The essences are each a separate glass, through which the sun of being's light is passed -- each tinted fragment sparkles in the sun: a thousand colors, but the light is one.
- Jami (15th Century)
Here is a simple method used to create computer-generated kaleidoscopes, and the concept should be easy for most personal computer users. The kaleidoscopes are composed primarily of triangles, with a few circles also scattered about.
Rather than use triangles, readers may easily apply these principles to a collection of points, if this is easier.
Start by randomly selecting 3 vertices of a triangle. We will call the first three vertices
( x1 , y1 ), ( x2 , y2 ) and ( x3 , y3 ).
Readers may draw this triangle using any computer graphics package available (or if they have patience, they can draw it by hand!). This is the "parent" triangle. All the reflected versions are called "children." Readers can calculate the positions of the children points rather easily. Just take each of the three (x,y) coordinates, and negate them, or shuffle them in all the possible ways:
(x,y), (-x,y), (x,-y), (-x,-y), (y,x), (-y,x), (y,-x), (-y,-x).This produces a kaleidoscopic symmetry. To insure that the initially selected random points of the triangle fall within the triangular region of the plane, we actually swap "x" and "y" whenever a randomly selected "x" is greater than a "y" in an (x,y) pair. The pseudocode shows how to do this.
ALGORITHM: How to Create a Computer Kaleidoscope.
DO FOR i = 1 to 40
x1 = random; y1 = random
x2 = random; y2 = random
x3 = random; y3 = random
/* Confine initial pattern to lower right quadrant */
if ( x1 > y1 ) then (save=x1; x1=y1; y1=save;)
if ( x2 > y2 ) then (save=x2; x2=y2; y2=save;)
if ( x3 > y3 ) then (save=x3; x3=y3; y3=save;)
DrawTriangleAt(x1,y1,x2,y2,x3,y3)
/* Create 7 reflected images /
DO FOR j = 1 to 7
Flip (x,y) Points as Described in Text
DrawTriangleAt (x1,y1,x2,y2,x3,y3)
END
END
The same basic idea applies to the positioning of the small circles in the designs. Colors may be chosen randomly.
We have access to special purpose graphics hardware that can render many thousands of triangular facets in a second, but even with simpler computer systems, beautiful pictures can result fairly quickly. For additional beauty, the patterns described by the above (x,y) shufflings can be translated several times in a checkerboard pattern, as we have done for the figures on the kaleidoscope pages. For additional beauty, we have also interpolated color across the facet of each triangle. This means that if one vertex is randomly chosen to be red, and the other two green, then various shades between red and green will show on the facets.
As we stare at the patterns moving and unfolding on the graphics screen we can only wonder what Sir David Brewster in 1817 would have thought. Would the modern-day electronic kaleidoscopes, involving no physical mirrors, be infringements on his patent? Probably not, but like the real thing, the electronic kaleidoscope's colored pieces make a fantastic "wallpaper for the mind." Readers are encouraged to design a similar system and to try applying different kinds of symmetries. Readers may also try designing a 3-D version, where the triangles and circles are scattered about in three dimensions, and not limited to a plane. Finally, assymetry may be introduced into the designs by choosing NOT to reflect certain triangles. Here is an interesting example where the computer allows experimenters to design a non-physical kaleidoscope which could not be created using mirrors in the real world.
For readers interested in learning more about traditional kaleidoscopes, consider Through the Kaleidoscope and Beyond by Cozy Baker (Beechcliff Books, 1987). The book describes handcrafted models, and contains beautiful color photographs. It also has a list of current kaleidoscope makers and shops. Another interesting book is Computers in Art, Design and Animation by Lansdown and Earnshaw (Springer, 1989). A chapter in this book discusses computer kaleidoscopes operating on digitized images of people and scenery. Another source for information on kaleidoscopes is The Family Creative Workshop series of books. This series published in 1974 by Plenary Publications in New York has a fascinating chapter on how to create various kaleidoscopes using mirrors, acrylic strips, and related.
I describe computer-generated kaleidoscopes in more detail in my book Mazes for the Mind: Computers and the Unexpected (St. Martins Press: New York).
Cliff