# Orbifold Pinball

An undergraduate research project by Yevgeniy Dodis, at the Geometry Center Summer Institute 1994. Mentor: Paul Burchard.

## What Is Orbifold Pinball?

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.)

## Things to Try in Orbifold Pinball

Here are some challenges you can try:

1. Imagine edges named a, b, and c drawn between the 3 bumpers. Try to make the ball roll across the edges in a predetermined sequence, for example, abaca.

2. Try to bounce the ball between two of the bumpers as many times as you can.

## Why Three Bumpers?

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.

## What Does It All Mean?

See the report, The Mathematics of Orbifold Pinball, for the mathematical story behind this game. In this report, the game board is revealed as an `orbifold', and the bumpers become `cone points of order 2'.

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.

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University of Minnesota
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