heilbronn, a German mathematician who fled his country before World War II and settled in England, thought of these triangles in the late 1940s when he saw a group of soldiers outside his window. The soldiers didn\u2019t appear to be in formation, which got him thinking: If there are n<\/em> soldiers inside a square, what is the size of the largest possible smallest triangle defined by three of them? Heilbronn wondered how one might go about arranging the soldiers (or, for mathematical simplicity, points) to maximize the size of the smallest triangle.<\/p>\n The problem is simple to state, but progress on the Heilbronn triangle problem, as it came to be called, has been halting, and results dried up entirely in the 1980s. Then this past May, three mathematicians \u2014 Alex Cohen<\/a>, Cosmin Pohoata<\/a> and Dmitrii Zakharov<\/a> \u2014 announced a new cap<\/a> on the size of the smallest triangle. \u201cI think it\u2019s a stunning result,\u201d said Anthony Carbery, a mathematician at the University of Edinburgh.<\/p>\n Researchers kept working on the Heilbronn triangle problem over the years, despite the long wait for progress, motivated by its tangle of links to other areas of math. \u201cThe things it\u2019s connected to make it come alive,\u201d said Pohoata, a professor at Emory University in Atlanta. It\u2019s closely related to problems about intersecting shapes, which in turn connect to both number theory and Fourier analysis \u2014 the study of complicated functions constructed from simple waves.<\/p>\n Cohen, a graduate student at the Massachusetts Institute of Technology, stumbled upon this web of connections last year. He\u2019d been perusing an old survey of the Heilbronn triangle problem by Klaus Roth, another refugee from the Nazis who fled to Britain as a boy. (Roth, who died in 2015, was the first British mathematician to win the Fields Medal.)<\/p>\n Cohen visualized the ideas from Roth\u2019s survey with a simple picture: a square crisscrossed by two thick strips, with a thin line down the middle of each. As he studied his diagram, Cohen realized that it might connect to ideas that his adviser, Larry Guth, had brought up in a recent reading group meeting. But Guth hadn\u2019t been talking about triangles at all. <\/span><\/p>\n \u201cI realized very quickly that these two methods were essentially equivalent,\u201d Cohen said. \u201cI got really excited about the triangle problem.\u201d<\/p>\n One day in the common room of MIT\u2019s math department, Cohen unexpectedly discovered that Pohoata, who had come to give a talk, and Zakharov, a fellow graduate student at MIT, had also been working on the Heilbronn triangle problem. What\u2019s more, they had found the same link. The three began collaborating. Seven months later, they made their breakthrough. Their paper brings in even more new areas of mathematics. \u201cThey use a huge amount of machinery and different insights,\u201d said Thomas Bloom of the University of Oxford, who said he expects the new paper to \u201cprompt a renaissance\u201d of progress on the triangle problem.<\/p>\n By placing three points very close together, you can easily make the smallest triangle in an arrangement arbitrarily small. (In the most extreme case, three points in line with one another form a triangle with zero area.) But trying to keep the smallest triangle big is trickier. As you keep adding in more dots, the smallest triangle is forced to be pretty small \u2014 new dots can only be so far from existing ones. It\u2019s relatively easy to show that the smallest triangle can\u2019t have an area any bigger than 1\/(n<\/em> \u2212 2) by splitting the square into nonoverlapping triangles.<\/p>\n But Heilbronn thought that the limit was even tinier than that. He guessed that no matter how the dots were arranged in the square, there couldn\u2019t be a smallest triangle with an area larger than around 1\/n<\/em>2<\/sup>, a number which shrinks much faster as n<\/em> grows.<\/p>\n He was wrong.<\/p>\n In 1980, the Hungarian mathematicians J\u00e1nos Koml\u00f3s, J\u00e1nos Pintz and Endre Szemer\u00e9di found a pattern<\/a> of dots whose smallest triangle had an area ever so slightly larger than 1\/n<\/em>2<\/sup>. In a separate paper published around the same time, they also showed that it\u2019s impossible to arrange n <\/em>dots to create a smallest triangle that is bigger than around 1\/n<\/em>8\/7<\/sup>. When n<\/em> is large, this is much smaller than 1\/n<\/em>, but much bigger than 1\/n<\/em>2<\/sup>.<\/p>\n Those results stood for over 40 years. \u201cImproving [the bound] in either direction was remarkably hard and required a lot of technical analysis and ingenuity,\u201d Bloom said.<\/p>\n \u201cYou very, very quickly get bogged down by a complete quagmire of stuff,\u201d Carbery added.<\/p>\n While the construction discovered in 1980 remains the one with the biggest known smallest triangle, Cohen, Pohoata and Zakharov have, for the first time in four decades, succeeded in lowering the upper bound.<\/p>\n By the time he met Cohen, Pohoata had already been working on the Heilbronn triangle problem for two years. In the summer of 2020, he put summer research students at Yale University to work on higher-dimensional versions of the problem \u2014 for example, narrowing down the largest smallest-volume shapes that appear among dots scattered in a three-dimensional cube.<\/p>\n As part of that project, Pohoata revisited all the prior work on the problem. Back in 1951<\/a>, Roth had split the search for small triangles into two parts: First find a pair of points to form the triangle\u2019s base, and then find a third point to complete the triangle. The strategy essentially framed the search for a large smallest triangle as the study of intersecting dots and rectangles \u2014 an approach that was refined in 1972 by Wolfgang Schmidt.<\/p>\n On reading Schmidt\u2019s paper, Pohoata identified a connection to the high-low method \u2014 a technique that Guth and collaborators had developed in 2017 for estimating the overlap between a collection of rectangular strips and a collection of disks. \u201cThat was an important psychological moment for me,\u201d he said.<\/p>\n In 2021, Pohoata brought up his ideas with Zakharov. The two had begun publishing together when Zakharov was still an undergraduate in Moscow. \u201c[Zakharov] was doing remarkable things as if he was a senior researcher at a young age,\u201d said Jacob Fox, a mathematician at Stanford University.<\/p>\n Zakharov was initially pessimistic about the Heilbronn triangle problem. \u201cI thought that, well, this 8\/7 stayed up there for 40 years, so who am I to crack it?\u201d he said. \u201cI mostly wanted to understand how it works.\u201d<\/p>\n After running into one another in October 2022, Cohen, Pohoata and Zakharov soon pinpointed the obstacle that Koml\u00f3s, Pintz and Szemer\u00e9di had unknowingly faced. \u201cThere\u2019s a very specific arrangement of the points that leads to this worst-case scenario where they can\u2019t do better than 8\/7,\u201d Cohen said. \u201cThe points can either be concentrated or spread out. The worst case is when it\u2019s some combination.\u201d That arrangement had points spread out on a large scale, but if you zoomed in on tiny sub-squares within the unit square, you\u2019d see orderly patterns.<\/p>\n Cohen, Pohoata and Zakharov realized they could make headway by studying the dimension of the small clusters of points. To nonmathematicians, dimensions are always whole numbers: A sheet of paper is two-dimensional; a clay brick has three dimensions.<\/p>\n Things can get weird when you consider the dimension of a set of points. A single point is normally considered zero-dimensional. But two finite sets of points can have completely different structures. One might have 10 points marching obediently in a ramrod-straight line, while the other has 10 points sprinkled across the entire unit square.<\/p>\n To capture the structure of even the weirdest sets of points, the early 20th-century mathematician Felix Hausdorff came up with a new notion of dimension. According to this definition, 10 points in a line are one-dimensional, while 10 points spread evenly over a square are two-dimensional. But in this world, dimensions don\u2019t have to be integers, and a one-dimensional set can be not linear but fractal, sporting infinite layers of intricate patterns. Depending on the details of these patterns, collections of points can even have a dimension that is bigger than 0 but less than 1.<\/p>\n Cohen, Pohoata and Zakharov uncovered a 1953 theorem<\/a> by John Marstrand that rephrased Koml\u00f3s, Pintz and Szemer\u00e9di\u2019s estimate in terms of the Hausdorff dimension \u2014\u00a0but only for dimensions greater than 1. In order to improve the estimate, Cohen, Pohoata and Zakharov would need to find some way to generalize Marstrand\u2019s result to sets whose dimension was smaller than 1.<\/p>\n Cohen, Pohoata and Zakharov didn\u2019t have to think for long. As it happened, a paper by Tuomas Orponen<\/a>, Pablo Shmerkin<\/a> and Hong Wang<\/a> that had just been posted online<\/a> extended Marstrand\u2019s 70-year-old theorem to sets whose dimension was less than 1.<\/p>\n Cohen didn\u2019t learn about the paper until February. Once he did, he quickly relayed it to Pohoata and Zakharov. By late May, they\u2019d posted their paper online, proving that the smallest triangle among n<\/em> points in a unit square can\u2019t ever be bigger than 1\/n<\/em>8\/7 + 1\/2000<\/sup>.<\/p>\n Shmerkin read the triangle paper on a whim after seeing it announced on Twitter. He hadn\u2019t even been aware of the Heilbronn triangle problem before then, so he was surprised when he spotted the reference to his proof. \u201cThis is not a direct application of what we do. There is a lot of insightful, creative and technical work in it,\u201d he said. \u201cFor me, it was a great feeling.\u201d<\/p>\n Bloom, too, was impressed. \u201cI could have looked at that paper for a long time and never thought, oh, this applies to the triangle problem.\u201d<\/p>\n While the new result improves Koml\u00f3s, Pintz and Szemer\u00e9di\u2019s exponent by only a tiny fraction, it has revived the long-languishing Heilbronn triangle problem. \u201cYou might take a look at it and say, yawn, yawn, yawn, it doesn\u2019t look so different from what was known in 1982. But an awful lot of time has passed since 1982,\u201d Carbery said.<\/p>\n By incorporating the high-low method and Orponen, Shmerkin and Wang\u2019s work, Cohen, Pohoata and Zakharov have unmasked a new set of ties between the Heilbronn triangle problem and the rest of mathematics. As Bloom put it, the triangle problem was thought of as \u201ca really nice, really hard problem that we don\u2019t know what to do. But they\u2019ve said it\u2019s connected to a huge amount of other stuff.\u201d<\/p>\n
\nThe Biggest Smallest Triangle Just Got Smaller<\/br>
\n2023-09-11 21:58:40<\/br><\/p>\nA Hypothesis Felled<\/strong><\/h2>\n
Convergent Evolution<\/strong><\/h2>\n
Making Connections<\/strong><\/h2>\n