Although it is a very attractive and interesting exercise to think of information as being fractal in nature, I don't believe that under close inspection the analogy fits, and indeed the two entities seem to be in direct conflict.
First of all, to adequately discuss this proposition, we need to define what information is and what fractals are.
What is information?
Qualitatively, information is a collection of interrelated concepts. You could imagine information as a diagram showing dots (representing concepts) connected by lines (representing association).
Quantitatively, the base unit of information is a bit. True or False. Yes or No. Information can be built out out of these atomic building blocks.
However, both of these interpretations represent the same thing. One is from the top looking down, and one is from the bottom looking up, but through many layers of abstraction, equivalent systems can be created.
What is a fractal?
The term fractal was first used by Benoit Mandelbrot, an IBM employee, in 1975. They are mathematical functions parameterised by two variables; one finitely bounded, one not. Fractals are most commonly shown as intricate, vividly coloured pictures, where the hue of the colour represents the value of the underlying function. These pictures are recursive which means that certain subsections of a fractal are the same as the its enclosing shape. As an example, here's two iterations of Sierpinski's Triangle:
/ \ / \
Here, if the fractal computation was continued, each of the sub-triangles would contain 4 sub-sub-triangles, they would all contain 4 sub-sub-sub-triangles and so on.
Obviously, for us to draw a fractal, we must omit a lot (i.e. infinite) amounts of detail, and only compute a finite number of iterations of the shape. However, this is just an approximation to the actual fractal, which would continue repeating itself ad infinitum.
Why are they different?
These entities both have two properties that must be noted:
- there is an atomic unit to information
- the diagram of interrelated concepts can contain cycles
- there is no limit at which fractals cease to hold infinte detail
- part, or all, of the structure is repeated in full
These properties show two fundamental differences between information and fractals. Whereas information can be fully expressed by atomic, finite units, to put an upper bound on the amount of detail to go into a fractal would by definition no longer be a fractal. Also, whereas fractals contain recursive structures that must be painstainkly explored without end, if a concept refers to an existing concept, all that is required is to place another interrelation onto our information diagram and halt.
There is another slightly more questionable difference: there is clearly infinite amounts of detail in a fractal, but is there infinite amounts of information? I would suggest that the universe can only hold so much state. The Heisenberg Uncertainty Principle tells us we cannot know the present in its entirety. There is a well defined limit on how much data we can store in, and extract, from elementary particles, and as the universe contains a bounded amount of energy, and hence a bounded mass, and hence a bounded number of elementary particles, surely we can only store a bounded amount of data in the universe?
In hobyrne's and wertperch's excellent existing writeups, the point is raised that in researching a topic, we seem to be greeted at each point by potentially vast amounts of detail on each sub-topic, and so on recursively. Indeed, this trend can be seen by looking at a noder's writeup list, and noticing that very often, a series of writeups tackle a similar subject, potentially in more and more detail, or focussing on particular aspects of the mother node.
However, it should be noted that you are not researching the same thing over and over in more and more detail, you are researching a separate topic that happens to be related to original material. This is an important distinction as rather than seeing our quest for knowledge as a non-terminating quest down into never ending detail, it is rather a winding path between interrelated concepts, where you will either get back to where you started or end up at the lowest level of detail there is.