Let us consider a system that can be cut or decomposed into a large collection of smaller subsystems. For instance, a picture on which we can move a small window, each position of the window is giving us a small image that is part of the picture. Of course, when the window is slightly moved the image we see is partially the same as previously, but in the end we will have explored the whole picture.
Then we choose a way to compare any two small images seen in different windows. For instance, using the corresponding light levels of the pixels it is possible to characterize the result of this comparison by a number. Say that this number, a sort of distance between the two small images, is zero if they are identical and grows as the difference between them becomes more and more important.
The nice mathematical fact is that the result that we will get at the end is almost independent of the way we choose to quantify this distance. Of course, to say that two small images are at a short distance in this sense, simply means that on this level of resolution they can be considered identical. This allows to classify all the small images in different types of patterns. At this point we can count how many different patterns are present in the collection of all windows.
Notice that two small images that were considered of the same type or pattern for a given resolution may or may not fall into different types if we increase the resolution by a certain factor. If the number of windows is large, it is possible to measure how fast the number of patterns is growing when the resolution increases. The complexity of the picture is just this measure!
You can imagine many different situations where this complexity can be estimated. In fact, all you need is a large collective system of which you are able to classify smaller parts according to a given notion of "resolution", that is to say, up to a certain level of detail. Then you let the resolution increase, in other words, you look to the small parts in more and more detail, and you count!
Complexity: The sculpture "Profile" by Beate Cruse on the wall near the ZiF main entrance
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