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 of, existence, periodic and/or orbits:
“No being exists or can exist which is not related to space in some way. God is everywhere, created minds are somewhere, and body is in the space that it occupies; and whatever is neither everywhere nor anywhere does not exist. And hence it follows that space is an effect arising from the first existence of being, because when any being is postulated, space is postulated.”
—Isaac Newton (1642–1727)
“How old the world is! I walk between two eternities.... What is my fleeting existence in comparison with that decaying rock, that valley digging its channel ever deeper, that forest that is tottering and those great masses above my head about to fall? I see the marble of tombs crumbling into dust; and yet I don’t want to die!”
—Denis Diderot (1713–1784)
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—Saf Lerman (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)