Wednesday, April 11, 2007

A Fractal Fairy Tale

Once upon a time, there was a giant who was fond of strolling the shore of his island each morning as the sun rose. One morning as he strolled he wondered how long his stroll was and asked his fairy godmother how far it was around his island. His fairy godmother gave him a magic, kilometer-long measuring stick and told him to measure it himself.

When he had done this, he went back to his fairy godmother and she gave him a magic, meter-long stick, and told him to measure it again. When he had done this, he was given a magic, millimeter-long stick and told to measure one last time.

Because the sticks were magic, they were ultimately precise and his measurements ultimately accurate; no errors or tolerances to worry about. However the giant found he recorded a longer distance as the sticks got shorter.

Being of a philosophical bent, the giant sat on a large boulder and thought about why. The creatures of the giant's island went about their lifes, tides waxed and waned, the sun sped its daily course across the heavens, seasons came and went, tectonic plates drifted about the planet like the skin on hot milk and the giant's ponderous ponderings finally came to a conclusion.

He decided that each stick, although magic, did not measure the distance absolutely, rather they averaged out the "wiggliness"of his island's coastline at its (the stick's) particular scale. So the giant's first stick ignored any coastal feature less the a kilometer in dimension and similarly for the other two measurements at their dimension.

The giant decided if he wanted to absolutely know how far he walked he would have to ask his fairy godmother to make his coastline conform to a formula from which the distance could be calculated. He also thought that it would be interesting for his island to have a coastline that was so "wiggly" that it would occupy all space but still have its present finite area.

However, deciding he liked his island just as it was, he returned the measuring sticks to his fairy godmother, thanked her for an excellent think, and wandered off down the tideline as the fairy godmother followed his meanderings with a gentle smile.

Monday, April 2, 2007

Wobbly Genes

Say you have a population of frogs in a pond. The number of adult frogs is governed by available froggy food, froggy predators, froggy diseases etc and also governs the size of the new generation of frogs. And these same constraints decide the number of adult frogs in the next generation at reproduction time. So you have a recursive system; the outcome of the previous generation sets the starting condition of the next. I hope you can see that such a system if you track its progress through a host of recursions is sensitive to the condition of the previous incarnation; its initial conditions.

This is the basic concept behind Chaos Theory ie does the flap of a butterfly’s wings in Brazil set off a tornado in Texas or sensitive dependence on initial conditions.

One of the implications of Chaos Theory is that the systems exhibiting this dependency are sensitive to impulses but after a period of instability as a result of such impulses they standardly find a new point of stability. Like a spinning top on a dimpled surface; a jog will move it from its orginal dimple and make it wobble but with enough spin it will eventually settle in a new dimple.

There are biology theorists that are using this characteristic to look at evolution afresh. They suggest that it is valid to look at genetics as one of these systems. A jog from a population’s environment will cause genes to move from a stable morphology, have a wobble-experiment with a bunch of different forms and finally settle to a new solution for survival (with the inference that the jog influences the system towards a successful but still unpredictable result). These theorists explain the potential biological forms available from a genetic set as a convoluted surface of stable dips and valleys (not necessarily three dimensions though ~evil grin~).

Apparently this concept does explain some of the more difficult problems with previous theories of evolution with regard to the mechanism of mutation and species differentiation.

Maybe there was or will be just such a jog to make humanity to think and act, not globalisation. >:{(}