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Mirror Systems

Taken from Cozy Baker's book, "Kaleidoscope Renaissance,"
published by Beechcliff Books, Annapolis, Maryland.

There are two major systems of mirrors in kaleidoscopes: the two-mirror, which produces one central image or one cluster of images, and the three-mirror, which produces innumerable images throughout the entire field of view. Both are set up in a triangular configuration - in a tube similar to a prism.

In the two-mirror system, the mirrors are arranged in a "V" with a third side that is blackened. The angle of the "V" determines the number of reflections. Alda Siegan explains the effects of the two-mirror arrangement as "somewhat similar to standing in front of a dressing mirror having a side leaf mirror. The close the angularity between the mirrors, the more reflected images of your face."

The most perfect symmetry and best images occur when the angle between the mirrors divides equally into 360 degrees.

60 degrees -- 6-fold symmetry -- 3 point star
45 degrees -- 8-fold symmetry -- 4 point star
36 degrees -- 10 fold symmetry -- 5 point star (shown at right)
30 degrees -- 12-fold symmetry -- 6 point star
22.5 degrees -- 16-fold symmetry -- 8 point star
20 degrees -- 18-fold symmetry -- 9 point star
18 degrees -- 20-fold symmetry -- 10 point star
15 degrees -- 24-fold symmetry -- 12 point star

To see if the angle you've chosen will create a perfectly symmetrical image, divide 180 by the number of degrees in your angle. If you get a whole number, then your image will be symmetrical. Also, this whole number will equal the number of points in the "star" that your image will make.

The three-mirror system can be arranged in any form of triangle, so long as the sum of the three angles equals 180 degrees. It produces a continuous field of honeycomb-like patterns.

The 60-60-60 degree equilateral triangle is the most common and produces the least attractive pattern. The 90-45-45 degree triangle gives a more interesting symmetry. But the most enjoyable image produced by the three-mirror system is the 30-60-90 degree triangle, which contains three types of symmetry: fourfold (from the 90 degree angle), sixfold (from the 60 degree angle), and 12-fold (from the 30 degree angle).

[Here's a puzzle for you to solve: Can you figure out the angle of the mirrors used in the three-mirror image at left? Hint: Use the "rules" for Two-Mirror Systems listed above. You'll find the final answer at the bottom of this page!]

Other systems, such as square four-mirror configurations, produce repeated square patterns, while four-mirror rectangular configurations produce repeated rectangular patterns. The images created are striped patterns, since the reflections move directionally up, down, right, and left.

Cylindrical tubes lined with a reflective material will produce a spiralling effect. Since there are no angles involved in this style, the reflection seems to climb through the tube assymmetrically.

Tapered mirror systems (shown at right) provide a spherical 3-D image when viewed through the larger opening.

The polyangular arrangement is the most elaborate and satisfying. It is a variation of the two-mirror design in which one of the mirrors can be adjusted, changing the angle of the "V" and thus the number of reflections. It is possible, therefore, to produce a wide range of symmetrical patterns.

It is also possible to build two or more separate mirror systems into the body of a scope, each with its own eyepiece and viewpoint of an object.

[Answer to the Three-Mirror Puzzle: The angles of the mirrors are 20 degrees, 80 degrees, and 80 degrees. How did we figure that out? Count the number of points on the largest, most central image. In this case, it is 9 points. Now divide 180 by the number of points (180 / 9) and you get 20. This is the number of degrees in one angle of this triangle of mirrors. Since all the angles in a triangle must add up to 180 degrees, subtract your answer from 180 to get 160. Now, we are lucky in this case in that the other two angles in this image are equal to each other, because then all you have to do now is divide the 160 remaining degrees by 2 to get 80 degrees for each of the other two angles!]

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