A distinctive feature of one class of fractal object is that of being defined recursively. Another term describing fractal objects of this class is iterated function system. The fractal object contains detail on both the macro- and micro-cosmic scale. The level of detail is defined by the number of iterations performed. The consideration set is also increased as the level of recursion or iteration of the function system is increased.
Low levels of iteration yield low detail. The consideration set is small and the overall quality of the system is very limited. High levels of iteration reveal enormous amounts of detail with a correspondingly high degree of quality and a very large consideration set.
For example, let us consider a cloud formation. A quick, cursory glance might invoke the image of elephant parading across the sky. Have you ever considered how much this cloud-elephant actually weighs?
Accounting for the amount of water in a cloud is not an easy task. I have heard recently of an estimate that the weight of water in a small cloud is approximately the same as the weight of one actual elephant. Let us equate the unit of measure of an elephant’s weight with iteration level one.
Now if we consider the macro-cosmic scale of a thunderstorm set of clouds, then our level of iteration increases to two, and the number of elephants’ weight of water increases to, let’s say, tens of thousands of elephants. Increasing the macro-scale further to iteration level three we consider a hurricane. Perhaps a hurricane equates to the weight of millions of elephants, give or take several thousand elephants. The unit of an elephant seems imprecise when considering a small cloud, however, when our consideration set expands to that of a hurricane, the elephant unit is very precise.
A similar recursive exploration of cloud systems may occur on the micro-cosmic level. Perhaps we are camped on the edge of quiet mountain lake. We awake early on a chilly morning to watch the sun rise. As the rays of the sun strike the lake a delicate fog arises from the water’s surface.
We notice the fine details of fog wisps. We examine the edge of the fog and try to detect the boundary between sky and water vapor. Our consideration set grows as we delve recursively into the detail at the interface between fog and non-fog. We can see individual wisps of water vapor streaming upwards no bigger than our arm.
Accounting for the amount of water in a single arm-sized wisp will obviously take a unit of measure much smaller than that of an elephant. An elephant-heavy cloud may eventually be formed from the fog rising from a large surface of water. However, here at the edge of the lake, intimately close to the interface between sky and water, we have a much more detailed perspective. Our consideration set is much smaller, yet we have also increased the overall number of levels of recursion of iterated function system we call a cloud.
In this way, the level of recursion or iteration of the function system is relative. It is a measure of dimension, or degree of separation between two different levels of consideration.
Therefore, an iterated function system that undergoes a large number of iterations results in a high level of detail. This level of detail is an expansion of the ability of the observer to consider both macro- and micro-cosmic features.
Karmatetra 