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 systemsŠ. 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 Š 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 developedŠ'
The reviewer further writes about Mandelbrot:
'First, he has enriched our geometric imagination
Š with computer graphics of stunning beauty Š
Secondly, he demonstrates that fractals are good
models for an impressive variety of natural
objects Š Thirdly, he emphasizes that fractals
imply an unconventional philosophy of geometry
[contrary to the conventional] "Newtonian"
picture Š 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.
--
Toyin Falola
Department of History
The University of Texas at Austin
1 University Station
Austin, TX 78712-0220
USA
512 475 7224
512 475 7222 (fax)
http://www.toyinfalola.com/
www.utexas.edu/conferences/africa
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