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?
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.
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?
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 1.0008. For a loosely coiled spiral like the one below, C is larger (1.0071). 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.
Finally, graph the log of the radius as a function of angle. The slope is .82/270 = .003. So C= 10.003 = 1.0071.