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9. Kaleidoscopes1 A term most commonly associated with a tubular toy, a kaleidoscope can be generally associated with a type of plane symmetry that will be investigated in some detail in subsequent sections. Invented by Sir David Brewster in 1813. Primary reference is: Sir David Brewster, The Kaleidoscope, Constable & Sons, 1819, reprinted by Van Cort Publishers, Holyoke, MA 1987. The word comes from the Greek words χαλοζ meaning beautiful, ειδοζ meaning form, and σχοπεω meaning to view. So “Beautiful form viewer”. Since optical focal length for humans is between 6 and 8 inches, commercial kaleidoscopes are often of those lengths. A kaleidoscope can be made using either two or three mirrors, so one must decide on appropriate angles between adjacent mirrors. For two mirror kaleidoscopes, a little experimentation suggests that angles of the form 180º/n, where n is a positive integer are best, since this reproduces exactly n copies of the region between the mirrors. Other angles will produce overlap or other distortion between the image and adjacent images. 1 Geometry by Discovery, David Gay and Symmetry, Shape and Space, Christine Kinsey and Theresa Moore m ! ABC = 59.95° C B A Contrast the above image with the one below where the angle is 50.17º: m ! ABC = 50.17° C B A In the first mirror arrangement (mirror angle = 60º), there are only six images while in the second, multiple images are produced with overlap occurring, thus distorting overall effect. It is instructive to examine how all of the images perceived by the eye, since there are always images that result from multiple reflections. In the figure below, the path of light from a point on an object between two mirror with mirror angle 90º to the eye has been traced, explaining the secondary reflection behind the two mirrors. Eye When three mirrors are used, the proper angles can be deduced from the following considerations. Adjacent mirror angles must be good two mirror angles, so it follows that the diagram below shows the general set-up. 180 n 180 180 m p Since the sum of the angles of a triangle is 180º, this leads to the 1 1 1 + + = 1. Starting with m = 2, one finds the only solutions equation m n p to be m = 2, n = p = 4, m = 2, n = 3, p = 6. and m = n = p = 3. !