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예식일 : What To Do About What Is Billiards Before It's Too Late
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His approach involved breaking the problem down into multiple cases and verifying each case using traditional mathematics and computer assistance. In their 1992 paper, Galperin and his collaborators came up with a variety of methods of reflecting obtuse triangles in a way that lets you create periodic orbits, but the methods only worked for some special cases. Then, in 2008, Richard Schwartz at Brown University showed that all obtuse triangles with angles of 100 degrees or less contain a periodic trajectory. By folding the imagined tables back on their neighbors, you can recover the actual trajectory of the ball. Pocketing the eight ball on a legal break shot is not a foul. That is, a laser beam shot from one point, regardless of its direction, cannot hit the other point. 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. 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. Billiards in triangles, which do not have the nice right-angled geometry of rectangles, is more complicated. Instead of just copying a polygon on a flat plane, this approach maps copies of polygons onto topological surfaces, doughnuts with one or more holes in them. Another approach has been used to show that if all the angles are rational - that is, they can be expressed as fractions - obtuse triangles with even bigger angles must have periodic trajectories. But obtuse triangles remain a mystery. This inscribed triangle is a periodic billiard trajectory called the Fagnano orbit, named for Giovanni Fagnano, who in 1775 showed that this triangle has the smallest perimeter of all inscribed triangles. Start with a trajectory that’s at a right angle to the hypotenuse (the long side of the triangle). A similar argument holds for any rectangle, but for concreteness, imagine a table that’s twice as wide as it is long. All three table sports are fun to play. Cutthroat pool is one of the most fun and interesting variations of billiards out there.

Shafts do get dirty just like other sports equipment out there. All billiards games require the basic equipment of a table, cue sticks, and balls. This is the time of 21st century, and lots of games are played throughout the world. Whichever option you choose, just be sure that you are happy with the look of the end result. The end of my perform out appeared to are available significantly additional instantly! In call shot games, the shooter may choose to call "safety" instead of a ball and pocket, and then play passes to the opponent at the end of the shot. Here’s what mathematicians have learned about billiards since Donald Duck’s epically tangled shot. A push shot is when the cue remains in contact with the ball longer than the brief moment needed to hit it. The story has been updated to reflect that though the smallest such polygon known to exist has 22 sides, it remains unknown if a smaller one can be constructed. Nobody knows. For other, more complicated shapes, it’s unknown whether it’s possible to hit the ball from any point on the table to any other point on the table.

It’s unknown if a shape with fewer sides exists. In 2019 Amit Wolecki, then a graduate student at Tel Aviv University, applied this same technique to produce a shape with 22 sides (shown below). 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 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. 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. You also win if your opponent accidentally pots the 8 ball, which is illegal. 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.

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