Michelle Obama pictured wearing Adire
A recurrent theme throughout my blog posts is the need for Africa to seriously harness science and technology for economic development and for Africans to embrace scientific thinking and develop a rational worldview. Given where we currently are this might seem like a tall order, but we should not despair. The capacity for rational thinking and scientific reasoning is one that all humans are endowed with. It should give us much encouragement to know that such capacity has been exhibited by Africans in times past (and present) and some evidence of this still exists in even some common objects around us.
Perhaps no field of scientific endeavor showcases the traditional African’s capacity for science than that of Fractal Geometry. Fractal Geometry is that branch of geometry that deals with the shape of many natural objects, as opposed to Euclidean Geometry (the one you learnt in secondary school) which deals with ideal shapes like circles, triangles, rectangles etc. that are surprisingly rare in nature (This should not be interpreted as a critique of Euclidean Geometry or that it is inferior. Both systems have their strengths and weaknesses). Think about it…how many things do you know in nature that are perfect circles, squares, triangles etc.? (Though critics would point to eggs, which are perfectly oval). Examples of things in nature that are considered fractals include algae, blood vessels, clouds, coastlines, crystals, DNA, earthquakes, heartbeat patterns, lightning bolts, mountain ranges, ocean waves, proteins, snowflakes, trees, etc. What all these things have in common is that if you look at a small part of the object, that small part looks like a small-scale version of the bigger object that it is a part of. Take the tree for example. If you look at a branch, it looks like a smaller version of the entire tree. Contrast that with a car, there isn’t any part of the car that looks like a small-scale replica of the entire car. Mathematicians refer to this repetition of patterns as self-similarity, and the change in size of the similarly shaped patterns, is referred to as scaling. Fractal analysis can be applied in a mind-numbing number of fields, from electronics engineering, to thermodynamics, biochemistry, video game design, finance, economics, fashion, medicine, neuroscience, geology, geography, archaeology etc. Fractal geometry is also intricately bound up with branch of mathematics/science known as “Chaos Theory”. Chaos Theory studies systems that are extremely sensitive to their starting conditions. Changing those starting conditions ever so slightly can lead to changes so drastic in the overall system that the system could end up being radically different from what it was in the beginning, making long term prediction of such systems’ behaviour impossible. The classic examples of chaotic systems are the weather and the stock market. This is why weather predictions are good for at best a few days ahead. Many academics in economics and finance claim that the stock market cannot even be predicted at all and that those that have seemingly made repeated killings in the market like Warren Buffet and George Soros are simply lucky monkeys. If you want to know more about this debate you can read up on the “Efficient Markets Hypothesis”. Chaotic systems make their appearance in many more areas of science and engineering. Often, in order to understand them better, scientists/engineers would often like a pictorial representation of a chaotic system. To do that, they program the mathematical equations that govern the behavior of chaotic systems into computers to generate graphics. The imagery generated is often stunningly beautiful, like this one:
As noted, many naturally occurring objects tend to be fractals. As a result, the movie and videogame industries have tapped into fractal geometry to give “special effects” a touch of realism. An example would be this scene from one of the Guardians of the Galaxy movies:
Or this videogame scene rendered by the legendary Unreal game engine:
The word,” fractal” was only coined recently, in 1975 by a French mathematician by the name of Benoit Mandelbrot. As a field of study, fractal geometry only took off in the west in the 17th century [1]. Africans, however have been doing fractal geometry long before then. Indigenous African architecture makes use of fractal design heavily. One example that probably stands out is the Great Wall of Benin (In Nigeria), destroyed by the British in 1897. It was estimated to be about 16,000km long. The only man-made artifact longer than that is the Great Wall of China, which is about 21,000km [2]. It took about 600 years for the people of Benin to complete it. Construction of the wall started out in the year 800 and wasn’t completed until around the year 1460 [3]. The walls guarded the city of Benin which, according to Portuguese visitors that visited in 1485, “was one of the most beautiful and best planned cities in the world” [4]. The design of the city and surrounding villages was based on fractal geometry. The use of fractal geometry in indigenous architecture abound across Africa. Other examples for instance, can be found in the indigenous settlements of the Kotoko people of the city of Logone-Birni in Cameroun, the Baila settlements in Southern Zimbabwe, and in the famous stone buildings of the Mandara Mountains in Cameroun [5].
Indigenes of the Sahel in Africa build windscreens to keep out strong wind and dust, utilizing fractal scaling patterns such that from bottom to the top of the windscreen, similar sections of straw of decreasing length are used to build the windscreen. The choices of the lengths of the different sections are not arbitrary. They show that the windscreen makers understand the mathematical relationship between wind speed and height (Of course, when I say this, I am not implying that they sit down and solve one equation like that, like you would in class. What I am saying is that mentally speaking, they have a decent mathematical model in their heads). Western mathematicians who have studied these windscreens have found that they closely fit what are known as power laws [6]. A power law is a mathematical relationship between two variables such that a change in one variable leads to a large change in the other variable because the change in the other variable is raised to a power (Like 2 raised to power 4, which equals 16. If 2 is changed to 3, 3 raised to power 4 becomes 81, a big jump from 16). The S-curve of technology adoption that I discussed in a previous post, is an example of a power law. The most common equations found in Wind Engineering Handbooks used by western engineers also happen to be power laws [7]. This shows that the windscreen makers in the Sahel clearly know what they are doing. Though the Sahel technique and western engineering give different answers, for instance, the value of a certain parameter derived by the Sahel technique was found to be roughly 3 times the value given by the western wind engineering handbooks, it was nonetheless judged to be a decent rough approximation by the western mathematicians studying the Sahel windscreens [8].
Another notable area where there is a preponderance of fractals is in African textiles. In Ghana, older kente patterns of the kind produced by handlooms in the village, and not the newer ones produced by automated machines that are exported, make use of a scaling technique that actually has an equivalent in western mathematics (For the mathematically inclined, the western mathematical method is called contractive affine transformation) [9]. Again, like the Sahel windscreen makers, the kente cloth makers do not apply the scaling technique by solving equations on paper or programming the equations into computers as westerners do. They use it by having a roughly right mental model in their heads and applying it to kente cloth design. Adire cloth makers from Nigeria aren’t left out. They also have a scaling algorithm (An algorithm is a mathematical procedure for solving a problem). Western mathematicians who have studied it have found it to be rather efficient [10].
Yet another application of fractal geometry which I was pleasantly surprised to discover was that African women hair plaiting styles. A lot of them are examples of what in the subject of geometry would be referred to as conformal mapping. In conformal mapping, a pattern is made to fit along the contours of a pre-existing structure…um…um…the structure in this case being the woman’s head. An example of this would be the Yoruba hairstyle called “Ipako Elede” (This translates to the nape of the neck of a pig. The pig’s bristles are known to have a similar scaling pattern). Americans call this hairstyle corn-rows and it is frequently sported by both male and female African-American hip-hop and NBA basketball stars. The mathematical structure of the Ipako Elede/Corn rows hairstyle might be better appreciated by looking at the screenshot of a piece of geometrical modeling software below:
The fractal property of self-similarity (repeating pattern) is pretty self-evident from the picture of the girl’s head. The scaling property (changing size of the repeating pattern) might not be so evident, but it is well-known among African women that with certain styles, the braid starts small and then gets progressively bigger, with each added plait. I think you can see that in at least one braid, the one that gets closest to her left ear. The repetitive nature of fractal patterns makes them a natural candidate to be artificially generated by computer programming. What you are seeing in the left side of the screenshot are Computer Generated Images (CGI) of the Ipako Elede braids. And because the braids consist of repeating plaits, all you have to do is specify the mathematical properties of the starting plait, the rules (algorithm) for generating the next plait from the preceding plait and the number of times the process is to be repeated (In computer programming this is called iteration) to be able to generate an entire braid from the starting plait. Like I intimated earlier, Ipako Elede is just one of many African hairstyles that exhibit fractal patterns. Others include Koroba, another Yoruba hairstyle and la tresse de fil, a hairstyle from Yaoundé, Cameroun. Perhaps looking at textiles and hairstyles (pun unintended) side by side would give you a better appreciation of the wide applications of fractal geometry:
It has been asked, particularly by westerners whether in African fractal design, there has been a deliberate attempt to grasp at underlying mathematical/scientific principles, indispensable for labelling anything as science or whether the designs are merely intuitive, with no attempt at understanding underlying order and simply made to have aesthetic appeal. Studies have concluded that African fractal designs occupy a large spectrum running from designs that are merely intuitive, to ones where like the windscreen example, there is a clear attempt to gain understanding of underlying scientific principles [11].
In future posts we will explore more African scientific and technological exploits. As a parting shot, I leave you with a fractal image of Africa:
BEFORE YOU GO: Please share this post with as many people as possible and please check out my book, Why Africa is not rich like America and Europe on Amazon. Thank you
References:
- Fractal Article on Wikipedia: https://en.wikipedia.org/wiki/Fractal
- Website about the Great Benin Wall: https://www.kingdomofbenin.com/the-benin-moat.html
- Ibid
- Koutonin, Mawuna. Mar 2016 ‘Benin City, the mighty medieval capital now lost without trace’ theguardian.com: https://www.theguardian.com/cities/2016/mar/18/story-of-cities-5-benin-city-edo-nigeria-mighty-medieval-capital-lost-without-trace
- Eglash, Ron. 1999 African Fractals: Modern Computing And Indigenous Design New Brunswick Rutgers University Press
- Ibid
- Ibid
- Ibid
- Ibid
- Ibid
- ibid
