Fractals have the property of self-similarity: a small part looks like the whole. We can see that in this image of a Koch curve, a fractal named for Swedish mathematician Helge von Koch. At top we see three big clusters, like the arms and head of a lumpy snowman. But each is made of similar clusters, and so on down to infinity. How did Koch construct such an infinitely wiggly curve?
The Koch curve begins as the simple seed shape seen here, just 4 lines. Then we shrink down the seed shape--as we say in mathematics we apply a scaling transformation --and replace its first line with a miniature version of itself. We do that for all 4 lines, and now we have the output of the first iteration.
In Africa, recursive loops are sometimes represented as a snake that bites its own tail. Just as contemporary computer science uses recursion to model the fractal structures in nature, Indigenous African knowledge recognizes and makes use of the "self-generating" properties of certain natural and social systems.
Another snake symbol in Africa represents well-being that lasts forever. They say that the snake is infinitely long, but it can fit in a finite space, because it has "coiled back on itself." As we will see, fitting infinite length into a finite space is also a good description of a fractal.
As we shrink down the ruler size, the length increases to infinity. But how fast does it increase? For a smooth Koch curve, it increases slowly. For a rough one it increases quickly. The slope of the line tells you how fast or slow. It is the fractal dimension.
A line has one dimension, a square 2, But fractals are in-between, so they have a fractional dimension! That is how they can fit infinity into a finite space. Their curve is so torturous that it is more than one dimensional. In fact it can get so wiggly that it actually does become a space-filling curve, in which case the fractal dimension will be 2.0 (a surface!).
In each iteration we replaced each line with a scaled-down seed shape. The amount it scales by is the scaling ratio. It is the ratio between the length of the line to be replaced, and the length of a baseline. The dashed line is the baseline. So the scaling ratio of the first segment is about 2.2/8, the length of the line segment, over the length of the baseline.