This applet draws 3 iterations of an iterated functions system of orientation preserving contracting similarities of the plane. The i-th similarity f_i is defined to be rotation by the angle a_i, followed by a scaling of by a factor of r_i, followed by a translation. The translation is defined by the following properties
f_1(0,0)=(0,0)
f_5(1,0)=(1,0)
f_i(1,0)=f_{i+1}(0,0) for i=1 to 4.
These restrictions allow one to choose a_1 to a_4 and r_1 to r_4 freely, while a_5 and r_5 are then determined.
Displayed is the image of the segment connecting (0,0) to (1,0) under three iterations of the similarities f_1,...,f_5. Shown in red are the images of the points (0,0) and (1,0) under one iteration of the mapping. Enjoy playing around!