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 Kaleidoscope Puzzle Change the pattern at right to any of the patterns at left. (The original pattern is at top left.) To change the pattern, click on a row, then click on the row you want to move it to. Do the same for other rows and/or columns and/or quadrants. If the pattern does not appear to change, you may be switching two parts of the pattern that are identical. Users of Internet Explorer may want to page back, then forward, to speed up the action of the kaleidoscope. Copyright © 2005 by Steven H. Cullinane.Copying for non-profit personal use by an individual is authorized.(In Firefox, click on "File" menu option, then on "Save Page As," then enter the file name "KaleidoscopePuzzle.html" (this name must be used) and select save-as type "Web page, complete.")

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Background:

For non-mathematicians...

See some selected quotations on Hesse's Glass Bead Game, and the way it combines music and mathematics.  Then see the combination of music and mathematics in Timothy A. Smith's Shockwave movie (or his pdf essay) combining patterns like those above with an analysis of a Bach fugue.

For mathematicians...

Kaleidoscopes are, of course, intimately related to reflection groups.  For the connection to the geometry of the above patterns, see The Diamond Theorem.  Switching two rows, columns, or quadrants in the "kaleidoscope" above is equivalent to performing an affine reflection (an automorphism of an affine space fixing a hyperplane pointwise) in a 16-point finite geometry. For some context, see Reflection Groups in Finite Geometry.

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