How come the whole is more than the sum of its parts?

In the paper Fractal Memory for Visual Form, in their application of the concept of iterated function systems aiming at constructing fractal objects, which appear as natural-looking images, it is mentioned that

"The resulting image is an attractor for the iterative system. One non-obvious point: the form of the attractor is independent of the starting image. I could have started with any image other than the square, and the Sierpinski Triangle would have resulted."

The fractal object produced, referred to as an attractor, being independent of the starting image? Any chosen object would produce the same attractor? The Sierpinski triangle? Provided that the procedure, the iterated function system, is the same? The iterated function system has a quality of its own? An attribute altering quality? The procedure being more significant than the part that makes up the whole? The particular iterated function will produce a certain form of an attractor despite the difference in the starting object? The iterated function systems have emergence qualities? What are the implications of such thoughts?

The whole, the fractal object produced being more than the sum of its parts. The attributes and properties of the whole different from these of the parts, the reason being the application of the iterated function systems procedures?

And the fractal geometry, that nature is built on, governed by "self-similarity" where the parts resemble the whole?

The human mind mirroring nature employs similar procedures along the lines mentioned below

" ... memory is not a passive list of attributes, but rather is a set of mental procedures. These procedures, when activated, allow a reconstruction of a semblance of the original experience."

Memories stored in reverberating neuron assemblies in the brain as mental procedures, ever ready when activated to construct experience.