Adinkra Spirals

Adinkra symbols embody traditional African ideas about geometry. Here we see two kinds of spirals. If we coil up a rope or hose, we get a linear spiral. The space between each revolution of the curve is the same. Ropes are not alive, they do not grow.

In contrast, the snail's shell is an exponential or logarithmic ("log") spiral. As the snail grows, its shell gets larger.

Which kind of spiral do you think is more common in Adinkra symbols?

linear spiral and rope Rope coil as a linear spiral
log spiral and snail Snail shell as log spiral
Adinkra Spirals: Dwennimmen
If you guessed the logarithmic spiral, you are correct. For example, this Adinkra symbol, Dwennimmen ("GIN-ee-mawn"), shows four spirals. The spirals are in pairs because this represents two rams butting heads. Its proverb says: “it is the heart, and not the horns, that leads a ram to bully.” In other words, we have to take responsibility for our actions. Notice that symbol makes use of the log spiral, just like the real ram’s horns.

Click to learn more about rams' horns and spirals

Adinkra Spirals: Examples From Nature
Animal horns are not the only logarithmic spiral from nature. Many shapes created by living organisms can be modeled using a log spiral. Some have several full rotations, like the snail shell. Others are just a small section (arc of a spiral), like the chicken's foot.
Adinkra Spirals: Sankofa
The adinkra symbol Sankofa ("sang-ko-fah") shows a bird looking backwards. The saying that accompanies the symbol is “You can always go back.” It is often used when talking about past cultural traditions; for example Ghanaians recovering traditions that were lost to colonialism, or more recently African Americans seeking cultural knoweldge that was lost in the slave trade.

In the case of the ram's horn symbol, Dwennimmen, a skeptic might say it is just copying nature without really thinking mathematically. But the Sankofa symbol comes in two different versions. The one at top might be said to just copy nature, but the version at the bottom is so abstract that you would never guess it stands for a bird unless you were told so. Just as today's scientists use log curves to model nature's shapes on computers, African artisans long ago began to use log curves to model nature's shapes in Adinkra. In the image at right we added a curve to make it easier to see how they moved from concrete to abstract. In the abstract version they have combined the spiral with another geometric idea, reflection symmetry, which we will discuss later.

This study of mathematical ideas in different cultures is called "ethnomathematics." When we want to translate words from one language to another--say from French into English--we often use a dictionary. For researchers in ethnomathematics, a computer simulation can be like having a dictionary, because it helps us translate the mathematical ideas from a culture that expresses them with carvings and cloth to a culture that expresses math in equations. By using the simulation tools on this website, you can create your own ethnomathematics investigation.
Sankofa Adinkra symbol
Sankofa with curve added
Abstract version of Sankofa symbol
Adinkra Spirals: Gye Nyame

Here are three more adinkra symbols that show the use of log spiral arcs to represent shapes in nature. Akoko Nan ("ah-ko-ko non") represents a chicken’s foot: "The hen treads on her chicks, but she does not kill them." Those in authority should nurture those they are in charge of, but not pamper them. Akoben ("ah-ko-ben") represents the horn that was sounded to bring warriors to battle. The symbol means both readiness-- to be on the alert-- as well as loyalty or devotion to a cause.

The symbol Gye Nyame ("jeh N-yah-mee") has a somewhat mysterious saying attached to it: "No one except God." It has two log spirals, one at each end.

Recall that the other adinkra symbols with log spirals were representing shapes of living organisms: Dwennimmen is based on the ram's horn; sankofa on the bird’s neck; Akoko nan on the chicken’s foot; and Akoben on the horn of a bull. But what do the logarithmic curves of the Gye Nyame symbol represent?

The knobs or lumps down the middle of the symbol represent the knuckles on a fist, a symbol for power. The full meaning of the the Gye Nyame symbol is, "No one except God has the power of life." The curves at each end are not representing any one particular living shape. Rather, they are a general abstraction for life itself. Developing a general abstraction or "invariant property" that fits all cases is fundamental to science. The idea that logarithmic scaling or "power laws" characterize patterns in biology is now widely accepted, although the reasons are still an active area of research. Let's look closer at these spiral shapes to see if we can get some clues.

Akoko Nan
Gye Nyame
Adinkra Spirals: Cell Growth

The simple idea of one cell splitting into two cells can help us understand log scaling. Each "daughter" cell splits into two more the next month, and so on. If we plot the sum of cells (S) each time period (t), we notice that it follows the curve S = 2t. You can see a simulation here. The equation is S = P(1+I)t where P is the principle (starting) number of cells, and I is the increase rate (the rate of cell reproduction). S = P(1+I)t = 1*(1+1)t = 2t.

Of course real cells don't all reproduce at the same time. Suppose only half reproduced each month (rate of reproduction 0.5), then the graph is less steep: S = 1.5t Its the same equation for a bank account: P is the Principle amount of dollars, just like the principle number of cells you start with. Interest rate I = 0.5, just like the cell reproduction rate. Compounded each month t, then total savings S= P(1+I)t = 1(1+0.5)t = 1.5t.

The image at bottom is a nautilus shell, split in half so you can see how the mollusk developed a new chamber as it grew. You can experiment with a simulation here. Each chamber in the simulation is 104% larger than the previous. If the size is proportional to the number of cells, that means the cells reproduce on average at a rate of 0.04.

In addition to growing by 104%, each chamber in the simulation is rotated 18 degrees from the previous. If you increase the growth rate-- say, 105%-- you will see that the spiral creates gaps--more like a horn than a shell. So if you increase the growth rate, you have to make the shell more "tightly coiled" to prevent gaps. The West African artisans who create Adinkra stamp carvings describe the growth rate of spirals in similar ways-- the terms they use in Twi, "aboapua awan" and "ntitim awan," translate into English as "tightly coiled" and "loosely coiled". Which is a better match to the nautilus shell?

Cell Growth Graph
Tightly coiled ("aboapua awan")
Loosely coiled ("ntitim awan")
Adinkra Spirals: Scaling Constant

If you said "tightly coiled" is a better match, you were right! Our eyes have put us on the right track; now lets see if numbers can help us make that more exact. The spiral’s equation in polar coordinates is radius = Ctheta, where theta is the angle and C is a constant. You can think of C as the “coil” parameter. For a tightly coiled log spiral like the one above, C is smaller: in this case 0.2. For a loosely coiled spiral like the one below, C is larger (0.5). In the simulation here you can explore the cultural curves of adinkra and biological curves of nature, using the blocks you see below the spirals.

We can also measure spirals in nature or culture to find the constant. First, take a protractor and measure the radius at selected angles. You can do that with the simulation here. Then take the log of each radius, as shown here.

Table of angle versus radius and log

Finally, graph the log of the radius as a function of angle. The slope is .82/270 = .003. So C= 10.003 = 0.5.

Log of the radius as a function of angle