This technique does not create new fractal attractors; instead it creates a new IFS which generates the same attractor. Hence the name. It is not however pointless, as the new IFS may be one that another IFS creation technique can be applied to, or even if that technique could be applied to the original IFS it may be one that gives are different result on application of that technique.
Consider a fractal F, with a uniform measure, generated by the set of transforms T, such that the images of F under the members of the set T are disjoint. If these exists a subset S of T such that the union of the images of F under the members of S is mapped to itself by a transform A then a new set of transforms U can be defined
U is also an IFS, whose attractor is the same fractal F.
The obvious application of the iso-fractal technique is when the image of S is cyclically or dihedrally symmetric. In the former case A can be a rotation, and in the latter either a rotation or a reflection.
Any triangle can be transformed to an equilateral triangle by a transform C. As an equilateral triangle is dihedrally symmetric then it be mapped to itself by a rotation or reflection D. In this case A can be C.D.C-1.
Any parallelogram can be transformed to an square by a transform C. As a square is dihedrally symmetric then it be mapped to itself by a rotation or reflection D. In this case A can be C.D.C-1. Note that although a square is D4 symmetric parallelograms are C2 symmetric, and rectangles and rhombuses are D2 symmetric. Hence, for some D a parallelogram, rectangle or rhombus is an acceptable substitute for a square, and may be more convenient.
No other applications have been encountered amongst the set of self-similar IFS fractals. However in principle there could be other applications. For example a wedge trapezium can be transformed to a mirror symmetric trapezium by the application of a skew transform, and the latter can be reflected in its axis of symmetry.
© 2001 Stewart R. Hinsley