Fractals, Chaos, and Complexity: The Mathematics of African Culture

In the Western world, mathematics is praised as one of the pinnacles of our cultural achievements. We see mathematics as a source of universal truth and even beauty. Our veneration of mathematics has influenced our culture so strongly that we assume we have a monopoly on mathematical culture. Although knowledge of simple arithmetic is understandably common in all cultures, we tend to assume that modern industrialized society is the only culture that has its  foundation in a profound understanding of mathematics. This is, however, not the case for many of the indigenous tribes of Africa. In fact, African culture has a strong tradition of mathematical thought which is entirely distinct from its European counterpart, and there is evidence that Africans were well-aware of certain mathematical concepts centuries before they were discovered in Europe. Nearly all elements of African culture, from its architecture to its art and religious philosophy, are influenced by the mathematics of fractals, complexity, and chaos.

Traditional African art is full of examples of fractal geometry. Fractals are shapes that have the property of self-similarity, meaning that they look the same on many scales. This property is clearly visible in many African paintings, sculpture, clothing, pottery, and jewelry. The abundance and complexity of fractal patterns in African art shows that they are used deliberately, they are not made unwittingly for aesthetic appeal. Fractals can also be found in African music, which contains complex syncopated rhythms that are self-similar on multiple scales. Even African hair styles, which consist of a variety of braiding patterns, are often fractal. A simple rule is used to produce each braiding pattern, allowing the patterns to scale for any size head. The presence of fractal geometry in African art indicates a strong cultural influence from mathematics.

The practical applications of fractal geometry were also known to the indigenous Africans. Fractal patterns can be found in African architecture, from the design of individual buildings to the layout of entire villages. In places like Mali and Niger, the indigenous people build huts with straw walls using a fractal scaling technique to maximize efficiency. When constructing the walls, they use long pieces of straw at the bottom and progressively shorter ones towards the top. This technique maximizes the tradeoff between the faster construction time of the longer pieces and the better wind protection created by the longer pieces. An analysis of the scaling technique revealed that scaling factor of the straw pieces was almost exactly proportional to the increase in wind speed at different heights. This kind of practical application of fractals also exists on larger scales in African villages, including the layout of the village itself. It can be seen, even in modern urban areas, that development occurs in fractal patterns on a large scale without any intention or top-down control. The fractal design of African settlements, however, is often fractal by design. This is often the result of repeated expansions on a progressively larger scale, which function as the iterations of the fractal pattern. The scaling properties of fractals allow the design of the settlements to be scaled up as the settlements grow in size. While Europeans were discovering the applications of Euclidean geometry, fractal geometry was already widely used by indigenous Africans.

Perhaps the most interesting example of mathematics in African culture is the use of deterministic chaos in religious practices. Deterministic chaos describes a system that is unpredictable despite being governed by precise rules. Weather systems, stock markets, and even brains are all examples of such systems. While European culture sought to reduce the complexity of these systems to simple rules, African culture embraced deterministic chaos as the ultimate explanation for the universe. This can be seen in religious practices such as the Ifa divination beads and the Bamana sand divination. These practices consist of using a set of rules recursively to produce an unpredictable sequence of codes and symbols. These symbols could then be interpreted as prophecies, similar to the predictions of a Tarot card reader. In addition to being a simple and elegant method of producing religious predictions, these practices are in contrast with Western monotheistic tradition. Christianity, Islam, and Judaism all provide a deterministic and predictable view of the universe, which has a clear beginning and end as defined by their respective holy books. This struggle for certainty distinguishes European culture from African culture. However, African culture also differs from other indigenous cultures in this area, such as the Native Americans. Religious myths in Native American culture, like in African culture, emphasize the unpredictability of the universe. However, their stories differ from the Africans' because their uncertainty is due to true probabilistic randomness, not deterministic chaos (this explains why gambling is prevalent in Native American culture). Deterministic chaos can even be found in African games such as Mancala, in which the effects of players' choices become unpredictable after simple rules are applied repeatedly. The combination of determinism and unpredictability is a distinguishing characteristic of African culture, and is one of the most profound examples of mathematical influence on African culture.

Understanding the influence of mathematics on African culture (and indigenous cultures in general), will provide insight into how the basic world view and philosophy of Africans differ from those of Westerners. They see the world not as a single entity with a single fate, but as a collection of interacting parts whose future is unpredictable. Their view of society is not one of a homogenous block controlled by a single leader, but of a complex system with organization on multiple scales. This difference in world view may explain the catastrophic consequences of colonization, the often violent meeting of Western and African cultures. The bottom-up organization of tribal African peoples was forcefully replaced by a top-down division of the continent into European-style states, whose borders often crossed ethnic boundaries. The lesson to be learned from these insights into African culture is that colonization was not the conquest of science and reason over primitiveness and nature, but rather the conquest of European mathematics over African mathematics (loosely speaking, ignoring the influence of India and the Middle East on mathematics). It is not clear to what extent mathematical knowledge enabled the domination of Western culture over so much of the world, but in any case, the achievements of African culture in mathematics should not be overlooked.