Tuesday, May 06, 2014

Patterns and Power Laws

In Geoff Lawton's Permaculture design course a lot of participants had problems with the video on patterns.

In it, he asserts that in nature, most if not all patterns in a whole raft of phenomena (river sizes, relative bone lengths, etc) occur in between 5 and 9 orders of size. ie there's a recognizable and limited number of classes of sizes of the object.

I've started to attempt to explain my take on it. I think, like on the video about entropy, he's 'onto something' but missing some of the explanation. And as both entropy and natural patterns are such big topics - we observe the phenomena and really want a good explanation for it - they can't be glossed over.

Draft the first - on Patterns in nature:

Patterns and power laws

What I take away from this discussion of pattern is that it is as a heuristic, or rule of thumb, when we observe natural patterns that they often show between 5 and 9 orders of size.

It’s not a “law” but a useful tool to use while looking for patterns. “Many” natural patterns show between 5 and 9 orders of size, so see if you can spot between 5 and 9 orders of size, before concluding the pattern you are observing has fewer or greater orders of size.

As for the scientific and mathematical rigour of the assertion …

A quick search came up with two links that might help understand “why” natural systems do this.

1) examples of the Fibonacci sequence http://jwilson.coe.uga.edu/emat6680/parveen/fib_nature.htm

2) a description of rank-size distributions http://en.wikipedia.org/wiki/Rank-size_distribution

From the description of rank-size Wikipedia says :

"(The) rule “works” because it is a “shadow” or coincidental measure of the true phenomenon.2 The true value of rank size is thus not as an accurate mathematical measure (since other power-law formulas are more accurate, especially at ranks lower than 10) but rather as a handy measure or “rule of thumb” to spot power laws."

Which I paraphrase as "many natural systems exhibit similar recurring patterns because they are following some power law relationship (in the mathematical sense) ie surface area increases as the square of the radius (radius “to the power of 2”) while volume increases as the cube of the radius (“to the power of 3”)

And because many natural systems must contend with these types of interactions – the two dimensional surface of water being pushed by wind that contains a three dimensional volume of liquid – similar patterns occur.

This is a good description of natural patterns.


And one of the “reasons” these patterns occur is because they are either energy maximizing (in the case of the distribution of leaves to capture sunlight) or minimizing (spheres are the minimum surface area to contain a given volume)

Natural selections favours organisms that use energy most efficiently, and physical and chemical systems follow the least energetic path (I.e water rarely flows uphill and products of combustion rarely reassemble into reactants)

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