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Spelling/Grammar Check, typos fixed: suprising → surprising, easily- → easily
# Connection between topological spaces and algebraic groups
# Connection between topological spaces and algebraic groups
# Ham sandwich theorem (easy to state, harder to motivate solution. Possibly there's something better.
# Ham sandwich theorem (easy to state, harder to motivate) solution. Possibly there's something better.
==== How can we tell the difference between a sphere and a torus? ====
==== How can we tell the difference between a sphere and a torus? ====
As is often the case is mathematics, we attempt to generalize these results on periodic orbits to other situations. What if the billiards table is not rectangular, but triangular? It's true, but extremely difficult to prove, that given any triangular billiards table, there exists a periodic orbit of '''some''' period. The existence of periodic orbits is an extremely subtle question. This can be generalized even further, to ask whether all polygons admit periodic orbits. What about polyhedra? Can you imagine some periodic billiards orbits on a cube-shaped "table" (no gravity! think billiards in space).
As is often the case is mathematics, we attempt to generalize these results on periodic orbits to other situations. What if the billiards table is not rectangular, but triangular? It's true, but extremely difficult to prove, that given any triangular billiards table, there exists a periodic orbit of '''some''' period. The existence of periodic orbits is an extremely subtle question. This can be generalized even further, to ask whether all polygons admit periodic orbits. What about polyhedra? Can you imagine some periodic billiards orbits on a cube-shaped "table" (no gravity! think billiards in space).
The orbits that aren't periodic turn out to be just as interesting as those that are. Imagine that you place the cue ball at random on your billiards table, and smack it in some random direction. Chances are it won't end up in a periodic orbit—putting the ball in a periodic orbit requires a very careful shot. So the ball will bounce around forever without retracing its path. But will it ever come back to the point where it started, perhaps moving in a different direction? The answer to this question is "no", but the Poincare Recurrence Theorem gives a somewhat satisfactory replacement: given any distance (e.g., an eighth of an inch), and any angle (e.g. two degrees), after some amount of time, the ball will be within that distance of its starting point, and directed within that angle of its initial path! This is a very suprising fact.
The orbits that aren't periodic turn out to be just as interesting as those that are. Imagine that you place the cue ball at random on your billiards table, and smack it in some random direction. Chances are it won't end up in a periodic orbit—putting the ball in a periodic orbit requires a very careful shot. So the ball will bounce around forever without retracing its path. But will it ever come back to the point where it started, perhaps moving in a different direction? The answer to this question is "no", but the Poincare Recurrence Theorem gives a somewhat satisfactory replacement: given any distance (e.g., an eighth of an inch), and any angle (e.g. two degrees), after some amount of time, the ball will be within that distance of its starting point, and directed within that angle of its initial path! This is a very surprising fact.
While this problem seems extremely elementary, sophisticated mathematical techiques from a range of fields of geometry and analysis have been brought to bear on it. Even so, many easily-stated questions remain open.
While this problem seems extremely elementary, sophisticated mathematical techiques from a range of fields of geometry and analysis have been brought to bear on it. Even so, many easily stated questions remain open.
# Periodic orbits in billiards in polyhedra. I think it should be possible to say some interesting things about this but with some real content. Possibly there's a better idea.
# Periodic orbits in billiards in polyhedra. I think it should be possible to say some interesting things about this but with some real content. Possibly there's a better idea.