Some elementary lessons from iterative fractal maps

The famous Sierpinski gasket was one of the first fractals we wrote code for when we got access to a computer. It impressed us enormously that an intricate object with self-similarity over all scales could be generated by a rather simple process:
1) Take three points that would define a triangle.
2) Take a fourth starting point.
3) Randomly pick one of the three vertices of the triangle. Plot the midpoint between the fourth starting point and the chosen vertex.
4) Repeat this process with that midpoint for a large number of iterations.
5) The points thus generated would converge to an attractor which is the Sierpinski gasket.

This process inspired us to use this procedure more extensively to develop other such fractal objects. We found an answer for this shortly thereafter in a book by the mathematician Barnsley precisely about this topic. In it Barnsley said something to the tune that we would stop looking objects in nature the same way once we see the fractals such as those for which he provided a recipe for construction. This had indeed already started happening to us with our encounter with the Sierpinski gasket. Notably, around that time we recorded a statement of Johannes Kepler (which we read somewhere), which was earlier expression of the same idea:
I believe the geometric proportion served the creator as an idea when he introduced the continuous generation of similar objects from similar objects.

The Hindu enacts such a concept when he performs śrauta rituals with complex rituals developing recursively with some modification from a simple “prakṛti”.

Coming back to Barnsley, he called these fractals Iterated Function Systems (IFS) and they are generated thus:
\begin{bmatrix} x_{n+1} \\ y_{n+1} \end{bmatrix} = \begin{bmatrix} a & b\\ c & d\end{bmatrix}_i \begin{bmatrix} x_{n} \\ y_{n} \end{bmatrix} + \begin{bmatrix} e \\ f \end{bmatrix}_i; \; p_i
where i=1..n

As one can see this map defines a set of affine transformations that are recursively applied to the starting coordinates (x_n,y_n). However, there can be a set of n such of matrices 1..n any one of which i can be applied to the point at a time with a probability of p_i. For each mapping of the point we have 7 numbers, 6 from the transformation matrices (a..f) and the 7th being the probability with which it is applied p_i.

Barnsley’s most famous fractal map was the fern leaf. Several variants of this have been generated (some by us and others whose creators are either Barnsley himself or others not known to us).


Figure 1: the first one is Barnsley’s original fern leaf

Given that a relatively small number of values to can generate a rather complex object by this procedure, Barnsley developed this as a method for compression of images. To us the IFS immediately struck us as being a mechanism for illustrating evolution: In the simplest scenario, the values in the above matrices and the probability of its choice can be seen as ‘genes’ which are prone to mutations with quantitative effects on traits. Thus, one can allow the matrices to mutate while we act as the proxy for natural selection.

The same is quite well-illustrated by another of the IFS fractals the leaf. In this leaf we have 4 matrices which gives us a total of 24+4=28 ‘genes’ which might be mutated. If we impose the conditions of continuity of the attractor region (a very real biological one) and reasonable bilateral symmetry then only 10 of those ‘genes’ can take mutations. Now, by letting them undergo mutation and selection we can see evolution occur in silico and spawn a variety of leaf forms close to what we actually see in nature.

ifs_leafFigure 2

One might learn two things from this:
1) If we give a person with no knowledge of how the matrices were constructed for generating a given shape, e.g. the leaf, he can mutate the ‘genes’ randomly and figure out how it works. After some mutagenesis he would soon discover that different ‘genes’ have different morphological roles and there is even a degree of hierarchy in terms of their effect on form. He would soon understand the difference between null mutations and change of function mutations. Thus, he can put together a developmental network for the role of the ‘genes’ in the development of the in silico leaf. In doing this on a very small scale he would have reproduced what geneticists have done to unravel the genetic networks for the development of real organisms like Caenorhabditis, Drosophila and the Danio (e.g. Nüsslein-Volhard and Wieschaus et al). Such efforts entirely dominated developmental biology and genetics for almost two decades.

2) Beyond doubt these efforts of the developmental geneticists contributed fundamentally to our understanding of biology. But in the end it spawned the field that is often irritatingly referred to as “evo-devo” — an attempt to marry the foundation of biology in the form of the evolutionary theory with developmental genetics. In our youth, even as we were discovering the beauty of IFS for ourselves, we were aligned towards a future as an evolutionary developmental geneticist. We even thought we should apprentice ourselves as a slave in one of those powerhouse developmental genetics labs, where we could bring our knowledge of evolution which was much ahead of theirs in those days. It was in this context the IFS experiments offered us an important negative lesson. We saw that the so-called evo-devo field was proceeding in a rather pedestrian direction. It was one where the protein or RNA products of the genes and were mostly faceless blobs (they would often literally be illustrated by researchers as such), with the primary expression of the studies being simple genome-demographic summaries from angle of genes they knew best and organism-specific variations of the maps of gene-interactions in development. To us this was not qualitatively telling us anything new about evolution of development beyond what we could glean from in silico experiments with systems such as IFS. Instead, we realized that we had to take a path that got directly to heart of what is unique to biology, i.e. the biochemistry of those gene products. This resonated with where we first began our science the chemistry of biological molecules. It was placing that within the framework of the evolutionary theory, which was to lead us closer to the foundations of biology than any of this other stuff.

In our childhood we had spent a lot of time simply gazing through the kaleidoscope. It brought home to us that one principle of aesthetics was symmetry. When we learned of IFS we saw how simple recursion was a second principle of aesthetics. We found that a very simple IFS could result objects of great beauty especially when combined with first principle symmetry. This led us to play with a simple map, namely one to generate the limit curve of a bifurcating tree. This can be done by choosing the below three matrices with equal probability:

\begin{bmatrix} x_{n+1} \\ y_{n+1} \end{bmatrix} = \begin{bmatrix} s \cos(\theta) & -s\sin(\theta)\\ s \sin(\theta) & s \cos(\theta)\end{bmatrix} \begin{bmatrix} x_{n} \\ y_{n} \end{bmatrix} + \begin{bmatrix} 0 \\ 1 \end{bmatrix}; \; p_1=\frac{1}{3}

\begin{bmatrix} x_{n+1} \\ y_{n+1} \end{bmatrix} = \begin{bmatrix} s \cos(\theta) & s\sin(\theta)\\ -s \sin(\theta) & s \cos(\theta)\end{bmatrix} \begin{bmatrix} x_{n} \\ y_{n} \end{bmatrix} + \begin{bmatrix} 0 \\ 1 \end{bmatrix}; \; p_2=\frac{1}{3}

\begin{bmatrix} x_{n+1} \\ y_{n+1} \end{bmatrix} = \begin{bmatrix} 1 & 0\\ 0 & 1\end{bmatrix} \begin{bmatrix} x_{n} \\ y_{n} \end{bmatrix} + \begin{bmatrix} 0 \\ 0 \end{bmatrix}; \; p_3=\frac{1}{3}

Here s is the scaling factor whereas \theta is the angle by which the branches of the bifurcating tree are rotated. Finally, we can rotate each fractal a fixed number of times to obtain a particular kaleidoscopic symmetry. In the first example below we have an overall symmetry of 6 with various \theta values and s=\frac{1}{\phi} the Golden ratio.

ifs_angle_curveFigure 3

In the second example we have a symmetry of 4 with similar rotations and scaling as above. One may note that the first case in this figure where \theta=\frac{\pi}{4} we get a structure that is immediately reminiscent of the floor plans towards which the central spire of the great Hindu temples (e.g. at Khajuraho) converged to before the coming down of the green curtain.

ifs_angle_curve_4Figure 4

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