Applied Mathematics in Pre-Colonial African Settlements
Recursion, infinity, Euclidean geometry, and Pre-colonial African Settlements!!!
I KNOW!!! Picking up the odd one out from the list above probably took you less than a fraction of a second. After all, even the science supports it. The perceiving system of a human being is very impressive. From about 3 months old, we start to develop a new law of pattern discrimination. Even the 1960s Machine Learning trained to automatically detect meaningful patterns could have realized that the Old man made a very serious typo in his opening line.
Euclidean geometry! That was your answer, right? Most likely not. Yes, the list above can be made into a very large number of different patterns, but let’s face it, 99% of the times, the term “Pre-colonial African Settlements” will come up as the odd one out, the last one picked, the black sheep.
That can be quite misleading though. At first glance, African architecture, or African designs in general, might seem so varied that one would conclude its structures have nothing in common. Although there is a great diversity among the many cultures of Africa, examples of advanced applied Mathematics can be found in every corner of the African continent.
So, about the buzzword, Fractal Geometry (all the coolkids are saying it these days :D), “what does it even mean?”. A fractal is a “drawing which also has self-similar structure, where it can be defined in terms of itself”. Fractals are characterized by the repetition of similar patterns at ever-diminishing scales. Think of recursion. As an example, I will use the famous Koch Fractal, which was probably the coolest thing before the invention of sliced bread.
- Start with a straight line. This is known as order 0 Koch fractal.
2. Now, to make an order 1 Koch fractal, erase the middle of the line above, and add two lines as shown below.
3. And order 2, repeat the process above on each and every of the 4 lines above to make a shape shown here:
And we can repeat the process above for higher orders, here is what order 10 would look like:
How cool! If we zoom in at any point on the pattern, we can also see similar structures of the fractal as well. Here is the recursion code I used to draw the fractals.
import turtle
fractal = turtle.Turtle()
wn = turtle.Screen()
def koch(t, order, size):
"""
Make turtle t draw a Koch fractal of ’order’ and ’size’
Leave the turtle facing the same direction.
"""
if order == 0:
t.forward(size) # The base case is just a straight line
else:
for angle in [60, -120, 60, 0]:
koch(t, order-1, size/3) # Go 1/3 of the way
t.left(angle)koch(fractal, 10, 1000)
wn.mainloop()
Now, back to the good part!!!
Circular fractals in African settlement architecture.
1937 Mary (Light) Meader
Much of Southern Africa has a Savannah climate, meaning short trees, and tall grasses. There are mostly arid plains where herds of cattle and other livestock are raised.
As an example of a settlement displaying fractals, we will use the Ba ila village shown above. This village has about 2000 inhabitants and has nearly a thousand huts. It is constructed in the form of a circle with the entrance on the down-wind side so that the cattle (5000!) when they come in shall not bury the place under dust. The kitchens are on the outside of the ring, then the dwelling huts, and a stockade, and around the inner edge of the big circle are a series of cattle kraals. The small circle in the middle is the chief’s stockade
Ring-shaped livestock pens, one for each extended family. Towards the back of each pen, we find the family living quarters, and towards the front entrance is gated entrance for letting livestock in and out. For this reason the front entrance is associated with low status (unclean, animals), and the back end with high status (clean, people).
The settlement as a whole has the same shape: it’s a ring of rings. Just like the livestock pen, the settlement has a back/front social distinction: the entrance is low status, and the back end is high status.
At the back end of the interior of the settle, we see a smaller detached ring of houses, like a settlement within a settlement. The chief’s extended family. At the back of the interior of the chief’s extended family ring, the chief has his own house, which actually has a ring like shape on it’s own.
If we were to view a single house from the above, we would see that it is a ring with a special place at the back of the interior: the house altar.:
Since we have a similar structure at all scales, this architecture is easy to model with fractals, recursion, and python turtle graphics.
We begin with a seed shape that could be the overhead view of a single house. This is created by active lines that make up the ring-shaped walls, as well as an active line at the position of the altar at the back of the interior. The only passive lines are those adjacent to the entrance.
In the next iteration, we have a shape that could be the overhead view of a family enclosure. At the entrance to the family enclosure we have only fencing, but as we go toward the back we have buildings of increasing size (which was a little hard to show with Python turtle graphics )
Finally, the third iteration provides a structure that could be the overhead view of the whole settlement. At the entrance we have only fencing.
Traditional African Settlements typically show this self-similar characteristic: circles of circles of circular dwellings, rectangular walls enclosing ever-smaller rectangles, and streets in which broad avenues branch down to tiny footpaths with striking geometric repetition. These structures can be easily identified when we compare the aerial views of these African villages and cities with corresponding fractal geometric simulation.
Patterns produced in different cultures can be characterized by specific design themes. In the Western World, cities are often laid out in a grid pattern of straight streets, and right angle corners. In many works of the Chinese art, we find hexagons used in extraordinary geometric precision, a choice that might seem arbitrary were it not for the importance of the number six in the hexagrams of their fortunetelling system (the l Ching), in the anatomy charts for the acupuncture (six spirits), and even in the Chinese architecture.
Overall, the presence of mathematics in African culture can be thought of in terms of a spectrum from unintentional to self-conscious. Termites mounds, for example, are excellent fractals (they have chambers within chambers within chambers) but no one would claim that the terminates understand the mathematics. An African artist who has no idea what the word “hexagon” mean could still draw one with great precision. But as we have already seen , the fractals in African architecture are much more than that. Their design is linked to conscious knowledge systems that suggest some of the basic concepts of fractal geometry.
Next time, we will find more explicit expressions of this indigenous mathematics in astonishing variety and form.
