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Word rank order
Figure 1.12.
Zipf's law. Word frequency is graphed as a function of rank order of frequency usage for the
8,728 most frequently used words in the English language. The inverse power law is that of
Zipf [ 42 ].
probable average rate of work expended . In this way the effort involved in making a
decision to solve a problem, that is, in traversing a path, is minimized.
Unfortunately, most people who quote Zipf's scientific contributions refer only to his
law regarding the relative frequency of use of words. Zipf's law is of the form ( 1.43 )
and is shown in Figure 1.12 , with y being the relative frequency of word usage in the
English language and x the rank of that word. In this example Zipf would argue that long
words are used less frequently by the general population than are short words because
they require more effort, so that ordering the words of a language by their frequency of
usage results in a decreasing function. The length of a word is proportional to the time
needed to produce the word, whether by writing, speaking, or otherwise, so that we may
associate correlation times with word length. If we assume there is no characteristic time
scale for language, or correspondingly no typical word length, then we may conclude
that language ought to follow an inverse power law. We present the mathematical basis
for this conclusion later. Zipf found that many languages have inverse power laws for
the frequency of word usage, but that the slopes vary slightly from language to language.
The second monograph to “explain” the importance of inverse power laws in the
natural, social and life sciences was Mandelbrot's now classic topic [ 21 ], in which he
introduced the concept of fractals. In our analyses it will become apparent that many of
Mandelbrot's ground-breaking concepts have found application even beyond the expan-
sive range he envisioned. A more recent topic, but one focused on the morphogenesis
of physiologic organs, is due to Weibel [ 36 ]. In Weibel's topic the fractal concept is
hypothesized to be a design principle in physiology, a concept that had previously been
 
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