Notes for the Lifebox, the Seashell, and the Soul - Rudy Rucker

Notes for The Lifebox, the Seashell, and the Soul, by Rudy Rucker
These exponent 1 and exponent 3 descriptions of the sizes of rocks become rather
similar if we graph them in a certain way. The idea is to use log-log graph paper.
A different type of power law arises with natural sounds such as the babbling of a
brook, the whispering of the wind in the trees, the crashing of the ocean, or the splashing of
rain. We can view these sounds as being made up of different-sized pieces in the sense that
they’re combinations of waves of different frequencies or cycles per second. The deep
sounds have low frequencies and the shrill sounds have high wavelengths. Here we view the
“size” of a wave as being its frequency, and we view its intensity as the amplitude of the
sound’s contribution. Now it so happens that most natural sounds obey the power law.
Sound component amplitude = 1 / frequency
Fractals can be characterized as systems that obey power laws with fractional
exponents. The Koch curve we can measure the size of a piece by its linear length, and the
intensity of that size by the number of components of that size. It turns out that we get levels
like this, but with the levels overlaid upon each other.
1 piece of length 1,
4 pieces of volume 1/3,
16 pieces of volume 1/9 ... and
4 L pieces of volume 1 / 3 L
Now it turns out that if D = log(4)/log(3), then 3 D = 4.
The nineteenth century Italian economist Vilfredo Pareto observed a kind of power
law in income distribution, with people’s income being proportional to the inverse of some
power of their rank. He came up with a simple kind of model to suggest how this could
come about.
In the more general case, suppose that each employee has K subordinates, and each
subordinate earns 1/L as much as his or her immediate boss.
If we generate the organizational chart down to N levels, viewing level 0 as being the
single supreme boss at the top, then the people at the Nth level will be getting a salary S of
(1/L) N = (1/L N ) times the supreme boss’s salary.
The number of people at the Nth level will be K N . Their rank R will also be on the
order of K N , because if we sum up the rows of employees ranking higher than the Nth level
people, we are looking at the sum with i ranging from 0 to N-1 of K i , which is (K N - 1) / K-1,
a quantity which is roughly K N-1 at the beginning of the row, but K N at the end of the row.
Now define D to be (log L) / (log K), then K D is L and if we say R is K N , then R D =
(K N ) D = L N .
Then S = 1/L N = 1/(K N ) D ~ 1 / R D or, quite simply,
S ~ 1 / R D .
Inverse Power Law Examples
Inverse power laws are all over the place; here’s a few more examples.
p. 88