Do mathematical equations, however simple, belong in any "intelligent" conversation. It takes a special kind of hubris to imagine that anyone but an engineer, scientist, or mathematician would try to read on when that space and indentation appear, followed by Greek and English (occasionally even Hebrew) letters and numbers -- as subscripts and superscripts and brackets, not to mention strange symbols like ∂, ∫, or √.
When it comes to concepts like fractals, a coffee table book might be acceptable -- pictures with an apparent, if questionable, aesthetic appeal based on some mumbo-jumbo created by somebody named Mandelbrodt. Didn't he have something to do with the IBM PC?
Finally, there's entropy. A post-impressionist or even modernist word, isn't it -- sort of a rationalization for Roaring Twenties degeneracy or Seventies consciousness-raising: "not with a bang, but a whimper."
There's been a kind of unconscious feeling among many that there's a connection between entropy and fractals. Google "entropy fractal". At the time of this post, I get 117,000 hits. Delve more deeply however into these links, and the connection (with few exceptions) is esoteric at best.
Fractals are peculiar components of natural or human-created structures, phenomena, or processes that show self-similarity at a variety of scales.
Entropy can be thought of as a measure of ignorance or, conversely, information.
In a fractal, repetition of a structural leit motif over multiple scales implies that the particular information nugget that describes the leit motif exists at each scale. In a human creation the leit motif might be constructed by algorithm or formula.
The technique of maximum entropy, popularized by physicist Edwin Jaynes (1922-1998), is a method of interpreting a phenomenon (or process) for which some mathematical structure is inferred and some "inadequate" constraints are known. By application of the technique (or principle), the most probable state of the phenomenon given the constraints may be derived.
In the case of fractals, their occurrence suggests a possible information content constraint at each scale. In a particular geographic area (e.g., Southern California) earthquake magnitudes and frequencies show fractal behavior over a broad range of earthquake energies. This implies that the processes that generate earthquakes (sudden fracturing of rock under dynamic stress) operate in a consistent manner over a wide scale. Small areas have the same kind of distribution of earthquake magnitudes and frequencies as large areas. Small earthquakes occur more frequently than large earthquakes.
It took Pastor-Satorras and Wagensburg (link) to demonstrate the mathematic connection between entropy and fractals. However, intuition can be used to understand the rationality of the connection (without equations).
Knowing something concrete about a process, having some knowledge of the likely structure of the process, the "most probable" state of the process might be inferred by applying the principle of maximum entropy. In the case of a chaotic process which demonstrates scale-invariant self-similarity we may well recognize entropy (the most probable) at work.
