Existence of Periodic Orbits
In ordinary polygonal billiards, the existence of periodic orbits is a major unsolved problem. For instance, it is unknown if every triangular shaped table has a periodic billiard path. More progress has been made for outer billiards, though the situation is far from well-understood. As mentioned above, all the orbits are periodic when the system is defined relative to a convex rational polygon in the Euclidean plane. Moreover, it is a recent theorem of C. Culter (written up by S. Tabachnikov) that outer billiards relative to any convex polygon has periodic orbits—in fact a periodic orbit outside of any given bounded region.
Read more about this topic: Outer Billiard
Famous quotes containing the words existence, periodic and/or orbits:
“There are in our existence spots of time,”
—William Wordsworth (1770–1850)
“It can be demonstrated that the child’s contact with the real world is strengthened by his periodic excursions into fantasy. It becomes easier to tolerate the frustrations of the real world and to accede to the demands of reality if one can restore himself at intervals in a world where the deepest wishes can achieve imaginary gratification.”
—Selma H. Fraiberg (20th century)
“To me, however, the question of the times resolved itself into a practical question of the conduct of life. How shall I live? We are incompetent to solve the times. Our geometry cannot span the huge orbits of the prevailing ideas, behold their return, and reconcile their opposition. We can only obey our own polarity.”
—Ralph Waldo Emerson (1803–1882)