Fractals and Benoit Mandelbrot: Lessons for society

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, , published by Pluto Press.
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