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Transcript
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.
!