Thursday, September 2, 2010

Measuring those awkward things

Imagine what the world would be like if only knew about whole numbers... no fractions or decimals. It would be very hard to live the life we do now without using continuous 'real' numbers, I think you would agree. Imagine the difference it makes to every part of study and work to be able to use these in-between, fractional numbers.
Now imagine yourself in the future looking back at today, a time when we only ever measure things using a whole number of dimensions. We can only ever measure the length of a river by pretending it is measured in metres^1, we can only count the branches on a tree by choosing a minimum size to be considered a branch.
In the real world most things do not have an integer number of dimensions, and pretending they do leads to problems in measuring them, an example of this is if you look up the length of Britain's coastline, you should find the quoted figures vary hugely.
Most things in the real world such as clouds, trees, land, water surface have a fractional number of dimensions.
OK, it seems reasonable to argue that nothing in reality is a fractal because nothing keeps its detail to infinite depth (to its atomic structure and beyond). However, the same argument can be used for whole numbers of dimensions, nothing in nature is made of absolutely straight lines or perfectly smooth curves either. The fact is that in both cases we approximate the real world within the range that is important to our measurements.
How do fractal measurements work? well the theory seems to already be known, but not apparently well used. You can use the Hausdorff measure to find the 'size' of a fractal structure, so if a coastline has fractal dimension 1.2, then you can measure its size in metres^1.2, e.g. Britain may be 200,000m^1.2.

So what would the world be like if we all started using fractional dimensions? Here are some areas with made up examples:
measure the area of mountainside in metres^2.3
count the boulders on a scree slope in metres^0.5
measure the size of a river system in metres^1.4
measure the cloud cover in humidity^1.3
measure the wind levels in (metres per second)^1.2
Town planning:
measure the number of roads in metres^1.4
count the moon's craters in metres^0.5
count the number of asteroids in the asteroid belt in kg^0.6
measure lung size in metres^2.5
measure the number of neurons in the brain or the size of the cortex (both fractal)
count the number of animals on the planet in kg^0.8
measure the size of a forest by more than just the number of trees, include all smaller vegetation.
measure the speed of a chemical reaction in seconds^0.5
compare the size of various algebraic fractals
stockmarkets fluctuate in a fractal distribution, measuring the size of different distributions is helpful
company pay follows a power law, so is fractal, so can be measured.

In fact, the point of this long post is really to say that pretty much everything follows a power law of some kind, so the only sensible way to measure it is a with a fractal measure. Something that as a whole we really don't do at the moment.

Next post I will go through the maths of measuring things with power laws.

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