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MatheMUSEments
Mirror, Mirror
By Ivars Peterson
Muse, March 2000, p. 18.
Look into a mirror, what do you see? One image of yourself.
But if you stand between two mirrors that are parallel
to each other, one in front of you and one behind, you see countless
images of yourself. Light rays bounce back and forth between the
mirrors to create that endlessly repeated pattern.
What happens when two mirrors are not parallel to each other?
You can find out by experiment.
Carefully join two small mirrors with a strip of tape. Stand
the pair of mirrors on a table so that the tape is vertical and
the reflecting surfaces face you. Make the angle between the
mirrors 90 degrees, and place an object between them. Count the
number of reflections you see.
You may have noticed such an arrangement of mirrors in the
corner of a fitting room at a clothing store. You see the object
(you in new clothes) and three images of the object.
What happens to the number of reflections as you make the
angle between the mirrors larger? Smaller?
At certain angles, the patterns are especially beautiful. When
the angle is 60 degrees, for example, you see the object plus
five images, with no overlap. That's the kind of pattern you
typically see in a kaleidoscope.
Suppose you looked into a special kaleidoscope and found a
pattern that consisted of one object and seven images of that
object. Can you figure out what the angle between the mirrors
inside the kaleidoscope must be?
You can experiment with your hinged mirrors, or you can use a
mathematical formula to answer the question. Simply take the
number of images you want, add one for the object itself, and
divide 360 degrees by this number. Three hundred sixty divided by
8 is 45. So the angle between the mirrors must be 45 degrees.
Artists, toymakers, and other people have built all sorts of
kaleidoscopes, including some you can actually climb inside and
some that are works of art.
The possibilities are limitless. Instead of two mirrors, you
can use three, fastened together to form a triangular enclosure,
to create an endless tapestry of repeated shapes. Instead of a
simple object, you can use pieces of colored glass, beads, coins,
or even lines and shapes drawn on paper.
Mess around with a computer kaleidoscope at aleph0.clarku.edu/~djoyce/pix/kaleido.html.
The Exploratorium in San Francisco has a
kaleidoscope that you can really get into:
www.exploratorium.edu/imagery/stills/Duck_Into_Kaleidoscope.jpg
The kaleidoscope illustrations for this article
were created using KaleidoMania! software. See
www.keypress.com/catalog/products/software/Prod_KaleidoMania.html.
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