First, a primer, then a question:
The Border Between Chaos and Order
2) fractional dimension
Self-similarity means that one part of the fractal is very similar to other parts of the same fractal. This can be seen in most fractal art . . . for example the fractal image above is a spiral made of smaller similar spirals, and each of those smaller spirals is itself made of similar smaller spirals, and so on, ad infinitum.
Start with a straight line . . . that has one dimension. Then make the line increasingly twisted in more and more complex ways . . . if the line was infinitely twisted it could fill an area and would thus be two-dimensional. Because of the principle of self-similarity (infinite complexity), a fractal line is part-way between one and two dimensions, so it is a fractal line that is on the way towards filling a space, because the wiggles on the line themselves have smaller wiggles, and those wiggles in turn have smaller wiggles and so on.
This might seem like mathematical abstraction but it has very practical results. For example, take the coastline of an island . . . look at it from far away and lay a piece of string along the coastline, and you will arrive at a length for that coastline. Then zoom in and you will see that where the coastline appeared to be a simple shape from far away, the line along the coast has a lot more detailed wiggles the closer you get. You could continue this increasing detail down to grains of sand along the coastline, and if you lay the piece of string around all the details, you get a LONGER measurement than you did from the initial far view!
There is a border between two countries in Europe that the two countries measure more than 10% differently, because they measure it at different scales!
The property of self-similarity is a very useful one, because it means that the data which represents a fractal need only define the basic motiv once, then define how that motiv is changed when it appears at the next scale down (in practice, the basic motiv is defined BY its relationship to its other versions). So an infinitely complex fractal can be represented by a very small amount of data. This is used in practice in JPEG image compression which uses a type of IFS fractal formula to compress images effectively.
It is a common misconception that fractals are “chaos”. While this is true in a few examples (such as white-noise, which is a fractal), the majority of fractals lie on the border between chaos and order.
So here’s my question: If an infinitely twisting one-dimensional line becomes two-dimensional by filling in space, does this hold all the way up the dimensional scale? From 2-D, to 3-D, to 4-D etc.? Higher order dimensions as opening space, over and over again, through fractals? . . . I’m in over my decidedly non-mathematical head here, nevertheless the question startles me. Stated slightly differently: is what moves us from one dimension to the next higher order dimension always a fractal that fills to infinity?
Is this a real question?