Evolution of Math into Art using Möbius Transformations
We start with an iterated function system defined by a simple replacement rule that uses Möbius Transformations to map the unit circle C into four circles tangent to C and pairwise tangent. The limit set is a Cantor Set, totally disconnected and very uninteresting. Gradually our IFS evolves. (Figures are at the bottom of the page.) A fifth circle tangent to the other 4 is added at the center of C. Next each of the original transformations is composed with a transformation from the Unit Circle Group. A transformation from this group maps C to itself and the interior of C to itself. By varying the parameters a variety of interesting and artistic pictures and limit sets appear. You can see two animations made by letting the parameters change in small increments.
Next we increase the number of circles to any number up to 12. Finally transformations from the Unit Circle Group are used to generate pictures of iterated chains of Steiner Circles
Next we have an interactive Flash Program that lets you choose the number of circles and the parameters. You must be careful to put in values in the correct range. Please read the instructions below so you understand what the program does.
The parameters for the Unit Circle Group are complex numbers u and v, satisfying u*u-v*v = 1. This means that |u| must be greater than or equal to 1 in magnitude; we call this mag(u). You may also enter the argument of the two parameters u and v; actually you enter a decimal multiple of 2 pi. That is, for example, if you enter .25 for arg(u), the program will read it as pi/2. The program uses Möbius transformations to map the unit circle into n+1 circles. n of the circles are tangent to the unit circle and arranged around the circle in such a way that each of the circles is tangent to the two adjacent circles. The (n+1)st circle is has its center at the origin and is tangent to the first n circles. So when you are asked how many circles, you may enter as few as 3 or as many as 12. (You will actually get one more than you requested, that is the one in the middle).
The transformations are then iterated using recursion; that is: the first step gives you the first big circle; the second step gives you the n+1 images of the first circle; the 3rd step gives you the (n+1)-squared images of the n+1 circles, etc. Due to the limitations of Flash, the maximum number of steps is 4. In fact if we use as many as 12+1 circles and 4 steps, most computers cannot handle the number of circles produced. (How many circles would that be?) So, as a safeguard, if you type in 10 or more circles and 4 steps and "show all steps, y or n", the program restricts you to n (that is you cannot see all stages for high numbers of circles and the maximum number of steps). If you have an old or slow computer, do not try large numbers!
References:
Tristan
Needham, "Visual Complex Analysis", Oxford University Press, 1997
David Mumford, Caroline Series and David Wright, "Indra's Pearls",
Cambridge University Press,2002
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