The Fractal Author

February 22, 2016

Since the publication of Benoit Mandelbrot's 1982 book, The Fractal Geometry of Nature, scientists have been finding fractals nearly everywhere that they look. Two examples of natural fractals, one living and the other inorganic, are shown in the figure.

Examples of fractals in nature. Left, Romanescu broccoli, and right, copper dendrites. Romanescu broccoli is also known as Romanesque cauliflower, since it resembles that vegetable. The copper dendrites, produced by the chemical reaction between aluminum and copper chloride, are shown (in the original image) at twenty times magnification. (left image by 'Rlunaro,' and right image, by Paul's Lab, both via Wikimedia Commons.)

Some non-physical objects also exhibit a fractal structure; that is, they have the same appearance when viewed at different scales, a concept known as self-similarity. The structure of music and literature is fractal over at least an order of magnitude when we analyze the structure of the simplest elements of each. That would be the pitch of each note in a piece of music, and individual letters making up a particular text.[2]

One suggested fractal analysis for music is to consider the frequency of occurrence F of the interval i between two successive notes to be proportional to i raised to a fractal dimension, -D, as below.[2] The intervals can be calculated by assigning C=0, C#=1, D=2, etc.

F ∝ i^-D

Examining literature as just a composite of letters can be done similarly if we associate each letter by its position in the alphabetic sequence; that is, a=1, b=2, etc. One such analysis of the Shakespeare corpus found a significant difference is fractal dimension D between such works as Hamlet (D=0.4500) and Anthony and Cleopatra (D=0.5516).[2] This is a clear indication of a change in writing style of Shakespeare between the c. 1600 date of Hamlet and the c. 1607 date of Anthony and Cleopatra.

One way for actresses to get major roles. This c. 1885 photograph shows Sarah Bernhardt (1844 - 1923) as Hamlet. She reprised the Hamlet role on stage in 1899, and in a short film clip, "Le Duel d'Hamlet," depicting a scene from the play, in 1900. (United States Library of Congress image, via Wikimedia Commons.)

Such an analysis of literature with recourse to just the alphabet in which it's written is a step removed from the actual writing process, since an author uses words to write a book or play, and he or she doesn't pay much attention to the letters except to correct spelling. Physicists from the Institute of Nuclear Physics of the Polish Academy of Sciences (Cracow, Poland), have recently done a fractal analysis of literary works based on word count as the determinant of sentence length. They did an analysis of sentence length variability (SLV) of 113 famous literary works of various genres written in several languages.[4]

Many of the works, written in English, French, German, Italian, Polish, Russian and Spanish, were created by such famous authors as Balzac, Arthur Conan Doyle, Charles Dickens, Fyodor Dostoevsky, Alexandre Dumas, Umberto Eco, James Joyce, Marcel Proust, Shakespeare, Tolkien, and Virginia Woolf.[4] In order to attain good statistics, each of the selected works contained at least 5,000 sentences,[4] which was certainly no problem for Proust, whose "À la recherche du temps perdu" ("Remembrance of Things Past") is more than three thousand pages long.

One of their findings was that literature is not just fractal. Some of it is multifractal; that is, the fractal structure has its own self-similarity. One literary genre, so-called "stream of consciousness" literature, was found to be exceptionally multifractal in nature.[4] The Bible, in particular, the Old Testament, was found to have this multifractal nature, but it's never been associated with stream of consciousness.[4] In my opinion, this might be a consequence of the Bible having multiple authors, even within its individual books.

Multifractality in literature. Works traditionally belonging to the stream of consciousness genre are marked in red. In this graph, points closer to unity on the vertical axis have more of a multifractal nature. The horizontal axis specifies the Probability that a long sentence would be immediately followed by a long one, and a short one would be followed by another short sentence. A value of 0.5 on the horizontal axis expresses a lack of such a tendency. (Via the Institute of Nuclear Physics Polish Academy of Sciences. Click for larger image.)

James Joyce tops the multifractal list with Finnegan's Wake. Says Stanislaw Drozdz, a professor at the Institute of Nuclear Physics of the Polish Academy of Sciences,

"The absolute record in terms of multifractality turned out to be Finnegan's Wake by James Joyce. The results of our analysis of this text are virtually indistinguishable from ideal, purely mathematical multifractals."[4]

Fractal character of War and Peace contrasted with Finnegan's Wake. (Via the Institute of Nuclear Physics Polish Academy of Sciences.)

There are some counterexamples, since Ayn Rand's "Atlas Shrugged" and Proust's "À la recherche du temps perdu" are considered to be stream of consciousness works, but they show very little multifractal character.[4] Drozdz sees this method as possibly a better way to categorize literary genres, since such an impartial, mathematical approach is less subjective.[4]

This graph illustrates that the multifractal character of Finnegan's Wake is essentially ideal. The function, f(α) is defined in ref. 4.[4] (Via the Institute of Nuclear Physics Polish Academy of Sciences.)

References:

1. Benoit B. Mandelbrot, "The Fractal Geometry of Nature," W. H. Freeman and Company, 1982, ISBN-13: 978-0716711865 (via Amazon).
2. Ali Eftekhari, "Fractal geometry of literature: first attempt to Shakespeare's works," arXix, August 17, 2004.
3. Stanisław Drożdż, Paweł Oświ¸cimka, Andrzej Kulig, Jarosław Kwapień, Katarzyna Bazarnik, Iwona Grabska-Gradzińsk, Jan Rybicki, Marek Stanuszek, "Quantifying origin and character of long-range correlations in narrative texts," Information Sciences, vol. 331 (February 20, 2016), pp. 32-44.
4. The world's greatest literature reveals multifractals and cascades of consciousness, Henryk Niewodniczanski Institute of Nuclear Physics Polish Academy of Sciences Press Release, January 21, 2016.