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예식일 : Explainer: what is Chaos Theory?
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Patrons wait for the gates to open near the first fairway before the start of the 80th Masters Golf Tournament at the Augusta National Golf Club. In 2014, Maryam Mirzakhani, a mathematician at Stanford University, became the first woman to win the Fields medal, math’s most prestigious award, for her work on the moduli spaces of Riemann surfaces - a sort of generalization of the doughnuts that Masur used to show that all polygonal tables with rational angles have periodic orbits. Red Baron's Antiques grew out of a very different sort of business. Jordan Spieth hits out of the rough along the 17th fairway Saturday during the third round of the Masters. A key method for analyzing polygonal billiards is not to think of the ball as bouncing off the table’s edge, but instead to imagine that every time the ball hits a wall, it keeps on traveling into a fresh copy of the table that is flipped over its edge, producing a mirror image. Lay out a grid of identical rectangles, each viewed as a mirror image of its neighbors. Since each mirror image of the rectangle corresponds to the ball bouncing off a wall, for the ball to return to its starting point traveling in the same direction, its trajectory must cross the table an even number of times in both directions.

Buy 9 x 12-inch squares of felt: You'll need three times the number of squares as there are children. In 2016, Samuel Lelièvre of Paris-Saclay University, Thierry Monteil of the French National Center for Scientific Research and Barak Weiss of Tel Aviv University applied a number of Mirzakhani’s results to show that any point in a rational polygon illuminates all points except finitely many. In 2018, Jacob Garber, Boyan Marinov, Kenneth Moore and George Tokarsky at the University of Alberta extended this threshold to 112.3 degrees. But in 1995, Tokarsky used a simple fact about triangles to create a blockish 26-sided polygon with two points that are mutually inaccessible, shown below. The key idea that Tokarsky used when building his special table was that if a laser beam starts at one of the acute angles in a 45°-45°-90° triangle, it can never return to that corner. In 1958, Roger Penrose, a mathematician who went on to win the 2020 Nobel Prize in Physics, found a curved table in which any point in one region couldn’t illuminate any point in another region. Adjust the original point slightly if the path passes through a corner.

6. When folded back up, the path produces a periodic trajectory, as shown in the green rectangle. By folding the imagined tables back on their neighbors, you can recover the actual trajectory of the ball. The hypotenuse and its second reflection are parallel, so a perpendicular line segment joining them corresponds to a trajectory that will bounce back and forth forever: The ball departs the hypotenuse at a right angle, bounces off both legs, returns to the hypotenuse at a right angle, and then retraces its route. In Wolecki’s 2019 article, he strengthened this result by proving that there are only finitely many pairs of unilluminable points. There have been two main lines of research into the problem: finding shapes that can’t be illuminated and proving that large classes of shapes can be. This is called the illumination problem because we can think about it by imagining a laser beam reflecting off mirrored walls enclosing the billiard table. There may be isolated dark spots (as in Tokarsky’s and Wolecki’s examples) but no dark regions as there are in the Penrose example, which has curved walls rather than straight ones. If two or more balls are equal distance from the head string, the shooter may designate which of the equidistant balls is to be spotted.

Billiards in triangles, which do not have the nice right-angled geometry of rectangles, is more complicated. His jagged table is made of 29 such triangles, arranged to make clever use of this fact. Suppose you want to find a periodic orbit that crosses the table n times in the long direction and m times in the short direction. If you reflect a rectangle over its short side, and then reflect both rectangles over their longest side, making four versions of the original rectangle, and then glue the top and bottom together and the left and right together, you will have made a doughnut, or torus, as shown below. Billiard trajectories on the table correspond to trajectories on the torus, and vice versa. Rather than asking about trajectories that return to their starting point, this problem asks whether trajectories can visit every point on a given table. Three Point Break Rule is used, if no ball is pocketed, three balls must touch the head string, or the break is considered ‘illegal break’. Baulk is the rectangular area of the table that is bordered by the baulk line and the three cushions at the head of the table. As you might remember from high school geometry, there are several kinds of triangles: acute triangles, where all three internal angles are less than 90 degrees; right triangles, which have a 90-degree angle; and obtuse triangles, which have one angle that is more than 90 degrees.

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