Buffalo Bull

Issue E.4 — Friday, 2012.01.28

Of Peas and Power Laws

Pea plants played an important role in the development of biology. In the 19th Century C.E., Gregor Mendel cultivated 29,000 pea plants in a series of experiments to determine the rules governing the inheritance of traits such as pod, flower and seed color; whether the peas were wrinkled or smooth; and pod shape. Mendel formulated his laws of inheritance, which became the foundation of the new science of genetics.

Mendel's work is well known: it is taught to most schoolchildren. Less widely known is another result of pea plant observation, in economics.

In the early 20th Century C.E., Wilfredo Pareto observed that 80% of the peas in his garden were contained in only 20% of the pods. He also observed that 80% of the land was owned by 20% of the population. Generalized, this became known as Pareto's Law, or the 80-20 Rule: 80% of the effects are due to 20% of the causes. Pareto's Law is approximate, but has been observed to be accurate in many situations not only in economics:

Of course, Pareto's Law isn't universal. For instance, within internet communities (such as chat rooms), almost all of the content is produced by, or edited by, only 10% of the members; this might be called a 99-10 rule. Note that the two numbers in a rule need not add to 100: although 80 and 20 add to 100, 99 and 10 add to 109. The sum of 100 in Pareto's law is only coincidentally equal to 80+20, and the 100-100 rule always holds.

Power Laws

Pareto's Law is an example of a power law, which is a class of mathematical relationship. A power law is one where the frequency of some event is proportional to some power of a characteristic of that event. In other words, if a relationship has the form ƒ(x) = αxk, then that relationship expresses a power law.

Power laws exhibit scale invariance, which means that they hold even after multiplying the scale by a constant factor. For example, when we say that 80% of the wealth is held by 20% of the people, then it doesn't matter whether the wealth is measured in dollars or marks or yen. A conversion from one currency to another is an example of a scale shift, a multiplication by a constant factor. Scale invariance holds by simple algebra:

    ƒ(cx) = α(cx)k = ckƒ(x) ∝ ƒ(x)

In other words, a linear scale shift in one axis corresponds to a (different) linear scale shift in the other axis.

The factor α is called the Pareto index. It must be within the interval [1, ∞]. The 80-20 rule holds when α is approximately 1.16096404744... (exactly, α = ln4 5)

For wealth distribution, the 80-20 rule isn't a law of nature. It simply happens to approximate the real data. However, wealth can be more or less evenly distributed compared to the 80-20 rule. Perfectly even wealth distribution (everyone having the same wealth) occurs when α = ∞. Inequality increases as α approaches unity. The United States is more or less average, along with China and Argentina. There are countries with more equality (such as most of Europe, Japan, Australia, Canada, India, and Russia), and countries with more inequality (such as Mexico, Brazil, and South Africa). The long term trend (over several decades) is toward more inequality, although some countries are moving toward more equal distribution of wealth.

The 1% and the 99%

The recent “Occupy” movement draws attention to the “1%” and the “99%”. As it turns out, in the United States, about 1% of the population has about half the wealth. The numbers are a little different for income, rather than wealth, but don't trust the income numbers, because they're easier to fudge. It's relatively simple to cook the income books and make them look more equal, by including or excluding items, or by changing the way they are measured or computed. The wealth numbers aren't entirely accurate, either, but they're a little better most of the time (depending on who's passing out the data).

The “1% owns half” rule can be derived easily from repeated application of Pareto's law:

In other words, it's closer to 0.8% owns half, but 1% is easier for most people to remember, and makes a niftier slogan.


Fractals are geometric objects which can be broken into parts, with the smaller parts being scaled-down versions of the whole. These smaller parts can, in turn, be broken down again, and each of those sub-parts is also equivalent to the whole. In other words, if you look at a fractal under a microscope, what you see is like what you see when looking at the entire object. They are named from Latin fractus, “broken” or “fractured”.

This property, called self-similarity, is the same property that was used in the above example computing the “1% owns half” rule: the 80-20 rule applies to society as a whole, but it also applies to each income stratum separately, and to each substratum within those, and so on. (In geometry, when two objects are said to be “similar”, they are alike in proportions and angles, but not necessarily the same size.)

The power law ƒ(x) = αxk defines a line on a two dimensional plane. For example, the axes might be labelled “wealth” and “frequency”. Although both are self-similar, the power law object is simply a line, whereas a fractal has a fine structure at any given magnification. In other words, the closer you look, the more detail you see, ad infinitum. In fact, a fractal curve can fill a plane, something an ordinary, “normal” line cannot do.

An ordinary line on a plane is considered a one-dimensional object: it has length but not width. A fractal curve is one dimensional, but can cover an area. This led mathematicians to define a new kind of meaning for the concept of “dimension”. The older, traditional kind of dimension is called the topological dimension. Topological dimensions are integers: zero, one, two, three, and so on. The new kind, which is needed to describe fractals, has various precise definitions, but they are referred together as definitions of the fractal dimension. Unlike topological dimensions, fractal dimensions do not have to be whole numbers. They don't even have to be rational. A given fractal might have a dimension of, for instance, 1.518... (the decimal fraction continuing infinitely).

It is important to note that fractals occur in the real world, in nature, and not just in the minds of mathematicians. I'm not sure that physicists have completely grappled with this, but when they do, I imagine the lights will be on late in the physical sciences buildings at more than one university. Somehow, though, I don't expect to see a movie named anything like Attack by Aliens from the 7.4182...th Dimension at theatres anytime soon. It's just not a very catchy title.

Calculations of fractals, that is, the determination of the structure of fractal objects, is typically done using rule sets and algorithms known as generators. There are various kinds of generators. One such generator is based on a structure and rule set known as a strange attractor.

Chaotic Systems

Certain kinds of systems are called chaotic; that is, they exhibit chaos. There are specific requirements for a system to be called chaotic; mathematicians would state these requirements as:

It might be easier to think in the following, very rough and approximate, everyday language equivalents:

Despite their description as “chaotic”, these systems are not random. That is, future states are determined by the initial state. However, it is usually infeasible to predict their future states with exactitude. This is usually because there is too much computation involved. Sometimes, extremely small round off errors may cause errors in the results, rendering them unreliable or useless. At other times, the massive number of computations may result in technical impossibility, even though, theoretically, the predictions ought to be computable.

Not all chaotic systems are fractal, but many of the most interesting ones are. This includes those based on strange attractors.

In the real world, chaotic systems are used to model and predict various phenomena, including weather, earthquakes, financial markets, and genetic evolution. With the last of these, we return full circle to the other branch of science originating among pea plants.

Frivolous Addendum

From Chaos, God made both Heaven and Earth;
Initial conditions attended their birth.
From the beginning, around strange attractors,
Land, sea, and air, and all living actors
Move through a phase space evading prediction
To follow a course more peculiar than fiction.
Based on equations which no one can solve,
The Universe turns, and always evolves.
Dense orbits trace every possible space,
Then Chaos returns at the end of the race.

Where are the electric monks when you need them?

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