Angled mirrors
The situation changes dramatically the moment the mirrors are no longer parallel, maybe even perpendicular. Lets have a look at the following experimental setup: We place Dr. Stickler inside an angle made from adjustable mirrors. In the applet above, Dr. Stickler is initially placed between two perpendicular mirrors. The result is one mirror image in every mirror. The fourth Dr. Stickler is a reflected mirror image. Because of the 90° angle, the mirror images overlap after a finite number of steps, resulting in a finite number of overall images (four in this setup). Changing the angle (by dragging the ends of the mirrors with the mouse), the reflections will no longer line up exactly. Only very special angles will result in a finite number of discrete mirror images. The "fitting" angles have to be a divisor of 180°. This case leads to angle segments of equal size, which fill out the full circle - like pieces of a pizza or cake which has been cut into pieces of equal size. So the fitting angles are: 180°, 90°, 60°, 45°, 36°, 30°,... (do try this out!). These angles play an important part in the theory of reflection groups. The whole thing not only works on the computer, generating beatiful pictures, but also in reality. Here are a few Photos, created with an adjustable angle mirror and an underlying post card. They depict, in order, angles of 90°, 60°, 45° and 36°, resulting in 4, 6, 8 or 10 sectors respectively. The images in each sector are by turns identical or reflected versions of the original.
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Dr. Stickler in the hall of mirrors
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Kaleidoscopes 