the PC apeman - 22 December 2010 01:22 PM

Ahem.

the PC apeman - 20 December 2010 08:56 AM

With an infinite number of iterations, we can continuously zoom in on this fractal and always find self-similarity. However, we don’t need to resort to infinite iterations to see my criticism of superficiality.

When we construct an object from 4 parameters (plus function instructions), we have a starting point. When we iterate this 1 time with a reduced self similarity, we have a fractal which already looks different from the original parameters. Self similarity is already lost by the first iteration, which produces an object with many more facets. The greater the number of iterations the less similarity the whole has to the original object in both size and shape, yet each iteration itself is an identical but “smaller” version of its predecessor.

If we stopped iteration somewhere and zoomed in on the outermost layers of the fractal structures, we would eventually see our old 4 part object reappear (the starting point of our fractal), but at a greatly reduced size.

If we disassembled the fractal itself (reverse the iteration process) each iteration reversal would yield a simpler and smaller overall shape with bigger constituents parts, until we arrive back at our 4 part original size starting point. From that point you cannot apply a further fractal reversal.

The confusing issue here is that the more iterations, the more complex and bigger an object (fractal) becomes, while the parts become exponentially more numerous, and smaller in size (including the original parameters of our start object).

If I understand Loll, if we were to iterate any fractal set until the parts become sub atomic in size and beyond and a zero value is reached for each part of the outermost boundaries, we reach the edge of the physical universe. The whole universal fractal is bounded by parameters which are so small they have no longer any properties at all (event horizon), yet the process of iteration continues at quantum level (theoretically into infinity). Thus while the properties of the universe keep getting more and more complicated and its size increases, the original parameters of our starting fractal has become so small that its shape is no longer of importance. This might explain the apparent increase in universal expansion without physical properties. It becomes a purely mathematical (logarithmic) function.

We can only see the physical part of the fractal and it works flawlessly (it is a logarithmic mathematical function). There is no reaon to assume that at Planck level this function breaks down. In any case, to call any part or function of this process “superficial” seems dismissive of the potential mathematical value of fractals. Fractals may provide an answer to the similarities of shapes and functions found throughout the physical universe and beyond.

In the case of the Romanesco, each plant shows remarkable similarities to all other Romanescos. This may be considered genetic, but then, cell division is a fractal process (self similar iteration without reduction in size) and apparently each plant has similar genetic fractal growth instructions which account for the apparent similarities of growth patterns in mature plants. Humans are also fractally constructed (with minor differences in genetic growth instructions), thus our similarity as well as our minor differences. Cell structures, neural and arterial networks, the brain, all are fractal and show recognizable similar properties in all humans. This is not a superficial resemblance. If we clone an organism (an exact iteration of the original chromosomal instructions), it grows identical to its parent. Any superficial differences among living things are from externally introduced changes which modify the fractal structure without changing its fundamental function.

Dang, if only I was able to express myself in formal scientific language. Please bear with me…