But note that when Hume says "objects", at least in the context of reasoning, he is referring to the objects of the mind, that is, ideas and impressions, since Hume adheres to the Early Modern "way of ideas", what is billiards the belief that sensation is a mental event and therefore all objects of perception are mental. That is, a laser beam shot from one point, regardless of its direction, cannot hit the other point. We ask if, given two points on a particular table, you can always shine a laser (idealized as an infinitely thin ray of light) from one point to the other. For example, it can be used to show why simple rectangular tables have infinitely many periodic trajectories through every point. Rather than asking about trajectories that return to their starting point, this problem asks whether trajectories can visit every point on a given table. Adjust the original point slightly if the path passes through a corner. Draw a line segment from a point on the original table to the identical point on a copy n tables away in the long direction and m tables away 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. Only the little man remained seated at the window, with his eyes fixed upon the bank of murky clouds which lowered over the sea. Time goes so fast I always fancy that I arrived only the evening before." Sometimes they get up a little race, and the ladies are disposed to take part in it, "for they are all very spry and able to run around the drawing-room five or six times every day." But they prefer indoors to the open air; in these days true sunshine consists of candle-light, and the finest sky is a painted ceiling,-is there any other less subject to inclemencies, or better adapted to conversation and merriment?

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. His approach worked not only for obtuse triangles, but for far more complicated shapes: Irregular 100-sided tables, say, or polygons whose walls zig and zag creating nooks and crannies, have periodic orbits, so long as the angles are rational. His jagged table is made of 29 such triangles, arranged to make clever use of this fact. 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. To put it another way, if we placed a light bulb, which shines in all directions at once, at some point on the table, would it light up the whole room? He has had a tapestry frame put up in the drawing-room; at which he works, I cannot say with the greatest skill, but at least with the greatest assiduity…

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. Proving results in the other direction has been a lot harder. 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. Because there are many different types of billiard tables, the weight will also differ per table. In Wolecki’s 2019 article, he strengthened this result by proving that there are only finitely many pairs of unilluminable points. Whereas finding oddball shapes that can’t be illuminated can be done through a clever application of simple math, proving that a lot of shapes can be illuminated has only been possible through the use of heavy mathematical machinery. 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. 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.

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