Fractals and Benoit Mandelbrot: Lessons for Society
by Prof Horace Campbell (Syracuse University)

It was announced this week that Benoit Mandelbrot passed away at the age of 85. One news source called him a 'maverick' mathematician. It was Mandelbrot who introduced the word 'fractals' to the Western world to capture an aspect of mathematics that had been resisted by the Western academy because of a worldview that would not deal with an 'alien' concept of uncertainty and the infinite complexity of nature. We want to use the news of his passing to bring to the fore the importance of fractals and fractal thinking in society.

According to the report on his passing by the New York Times, 'Dr. Mandelbrot coined the term "fractal" to refer to a new class of mathematical shapes whose uneven contours could mimic the irregularities found in nature.' In the era of quantum mechanics, complexity and chaos, the ideas behind fractal thinking could no longer be ignored and grudgingly, fractal geometry began to gain acceptance in the Western academy. We want to salute Mandelbrot for his tenacity in bringing the concept of fractals to the Western academy. While we commend Mandelbrot for his doggedness, we use this opportunity to state that before Mandelbrot coined the term 'fractal' and popularised it in the Western academy, the knowledge and application of this geometry of nature had always existed in the thinking of African peoples.

Fractal geometry was at the heart of the African ontology and knowledge system, from divination and architecture to hair weave and craft. More than 40 years ago, Claudia Zaslavsky exposed to the West her research on the African mathematical heritage. Her book, 'Africa Counts: Number and Pattern in African Culture' was a major contribution to the understanding of mathematics in everyday life in Africa. This analysis was carried to another level by Ron Eglash at the end of the 20th century.

In his research presented in the book 'African Fractals: Modern Computing and Indigenous Design', Ron Eglash was exposed to the fact that the knowledge and application of fractal had been alive for millennia in Africa. There are invaluable lessons to be learned for humanity by exploring further the heap of ideas surrounding fractals. Particularly, African societies, the African academy and the political leadership in Africa must pay close attention to exploring the transformational and revolutionary ideas embedded in fractals.

IMPRESSIVE CONTRIBUTION OF BENOIT MANDELBROT

There is no doubt about the tremendous contribution of Mandelbrot to the fields of mathematics and science. Almost every discipline in the Western academy has been affected by fractal geometry. For decades, Benoit Mandelbrot was at the forefront of explaining and writing about fractals. 'If you cut one of the florets of a cauliflower, you see the whole cauliflower but smaller. Then you cut again, again, again, and you still get small cauliflowers. So there are some shapes which have this peculiar property, where each part is like the whole, but smaller,' explained Mandelbrot. He argued that seemingly random mathematical shapes followed a pattern if broken down into a single repeating shape. The concepts of self-similarity and scaling in fractals enabled scientists to measure previously immeasurable objects, including the coastline of the British Isles and the geometry of a lung or a cauliflower. We now know that the seminal contribution of fractal mathematics led to technological breakthroughs in the fields of digital music and image compression. Computer modelling and the information technology revolution have been pushed by insights from fractal geometry. In his interviews and books, Mandelbrot argued that seemingly random mathematical shapes followed a pattern if broken down into a single repeating shape. This is what in fractals is called self-similarity. This concept of self-similarity is also linked to the other key elements of fractal concepts: scaling, recursion and infinity.

In fractals, this concept of infinity is also known as the Cantor Set. In the late 19th century, George Cantor (1845–1918) had provided a new approach for European mathematicians when he showed that it was possible to 'keep track of the number of elements in an infinite set', and did so in a descriptively simple fashion. Starting with a single straight line, Cantor erased the middle third, leaving two lines. He then carried out the same operation on those two lines, erasing their middles and leaving four lines. In other words he used a sort of feedback look, with end result of one stage brought back as the starting point for the next. The technique is called 'recursion' (Eglash, p. 8). This concept of infinity had for long, before Cantor, been part of the African divination system. In Africa, Eglash encountered some of the most complex fractal systems that exist in religious activities, such as the sequence of symbols used in sand divination, a method of fortune telling found in Senegal. The concept of infinity had a metaphysical link with infinity. This sand divination was to be later referred to as 'geomancy' in Europe (Eglash, p. 99–101). Eglash and others credited Mandelbrot with the conceptual leap in the application of fractal geometry from the simulations of natural objects.

The relevant point is that fractals existed in nature and before Mandelbrot there was Koch and Cantor. Before Koch and Cantor there were many people in Africa who understood fractal geometry and the explicit and implicit mathematical idea that was to be found in everyday life in Africa.

AFRICAN FRACTALS

It has been established that before Mandelbrot exposed the Western world to the application of fractals, these forms of knowledge had always existed in the ontology and creativity of Africans. The ideas about the infinite nature of the universe that are now central to particle physics were manifest in many African communities with the celebrated case of the Dogon people, which is the most widely known. Other aspects of advanced geometry and physics were present in the numeric systems of many societies, especially in relation to the Lusona drawings of the Chokwe people. When the colonial missionaries could not decipher the complex mathematics behind the Lusona they deemed the Chokwe to be the most backward and uncivilised in Africa. It is now known that the Dogon and Chokwe reflected a deep understanding of the mathematics of nature. African village settlements show self-similar characteristics, circle of circles, circular dwellings and streets in which broad avenues branch down to tiny footpaths with striking geometric repetition, distinguishable from the Euclidian layout. Ron Eglash presented his research findings in his book 'African Fractals' to show that African fractals emanated from a conscious knowledge system and not from unconscious activity.

It was during an aerial exploration of rural parts of Africa that Eglash grasped the central aspect of the architectural designs in terms of self-similarity and scaling of patterns. In his book he said clearly that, 'While fractal geometry can take us into the far reaches of high tech science, its patterns are common in traditional African designs and the concepts are fundamental to African knowledge system.'

Eglash's findings also include the use of sophisticated mathematical ideas in everyday objects. In the arid region of the Sahel, for example, artisans produce windscreens by utilising a scaling design that gives them the maximum effect – keeping out the wind-driven dust – for the minimum amount of effort and material. Abdul Karim Bangura, another scholar of African science and mathematics, in his review of Eglash's text noted that:

'Aerial photographs of various settlement compounds revealed that many were composed of circular structures enclosed in other circles, or rectangles within rectangles, and that the compounds were likely to have street patterns in which broad avenues branched into very small footpaths. As Eglash notes, at first he thought it was just from unconscious social dynamics. But during his fieldwork, he found that fractal designs also appear in a wide variety of intentional designs--carving, hairstyling, metalwork, painting, textiles--and the recursive process of fractal algorithms are even employed in African quantitative systems…. These results, Eglash concludes, are congruent with recent developments in complex systems theory, which suggest that pre-modern, non-state societies were neither utterly anarchic, nor frozen in static order, but rather utilized an adaptive flexibility that capitalized on the non-linear aspects of ecological dynamics.'

Since the writing of this review, Ron Eglash has not only written extensively on African Fractals but his widely watched presentation at the TED conference has brought the ideas of Fractals to an international audience.

When Eglash returned from Africa, one of his colleagues advised him to focus on scaling patterns in African hairstyles. In the conclusion on scaling, Eglash himself admitted: 'While it is not difficult to invent explanations based on unconscious social forces – for example flexibility in conforming designs to material surfaces as expressions of social flexibility – I do not think that any such explanations can account for this diversity. From optimisation engineering, to modelling organic life, to mapping between different spatial structures, African artisans have developed a wide range of tools, techniques and design practices based on the conscious application of fractal geometry' (p. 85).

Scaling and self-similarity are descriptive characteristics; one can see these in African designs. The idea is to grasp how these were intentionally designed so that we can have a better grasp of African fractals. Eglash then went on to look closely at African architecture, designs, art and village structure, cosmology and divination systems and sought to understand how all of these are linked to an African knowledge system. I have elsewhere used the term the African ideation system or worldview. The question for us is to understand how this is linked to political relations in Africa.

Of the five main elements of Fractals that were highlighted in his book – scaling, self-similarity, recursion, infinity and fractal dimensions – Eglash drew attention to the recursive processes that generate a feedback loop. Eglash gave three examples of recursion, namely, cascade, iteration and self-reference.

I was introduced to fractals and African mathematics by Sam E. Anderson, and I met Eglash in 1999 to engage him on this concept of African fractals. Ever since my meeting with Eglash, I have seen the revolutionary implications of fractal thinking and a fractal worldview. I have sought to further the understanding of the relationship between fractal optimism and politics in my book, 'Barack Obama and Twenty First Century Politics'. In this book, I sought to underline the importance of self-organisation and self-mobilisation as the basis for a new bottom-up politics that could unleash a new form of participatory democracy for the 21st century (based on the intentional activities of conscious humans). Fractal has been applied in many other fields. In the application of fractal to political science, elements such as recursion, cascading, self-similarity and memory help us understand the self-replication of genocidal violence, exploitation, militarism, masculinity and environmental plunder, among others. Thus, it becomes imperative for there to be a coordinated human intention to make a break with such traditions (negative recursion) and to establish a different legacy that would form a positive recursive loop for the transformation of society for posterity.

SELF-SIMILARITY, RECURSION AND SOCIETY

One lesson of fractal for African (and other) societies is the conceptual application of the ideas of self-similarity and self-referencing in recursion, and the imperative that this mode of thinking breaks the certainty and predictability of determinism. Determinism, simplicity and reductionism had migrated from the physical sciences to implant the artificial divisions in the academic disciplines that became the hallmark of the social sciences in the Western world. F. Kapra had warned against this certainty of Western thinking. In the book, 'The Turning Point', he argued:

'For two and a half centuries physicists have used a mechanistic view of the world to develop and refine the conceptual framework known as classical physics. They have based their ideas on the mathematical theory of Isaac Newton, the philosophy of René Descartes, and the scientific methodology advocated by Francis Bacon … Like human-made machines, the cosmic machine was thought to consist of elementary parts. Consequently it was believed that complex phenomena could always be understood by reducing them to their basic building blocks and by looking for the mechanisms through which these interacted. This attitude, known as reductionism, has become so deeply ingrained in our culture that it has often been identified with the scientific method.'

Humans now know that this reductionism of the 'scientific method' emanated from a European reading of science and human knowledge. With the advances in digital technology and genetic engineering, advances made possible by the application of fractal geometry, the promise of the future demands that humans have a deep appreciation of the inter-relationship between humans and nature so that we do not become slaves to technology. This demands from us the obligation to intervene as humans to reverse the headlong rush towards dehumanisation and the destruction of the planet earth. Fractal thinking and the understanding of the consequences of the reference points for progress demonstrates the necessity to make a break with the recursion of negative self-similar patterns such as conflicts and wars, domination, exploitation, militarization and religious and ethnic tensions. We can see that we are in a feedback loop of economic crisis, intensified exploitation, stock-market failures and conflicts. This kind of recursive process has a definite reference point which is the history of capitalism, racism, domination, oppression, greed and plunder. It is in examining the connection between the two (recursion and cultural categories) that the use of fractal geometry as a knowledge system (and not just unconscious social dynamics) becomes evident.

The next lesson of African fractals is for African educational institutions. African education must support research agendas that seek to unearth the richness of Africa and focus on positive aspects of the African knowledge system as an indispensable site of knowledge. The road to the re-establishment and reaffirmation of Africa as a site of knowledge has never been smooth, and may get rougher unless our scholarly tradition refrains from following a recursive path that is self-similar to that which attempted to deny and subjugate our intelligence and ontology. Three years ago, Paulus Gerdes and Ahmed Djebbar produced the important bibliography on 'Mathematics in African History and Culture'. This bibliography carried forward the traditions of Cheikh Anta Diop, who did so much to unearth and highlight the contributions of African mathematics to research and learning.

Diop studied in France at the same time of Mandelbrot. Diop moved to Paris in 1946 and studied nuclear physics and Egyptology. He submitted his thesis to the University of Paris in 1951, but could not find a committee to examine his work on the Egyptian contribution to math and science. It was after nine years that he was granted his doctorate by the University of Paris in 1960. It was not by chance that Diop was a physicist who had studied relativity and quantum physics. It was this study that brought Diop back to an awareness of the richness of African knowledge and intellectual traditions and although he did not use the term fractals, his research and work shared many points of convergence with Benoit Mandlebrot.

POPULARISING FRACTALS IN THE WEST

Just as how it was difficult for the ideas of Diop to be accredited in the French academy, so Mandelbrot's popularisation of the idea of fractals in the West was not an easy task. Mandelbrot attended school in France at the same period when the African scientist Cheikh Anta Diop was also studying in Paris. Between 1949–52, Mandelbrot wrote his Docteur d'Etat ès Sciences Mathématiques: Faculté des Sciences, Paris.

After receiving his doctorate in 1962 from France, Mandelbrot moved to the United States, where he pursued postdoctoral work. Mandelbrot followed a tortuous career between industry and the academy because of his view on complexity and infinity. It was not until he was nearly 75 years old that he was granted tenure in the mathematics department at Yale in 1999. His book, 'The Fractal Geometry of Nature', was first published in 1982.

Writing in the popular magazine the New Scientist, one reviewer said of the book:

'Fractal geometry is one of those concepts which at first sight invites disbelief but on second thought becomes so natural that one wonders why it has only recently been developed…'

The reviewer further writes about Mandelbrot: 'First, he has enriched our geometric imagination … with computer graphics of stunning beauty … Secondly, he demonstrates that fractals are good models for an impressive variety of natural objects … Thirdly, he emphasizes that fractals imply an unconventional philosophy of geometry [contrary to the conventional] "Newtonian" picture … Mandelbrot's essay is written in a personal, intense and immediate style.'

Mandelbrot wrote the book, 'The (Mis)behavior of Markets: A Fractal View of Risk, Ruin, and Reward'. In this book, Mandelbrot warned that markets are far riskier than society wanted to believe. From the gyrations of IBM's stock price and the Dow, to cotton trading and the dollar–euro exchange rate, Mandelbrot showed that the world of finance can be understood for its volatility. Contrary to the advice of stockbrokers, there was nothing certain about the future and stability of the stock market. The ideas of fractals were further popularised and published in Scientific American in 1999 under the title, 'A Multifractal walk down Wall Street.' In this article Mandelbrot argued that:

'Fractal patterns appear not just in the price changes of securities but in the distribution of galaxies throughout the cosmos, in the shape of coastlines and in the decorative designs generated by innumerable computer programs.

'In finance, this concept is not a rootless abstraction but a theoretical reformulation of a down-to-earth bit of market folklore – namely, that movements of a stock or currency all look alike when a market chart is enlarged or reduced so that is fits the same time and price scale. An observer then cannot tell which of the data concern prices that change from week to week, day to day or hour to hour. This quality defines the charts as fractal curves and makes available many powerful tools of mathematical and computer analysis.'

Despite the warnings about the fact that there was uncertainty in this branch of finance, a brand new group of financial wizards attempted to bring back the linearity and certainty of capitalist development and growth to predict the unlimited rise of the stock market. These wizards were to be called 'quants' on Wall Street, and they populated the area of speculation called the market for derivatives. Warren Buffet had called these derivatives 'financial weapons of mass destruction'. The world was brought face to face with the complexity and chaos of this branch of finance in 2008, yet the mindset of certainty and unlimited potential of capitalism has meant that the gurus of the world of quants have returned to the mythical world of unlimited profits.

In the New York Times report on the passing of Mandelbrot we are reminded by Mandelbrot himself that life is not linear and not based on a straight line:

'Dr. Mandelbrot compared his own trajectory to the rough outlines of clouds and coastlines that drew him into the study of fractals in the 1950s.

'"If you take the beginning and the end, I have had a conventional career," he said, referring to his prestigious appointments in Paris and at Yale. "But it was not a straight line between the beginning and the end. It was a very crooked line."'

The important point was that human intentions become an important aspect of human interactions with nature and it is this intentionality that existed in Africa that was brought out in the book 'African Fractals' by Eglash. The study of fractals illustrates the importance of the human intention to make a break when the recursive processes lead to militarism, destruction and greed.

While the quants have applied fractal geometry to the modelling for the derivatives market, it is only the conscious actions by citizens that can make a break from these financial weapons of mass destruction. This break with negative recursion and the establishment of a positive recursive loop is applicable to our education system, our leadership orientation, our engagement with the environment and in our relations as humans.

In this bid, we propose that there must be human intentions to make Ubuntu – shared humanity and respect for the environment –the reference point that would self-replicate and cascade itself across all sections of society. 

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17 October 2010

Could a rusty coin re-write Chinese-African history?
By Peter Greste BBC East Africa correspondent, Mambrui, Kenya
 See video short at: www.bbc.co.uk/news/world-africa-11531398#story_continues_2

It is not much to look at - a small pitted brass coin with a square hole in the centre - but this relatively innocuous piece of metal is revolutionising our understanding of early East African history, and recasting China's more contemporary role in the region.
  
"This is a wonderful and very important piece”
Professor Qin Dashu Peking University

A joint team of Kenyan and Chinese archaeologists found the 15th Century Chinese coin in Mambrui - a tiny, nondescript village just north of Malindi on Kenya's north coast.

In barely distinguishable relief, the team leader Professor Qin Dashu from Peking University's archaeology department, read out the inscription: "Yongle Tongbao" - the name of the reign that minted the coin some time between 1403 and 1424.

"These coins were carried only by envoys of the emperor, Chengzu," Prof Qin said.

"We know that smugglers would often take them and melt them down to make other brass implements, but it is more likely that this came here with someone who gave it as a gift from the emperor."

And that poses the question that has excited both historians and politicians: How did a coin from the early 1400s get to East Africa, almost 100 years before the first Europeans reached the region?

When China ruled the seas
The answer seems to be with Zheng He, also known as Cheng Ho - a legendary Chinese admiral who, the stories say, led a vast fleet of between 200 and 300 ships across the Indian Ocean in 1418.

It is now believed that China's Zheng He reached East Africa long before any European explorer.

Until recently, there have only been folk tales and insubstantial hints at how far Zheng He might have sailed.

Then, a few years ago, fishermen off the northern Kenyan port town of Lamu hauled up 15th Century Chinese vases in their nets, and the Chinese authorities ran DNA tests on a number of villagers who claimed Chinese ancestry.

The tests seemed to confirm what the villagers have always believed - that a ship from Zheng He's fleet sank in a storm and the surviving crew married locals, meaning some people in the area still have subtly Chinese features.

Searching for clues
It was then that Peking University organized its expedition to try to find conclusive evidence. The university is spending $3 million (£2 million) on the three-year project.

The dig suggests China's interest in Africa goes back a long way

Prof Qin's team chose to dig in Mambrui for two reasons.
First, ancient texts told of Zheng He's visit to the Sultan of Malindi - the most powerful coastal ruler of the time. But they also mentioned that Malindi was by a river mouth; something that the present town of Malindi doesn't have, but that Mambrui does.

"The Chinese had a very different approach from the Europeans to East Africa...they saw us as equals”
Dr Herman Kiriama National Museums of Kenya
The old cemetery in Mambrui also has a famous circular tomb-stone embedded with 400-year-old Chinese porcelain bowls hinting at the region's long-standing relationship with the East.

In the broad L-shaped trench that the team dug on the edge of the cemetery, they began finding what they were looking for.

First, they uncovered the remains of an iron smelter and iron slag.

Then, Mohamed Mchuria, a coastal archaeologist from the National Museums of Kenya, unearthed a stunning fragment of porcelain that Prof Qin believes came from a famous kiln called Long Quan that made porcelain exclusively for the royal family in the early Ming Dynasty.

The jade-green shard appears to be from the base of a much larger bowl, with two small fish in relief, swimming just below the surface of the glaze.

"This is a wonderful and very important piece, and that is why we believe it could have come with an imperial envoy like Zheng He," Prof Qin said.

The team is hoping to unearth more buried secrets

Re-writing history?
While the evidence is still not conclusive, it undermines Portuguese explorer Vasco Da Gama's claim to have been the first international trader to open up East Africa.
He arrived in 1499 on an expedition to find a sea route to Asia, and launched more than 450 years of colonial domination by European maritime powers.

"It... gives politicians a reason to say: 'Let's look East' because we've been looking that way throughout the ages."
Herman Kiriama National Museums of Kenya

"We're discovering that the Chinese had a very different approach from the Europeans to East Africa," said Herman Kiriama, the lead archaeologist from the National Museums of Kenya.

"Because they came with gifts from the emperor, it shows they saw us as equals. It shows that Kenya was already a dynamic trading power with strong links to the outside world long before the Portuguese arrived," he said.

And that is profoundly influencing the way Kenya is thinking about its current ties to the East.

It implies that China has a much older trade relationship with the region than Europe, and that Beijing's very modern drive to open up trade with Africa may in fact be part of a far deeper tradition than anyone suspected.

In 2008 China's trade with the continent was worth $107bn (£67bn) - more even than the United States, and 10 times what it was in 2000.

"A long time ago, the East African coast looked East and not West," said Mr Kiriama.

"And maybe that's why it also gives politicians a reason to say: 'Let's look East' because we've been looking that way throughout the ages."