Think of a monochrome picture, let’s say a white one. It can be completely described with a very small amount of information. You may say: 512 x 512 pixels of white (or any other color that can be described with a few numbers of any color coding system).
That you can describe such a picture with a few words means that it contains very little information. If you think of it as a file in your computer’s file system, it may be compressed into a very small file. Its information content is small. In the mathematical theory called “information theory”, this information content is called “entropy”. It is a measure for the amount of information needed to describe the data or a measure for the amount of disorder in it.
In the following picture, there are rounded edges and a white frame and this adds to the amount of information necessary to describe it completely, but the file could still be compressed a lot.
This picture is very repetitive, it has a high degree of order and thus a small entropy. On the other hand, its repetitiveness, called redundancy in information theory, is high. (In case you are interested in the mathematical detail, search the web for “information theory”.)
On the other hand, if you fill the pixels with random “noise” (as it is called in information theory), i.e. you determine each pixels color value using some random process, you get something like this:
Picture from (http://commons.wikimedia.org/wiki/File:White.noise.col.png)
Try to compress such a file and there will be hardly any compression at all. This means that it contains a very large amount of information. Not necessarily useful information about something, but you need a lot of storage space to store it because the pixels in it are not correlated. Having no correlation, no structure, means that you cannot compress the picture. You have to tell for each single point which color it has. If you look at the details, you may find some trace of a structure here and there but any such structure disappears if you look at larger areas. If you would increase the number of pixels, you would not be able to find any structure at all. Being random means that the color of the pixels are not correlated. There is no regularity.
If you look at a monochrome picture, you will notice it is very boring. If you look at the noise picture on the other hand, if you try to view it as a pattern of pixels you will not be able to discover much structure in it. It would be frustrating. You may also view it as just evenly covered with a noise pattern. Then it would look as boring as the monochrome picture.
Both kinds of pictures, the monochrome and the noise, are uninteresting. Between these two extremes are the interesting and the beautiful pictures.
Imagine a picture that is white but contains some black straight lines. The amount of information in such a picture is larger than in the monochrome picture. You would have to describe the position of the end points of the lines, maybe also how thick they are. But still the amount of information needed for a complete description is quite small. If you describe the picture in a pixel by pixel fashion, you would need the same amount of information as in the noise picture, but by discovering the lines, you can compress that information into a small amount. As the example shows, the discovery of structure may be seen as a process of information compression. Some scientists have described the process of perception as a process of information compression.
Think of a picture with parallel lines in regular distances. Here to describe the picture, it would be sufficient to describe one line and then to describe the others as parallels, just giving their distance. So if there is a regularity in the picture (or more generally: in the data) discovering this regularity may enable you to compress the file. In human perception, discovering a regularity means you can reduce the amount of information you have to process or memorize, and it becomes easier to spot the non-regular rest.
If the data is very regular, we can just compress it right away using a known procedure, something we may model as an algorithm. There is no challenge. The perceptual task is boring.
If there is too much complexity, we will not be able to compress the information. We will be confused or frustrated, or we will switch to ignoring the detail and thus perceiving the complex data as smooth noise. This in turn will be boring.
If there is some structure of a medium complexity, we will manage to discover regularities. We will manage to compress the information, but not by just applying a preexisting algorithm but by creatively constructing new knowledge on the spot. Succeeding in this task will generate a positive experience of success. My hypothesis is that this is the emotion we perceive as beauty. It is the reward for integrating novel information.
If the data contains a rich and varied structure so that whenever we manage to assimilate part of it, some will remain and our perceptive system remains engaged for an extended time, we will be rewarded with a prolonged experience of beauty. This requires structures of a medium amount of information content. There must be a lot of information so that it does not become boring but not too much so we are not overwhelmed by it. Unlike the monochrome picture, such a picture will resist compression into a very small file, but it will be compressible to some extent because it contains some regularity or order (redundancy).
Pictures with such a medium degree of order look less boring than a monochrome picture and more interesting or even than the confusing or boring noise picture. You might like them. Look at this, for example:
This picture has a middle amount of regularity and irregularity. There might be better pictures than this one (I choose it because it is open domain, but also because I don’t find it bad) but you will notice that it is less boring than the other two. It is somewhere in the middle and it gives our visual system some work to do over an extended time.
The connection between beauty and medium information (or entropy) content has been noted before, e.g. by the philosopher Max Bense. There is actually quite some scientific literature on this topic. However, I suspect here a link to creativity. Following the works of Kurt Ammon, I view creativity as the ability to break out of any given formal system or algorithm. In perception, that means to extend the preexisting knowledge to assimilate novel information. The reason that medium-information structures are beautiful is that they lead to repeated extensions of the analytical spaces our perceptive system consists of.
Composers learn to create auditory structures that have this property. This is what we call music. Photographers learn to find such structures and snap them. Artists, e.g. painters, learn to create such structures using canvas and paint.
If I am right, the possibility of art, especially pre-conceptual (or “abstract”) art in which novel beautiful structures are created by the artist, is a side effect of the way our perceptive system works, and that is the permanent extension and growth or our perceptive knowledge.