Visualizing the Dynamics of the Unit Circle Group

Go directly to the pictures

The Unit Circle Group is a subgroup of the group of Möbius Transformations. It is worth noting the conditional perception of mathematics: namely, transformation, dynamics, cyclicity, etc., associated with the circle, other figures, their intersections and parallel existence. This speaks to the "quickening", thanks to modules, equalities/inequalities and so on. This combination of mathematical equations and proofs should be involved in a more artistic explanation to make it clearer. More about the use of such methods in teaching, you can start buy custom college essays and satisfy interest. An elementof this group has the form T(z) =

with |a|2 - |b|2  = 1.

A transformation in this group maps the unit circle onto itself and maps the interior onto itself. Note that |a| must always be greater than or equal to 1.

If a = 1, then T is just the identity map; if |a| = 1, then b= 0 and T is a rotation through twice the angle Arg(a), ( 0 ≤ Arg(a) < π). In this case the only fixed point is the origin. If Arg(a) is a rational multiple of 2π then T has periodic points, otherwise T is chaotic. If =a and b are real, a >1  and b > 0, then 1 and -1 are fixed points of T. -1 is a “source” and +1 is a “sink”, that is, T  moves any point inside the circle  away from -1 and closer to +1. If |a| > 1, |b| is determined. If Arg(a) and/or Arg(b) are not zero, then points will be rotated as well as stretched.

Now we will construct a design made up of disjoint circles placed inside the unit circle and illustrate the dynamics of applying transformations from the Unit Circle Group by “continuously” varying the parameters |a|, arg(a) and arg(b). (Since |b|2  = |a|2 – 1, there are only 3 parameters that can be freely chosen).

 Original design One iteration: a = 1, b = 0

 One iteration: a = 1.25, arg b = 0 One iteration |a|= 1.25, arg(a) =π/4, arg(b) = 0 One iteration |a|= 1.25, arg(a) =π/8, arg(b) = 0

 After 4 iterations; a = 1, b = 0 4 iterations: a = 1.25, arg a = arg b = 0 4 iterations |a|= 1.25, arg(a) =π/4, arg(b) = 0 4 iterations:|a|= 1.25, arg(a) =π/8, arg(b) = 0

More examples of circle pictures