The following techniques apply to Iterated Function Systems composed of affine transformations. I haven't fully investigated to what degree they apply to other IFS systems, though I am confident that some of them can be so applied. They are illustrated with a self-similar attractor (the rep-4 equilateral triangle) and a self-affine attractor (the 3-ray starburst triangle).
Consider a fractal F, generated by the set of affine transforms U. A new set of affine transformations T can be defined such that
T is also an IFS, whose attractor is also F. Thus this technique does not generate new fractals. However it gives an alternative start point for the construction of other fractals by other techniques, and therefore opens up access to additional fractals.
Alternatively we can restrict the composition to a subset S of U, thereby defining a new set of affine transformations T such that
T is again also an IFS, whose attractor is also F. Again this technique does not generate new fractals. However it gives an alternative start point for the construction of other fractals by other techniques, and therefore opens up access to additional fractals.
Appropriate adjustment to the probabilities associated with the transforms of T is necessary to retain a uniform measure.
Consider a fractal F, generated by a set of affine transforms U, which is also generated by a set of affine transformations V. A new set of affine transformations T can be defined such that
Yet again T is an IFS, whose attractor is also F. This technique does not generate new fractals. However it gives an alternative start point for the construction of other fractals by other techniques, and therefore opens up access to additional fractals.
Note that this technique only works if the attractors of U and V have the same position, size and orientation. However the affine technique can alway be used to modify two similar fractals to have the same position, size and orientation.
Note that above, and in general, S=V.U is not the same as T=U.V.
In the same way as for the allo-composition technique we do not have to perform the composition for all elements of V. Defining a subset S of V we can define a new set of affine transformation T such that
Yes, you've guessed it: T is an IFS, whose attractor is also F. This technique does not generate new fractals. However it gives an alternative start point for the construction of other fractals by other techniques, and therefore opens up access to additional fractals.
Appropriate adjustment to the probabilities associated with the transforms of T is necessary to retain a uniform measure.
Given a set of affine transformations U, we can take a proper subset S of U, and define a second set of affine transformations T such that
In the case the attractor of T is not the same as the attractor of U. For these reason I call the attractor of T a meta- (different) figure, and this technique the metafigure technique. If the attractor of U has a similarity dimension of 2 and a uniform measure, then so does the attractor of T (with appropriate adjustment to the probabilities associated with the transforms of T). Self-similarity is also conserved. However rep-tile-hood is not conserved, as the resulting figure has elements of varying sizes.
An analogous partial pre-iso-composition technique does not in general conserve the properties of a uniform measure or of tiling the plane. I am not aware of a single example in which it does conserve these properties, but I am not possessed of a proof that it never does.
Consider distinct fractals F and G, generated by sets of affine transforms U and V. A new IFS T can be defined such that
In some cases, if F and G have a similarity dimension of 2 and a uniform measure then the attractor of T also has a similarity dimension of 2 and a uniform measure. Self-similarity is also conserved. Connectedness is commonly, but not universally, conserved.
I have not identified universal rules for when this technique is productive. I have however found empirically that it is separately productive for the homeopolymer members of the sets of hextals (fractals composed of elements placed at the centres of a grid of hexagons) and kingtals (fractals composed of elements placed at the centres of a grid of squares). In these cases the fractals F and G should have the same area, and have a relative rotation equivalent to the difference in the rotation of the fractal's elements with respect to the overall fractal. I have not investigated to what degree allo-composition can be extended to the heteropolymer members of these sets, or to other sets of fractals.
Both homeofractals and homeokingtals and many naturally defined subsets of these sets are closed under allocomposition. Note that all these sets have a countably infinite number of members.
It has been observed that a uniform measure and a similarity dimension of 2 are also produced for other (integral) ratios of areas. I have not identified the rules for these ratios, and note that the resulting fractal is typically not simply connected.
© 2001 Stewart R. Hinsley