The world's greatest literature reveals multifractals and cascades of consciousness

Sequences of sentence lengths (as measured by number of words) in four literary works representative of various degree of cascading character.

Credit: Source: IFJ PAN

 

 

James Joyce, Julio Cortazar, Marcel Proust, Henryk Sienkiewicz and Umberto Eco. Regardless of the language they were working in, some of the world's greatest writers appear to be, in some respects, constructing fractals. Statistical analysis carried out at the Institute of Nuclear Physics of the Polish Academy of Sciences, however, revealed something even more intriguing. The composition of works from within a particular genre was characterized by the exceptional dynamics of a cascading (avalanche) narrative structure. This type of narrative turns out to be multifractal. That is, fractals of fractals are created.

 

 

As far as many bookworms are concerned, advanced equations and graphs are the last things which would hold their interest, but there's no escape from the math. Physicists from the Institute of Nuclear Physics of the Polish Academy of Sciences (IFJ PAN) in Cracow, Poland, performed a detailed statistical analysis of more than one hundred famous works of world literature, written in several languages and representing various literary genres. The books, tested for revealing correlations in variations of sentence length, proved to be governed by the dynamics of a cascade. This means that the construction of these books is in fact a fractal. In the case of several works their mathematical complexity proved to be exceptional, comparable to the structure of complex mathematical objects considered to be multifractal. Interestingly, in the analyzed pool of all the works, one genre turned out to be exceptionally multifractal in nature.
Fractals are self-similar mathematical objects: when we begin to expand one fragment or another, what eventually emerges is a structure that resembles the original object. Typical fractals, especially those widely known as the Sierpinski triangle and the Mandelbrot set, are monofractals, meaning that the pace of enlargement in any place of a fractal is the same, linear: if they at some point were rescaled x number of times to reveal a structure similar to the original, the same increase in another place would also reveal a similar structure.
Multifractals are more highly advanced mathematical structures: fractals of fractals. They arise from fractals 'interwoven' with each other in an appropriate manner and in appropriate proportions. Multifractals are not simply the sum of fractals and cannot be divided to return back to their original components, because the way they weave is fractal in nature. The result is that in order to see a structure similar to the original, different portions of a multifractal need to expand at different rates. A multifractal is therefore non-linear in nature.
"Analyses on multiple scales, carried out using fractals, allow us to neatly grasp information on correlations among data at various levels of complexity of tested systems. As a result, they point to the hierarchical organization of phenomena and structures found in nature. So we can expect natural language, which represents a major evolutionary leap of the natural world, to show such correlations as well. Their existence in literary works, however, had not yet been convincingly documented. Meanwhile, it turned out that when you look at these works from the proper perspective, these correlations appear to be not only common, but in some works they take on a particularly sophisticated mathematical complexity," says Prof. Stanislaw Drozdz (IFJ PAN, Cracow University of Technology).
The study involved 113 literary works written in English, French, German, Italian, Polish, Russian and Spanish by such famous figures as Honore de Balzac, Arthur Conan Doyle, Julio Cortazar, Charles Dickens, Fyodor Dostoevsky, Alexandre Dumas, Umberto Eco, George Elliot, Victor Hugo, James Joyce, Thomas Mann, Marcel Proust, Wladyslaw Reymont, William Shakespeare, Henryk Sienkiewicz, JRR Tolkien, Leo Tolstoy and Virginia Woolf, among others. The selected works were no less than 5,000 sentences long, in order to ensure statistical reliability.
To convert the texts to numerical sequences, sentence length was measured by the number of words (an alternative method of counting characters in the sentence turned out to have no major impact on the conclusions). The dependences were then searched for in the data -- beginning with the simplest, i.e. linear. This is the posited question: if a sentence of a given length is x times longer than the sentences of different lengths, is the same aspect ratio preserved when looking at sentences respectively longer or shorter?
"All of the examined works showed self-similarity in terms of organization of the lengths of sentences. Some were more expressive -- here The Ambassadors by Henry James stood out -- while others to far less of an extreme, as in the case of the French seventeenth-century romance Artamene ou le Grand Cyrus. However, correlations were evident, and therefore these texts were the construction of a fractal," comments Dr. Pawel Oswiecimka (IFJ PAN), who also noted that fractality of a literary text will in practice never be as perfect as in the world of mathematics. It is possible to magnify mathematical fractals up to infinity, while the number of sentences in each book is finite, and at a certain stage of scaling there will always be a cut-off in the form of the end of the dataset.
Things took a particularly interesting turn when physicists from the IFJ PAN began tracking non-linear dependence, which in most of the studied works was present to a slight or moderate degree. However, more than a dozen works revealed a very clear multifractal structure, and almost all of these proved to be representative of one genre, that of stream of consciousness. The only exception was the Bible, specifically the Old Testament, which has so far never been associated with this literary genre.
"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," says Prof. Drozdz.
The most multifractal works also included A Heartbreaking Work of Staggering Genius by Dave Eggers, Rayuela by Julio Cortazar, The US Trilogy by John Dos Passos, The Waves by Virginia Woolf, 2666 by Roberto Bolano, and Joyce's Ulysses. At the same time a lot of works usually regarded as stream of consciousness turned out to show little correlation to multifractality, as it was hardly noticeable in books such as Atlas Shrugged by Ayn Rand and A la recherche du temps perdu by Marcel Proust.
"It is not entirely clear whether stream of consciousness writing actually reveals the deeper qualities of our consciousness, or rather the imagination of the writers. It is hardly surprising that ascribing a work to a particular genre is, for whatever reason, sometimes subjective. We see, moreover, the possibility of an interesting application of our methodology: it may someday help in a more objective assignment of books to one genre or another," notes Prof. Drozdz.
Multifractal analyses of literary texts carried out by the IFJ PAN have been published in Information Sciences, a journal of computer science. The publication has undergone rigorous verification: given the interdisciplinary nature of the subject, editors immediately appointed up to six reviewers.

Source:
The Henryk Niewodniczanski Institute of Nuclear Physics Polish Academy of Sciences. "The world's greatest literature reveals multifractals and cascades of consciousness." ScienceDaily. ScienceDaily, 21 January 2016. <www.sciencedaily.com/releases/2016/01/160121110913.htm>.

How many ways can you arrange 128 tennis balls? Researchers solve an apparently impossible problem

Tennis balls.Credit: © Hayati Kayhan / Fotolia

 

 

Researchers have solved an apparently overwhelming physics problem involving some truly huge numbers. In summary, the problem asks you to imagine that you have 128 tennis balls, and can arrange them in any way you like. The challenge is to work out how many arrangements are possible and -- according to the research -- the answer is about 10^250, also known as ten unquadragintilliard: a number so big that it exceeds the total number of particles in the universe.

Despite its complexity, this study also provides a working example of how "configurational entropy" might be calculated in granular physics. This basically means the issue of measuring how disordered the particles within a system or structure are. The research provides a model for the sort of maths that would be needed to solve bigger problems still, ranging from predicting avalanches, to creating efficient artificial intelligence systems.
A bewildering physics problem has apparently been solved by researchers, in a study which provides a mathematical basis for understanding issues ranging from predicting the formation of deserts, to making artificial intelligence more efficient.
In research carried out at the University of Cambridge, a team developed a computer program that can answer this mind-bending puzzle: Imagine that you have 128 soft spheres, a bit like tennis balls. You can pack them together in any number of ways. How many different arrangements are possible?
The answer, it turns out, is something like 10250 (1 followed by 250 zeros). The number, also referred to as ten unquadragintilliard, is so huge that it vastly exceeds the total number of particles in the universe.
Far more important than the solution, however, is the fact that the researchers were able to answer the question at all. The method that they came up with can help scientists to calculate something called configurational entropy -- a term used to describe how structurally disordered the particles in a physical system are.
Being able to calculate configurational entropy would, in theory, eventually enable us to answer a host of seemingly impossible problems -- such as predicting the movement of avalanches, or anticipating how the shifting sand dunes in a desert will reshape themselves over time.
These questions belong to a field called granular physics, which deals with the behaviour of materials such as snow, soil or sand. Different versions of the same problem, however, exist in numerous other fields, such as string theory, cosmology, machine learning, and various branches of mathematics. The research shows how questions across all of those disciplines might one day be addressed.
Stefano Martiniani, a Benefactor Scholar at St John's College, University of Cambridge, who carried out the study with colleagues in the Department of Chemistry, explained: "The problem is completely general. Granular materials themselves are the second most processed kind of material in the world after water and even the shape of the surface of the Earth is defined by how they behave."
"Obviously being able to predict how avalanches move or deserts may change is a long, long way off, but one day we would like to be able to solve such problems. This research performs the sort of calculation we would need in order to be able to do that."
At the heart of these problems is the idea of entropy -- a term which describes how disordered the particles in a system are. In physics, a "system" refers to any collection of particles that we want to study, so for example it could mean all the water in a lake, or all the water molecules in a single ice cube.
When a system changes, for example because of a shift in temperature, the arrangement of these particles also changes. For example, if an ice cube is heated until it becomes a pool of water, its molecules become more disordered. Therefore, the ice cube, which has a tighter structure, is said to have lower entropy than the more disordered pool of water.
At a molecular level, where everything is constantly vibrating, it is often possible to observe and measure this quite clearly. In fact, many molecular processes involve a spontaneous increase in entropy until they reach a steady equilibrium.
In granular physics, however, which tends to involve materials large enough to be seen with the naked eye, change does not happen in the same way. A sand dune in the desert will not spontaneously change the arrangement of its particles (the grains of sand). It needs an external factor, like the wind, for this to happen.
This means that while we can predict what will happen in many molecular processes, we cannot easily make equivalent predictions about how systems will behave in granular physics. Doing so would require us to be able to measure changes in the structural disorder of all of the particles in a system -- its configurational entropy.
To do that, however, scientists need to know how many different ways a system can be structured in the first place. The calculations involved in this are so complicated that they have been dismissed as hopeless for any system involving more than about 20 particles. Yet the Cambridge study defied this by carrying out exactly this type of calculation for a system, modelled on a computer, in which the particles were 128 soft spheres, like tennis balls.
"The brute force way of doing this would be to keep changing the system and recording the configurations," Martiniani said. "Unfortunately, it would take many lifetimes before you could record it all. Also, you couldn't store the configurations, because there isn't enough matter in the universe with which to do it."
Instead, the researchers created a solution which involved taking a small sample of all possible configurations and working out the probability of them occurring, or the number of arrangements that would lead to those particular configurations appearing.
Based on these samples, it was possible to extrapolate not only in how many ways the entire system could therefore be arranged, but also how ordered one state was compared with the next -- in other words, its overall configurational entropy.
Martiniani added that the team's problem-solving technique could be used to address all sorts of problems in physics and maths. He himself is, for example, currently carrying out research into machine learning, where one of the problems is knowing how many different ways a system can be wired to process information efficiently.
"Because our indirect approach relies on the observation of a small sample of all possible configurations, the answers it finds are only ever approximate, but the estimate is a very good one," he said. "By answering the problem we are opening up uncharted territory. This methodology could be used anywhere that people are trying to work out how many possible solutions to a problem you can find."
The paper, Turning intractable counting into sampling: computing the configurational entropy of three-dimensional jammed packings, is published in the journal, Physical Review E.
Stefano Martiniani is a St John's Benefactor Scholar and Gates Scholar at the University of Cambridge.


Source:
University of Cambridge. "How many ways can you arrange 128 tennis balls? Researchers solve an apparently impossible problem." ScienceDaily. ScienceDaily, 27 January 2016. <www.sciencedaily.com/releases/2016/01/160127053413.htm>.