5. KALEIDOSCOPE
 The kaleidoscope construction http://javaboutique.internet.com/Kaleidoscope/source.html ). I think it is fair to say that the patterns produced by this applet are kaleidoscope-like – there is none of the rotation effect that you would get from a real kaleidoscope. However, many interesting patterns can be generated. Another type of tessellation are the semiregular tessellations. These are also tessellations of regular polygons placed edge to edge, but here we allow more than one type of polygon. We insist that the polygons surround each vertex in the same way. There are in fact eight possible tessellations of this type, and they are illustrated below.
• Use the applet to see if you can discover any of the above semiregular tessellations. Some you would not expect to find in this way. Why is this?
# Because of the construction of the standard kaleidoscope, we would probably only expect to get tessellations involving equilateral triangles and regular hexagons (unless the original motif pattern had, for example, squares). It is easy to see that there are just three possible regular tessellations. For there must be at least three polygons meeting at each vertex. Further, the interior angles of the regular polygons are 60° (triangle), 90° (square), 120° (hexagon), and more than this for the rest. Since the total angle at a vertex is 360°, we deduce that these are the only possibilities. Isometries An isometry is a transformation which preserves lengths (distances). [From the Greek: iso equal, same; metron measure.] Any object which is mapped by an isometry has an image which is congruent to the original: it has the same size and shape.
It is known that there are exactly four different types of isometry in the plane, and that they can all be obtained as a ‘product’ of reflections: that is, take the original F (say) and find its image under reflection, then find the image of this image under another reflection and so on for a finite number of steps.
• We already have the original F and an image which illustrates reflection.
These four isometries form a group. See for example http://internal.maths.adelaide.edu.au/people/pscott/groups/gpf/
Bibliography General ... http://en.wikipedia.org/wiki/Kaleidoscope Make your own ... http://www.csiro.au/resources/ps1up.html Other applets ... |
In the old days, before television and computer games, people had to look elsewhere for their entertainment. One source was the kaleidoscope – perhaps the best known of all optical toys. The kaleidoscope was known to the ancient Greeks, but it was rediscovered and patented in 1817 by the Scottish scientist Sir David Brewster. The name ‘kaleidoscope’ comes from three Greek words that mean ‘an instrument with which we can see things of beautiful form’. After the publication in 1819 of Brewster’s Treatise on the Kaleidoscope, the kaleidoscope became an extremely popular toy. 
A kaleidoscope is made up of three inward facing mirrors forming a hollow triangle and usually placed in a long tube. One end of the tube is translucent and contains a selection of little pieces of coloured glass; the other end is covered with a hole in its center to form an eye piece. By looking through the eye piece and rotating the tube, you see beautiful changing patterns as the pieces of coloured glass tumble about and are reflected in the mirrors.
Squares are regular four-sided polygons which tessellate the plane: that is, they fit together, edge to edge, in a way which completely covers the plane, but with no overlappng.
A simple example of an isometry is a reflection in a line. Thus in the diagram at right, the letter F is reflected in the mirror line m to give a reversed letter F. The object and its image have the same size and shape. They are equidistant from the mirror line.
# The kaleidoscope (product of reflections) gives a pattern of Fs as shown at right.