Orbifold Pinball Orbifold Pinball
An undergraduate research project by Yevgeniy Dodis, at the Geometry Center Summer Institute 1994. Mentor: Paul Burchard.
Orbifold Pinball is a game in which you roll a ball on an unusual playing field. The game board is like ordinary pinball in that it features `bumpers' off which the ball can bounce.
But in this game, the board is `curved' in such a way that a ball barely missing the bumper will still be whipped around in nearly the same way as if it had actually hit the bumper itself. Given this requirement, the curvature of the board is designed to be as uniform as possible. (The board's shape is an example of what mathematicians call an orbifold.)
Here are some challenges you can try:
It turns out that the nature of the game changes dramatically when the board acquires more than two bumpers.
If the board has only two bumpers, the path of the ball will be fairly predictable. For example, it may be an ellipse or hyperbola with the two bumpers as foci. More generally, it can also be a spiral trajectory lying between these two extremes.
But with three or more bumpers, highly complex trajectories are possible! In addition, the trajectories may be highly sensitive to the initial position and direction of the ball. This application provides a board with three symmetrically-placed bumpers.
For a gentle introduction to orbifolds, symmetry groups, and the ties between them, see the Geometry and the Imagination course notes available on this Web server.
The Geometry Center University of Minnesota 400 Lind Hall 207 Church Street S.E. Minneapolis, MN 55455