I.2 Fractal Geometry and Chaos Theory

We are familiar with the normal concept of geometrical dimensions, such as lines, surfaces and volumes, which have one, two and three dimensions respectively. In mathematics, the dimension of a mathematical space is the minimum number of coordinates needed to specify any point within it. That s why our normal space have three dimensions . These normal dimensions are called Euclidean, and they are always complete integers. In relativity, we use the four dimensions of space-time, and strings theory can have more than ten dimensions, while the state-space of quantum mechanics is an infinite-dimensional function space. Therefore, the concept of dimension is not restricted to physical objects, but it can be easily generalized to any number and kinds of independent variables that describe the state of a system. In mathematics and physics, such as in Lagrangian or Hamiltonian mechanics, topological space is abstract and it can have any number of dimensions that are independent of the physical space we live in.

Unlike topological dimensions, the fractal dimension can take non-integer values, because it measures the complexity of the object in addition to its usual geometrical dimensions. For example, a curved line with a fractal dimension  is closer to a straight line than another curved line with a fractal dimension , because the latter line winds convolutely through space very nearly like a surface. We can even have lines with fractal dimensions that may equal  or , because they are curved in certain ways that can fill the plane or the volume, respectively.

The term fractal was coined by Mandelbrot in 1975, about a decade after he published his paper on self-similarity in the coastline of Britain. The length of coastlines depends on the scale of measurement. If one were to measure a stretch of coastline with a yardstick, one would get a shorter result than if the same stretch were measured with a one-foot ruler. It is impossible to precisely measure the length of the coastline, because the measured length increases without limit as the measurement scale decreases towards zero.

Therefore, a fractal dimension is an index for characterizing patterns or sets by quantifying their complexity as a ratio of the change in detail to the change in scale. For sets describing ordinary geometric shapes, the theoretical fractal dimension equals the set s familiar Euclidean or topological dimension. If the theoretical fractal dimension of a set exceeds its topological dimension, the set is considered to have fractal geometry.

As is the case with normal Euclidean dimensions, fractal dimensions are general descriptors that do not uniquely define patterns. So two completely different shapes may have the same fractal dimensions, just like a triangle and a square are both two dimensional. The value of of the dimension quantifies the pattern and how it behave under scaling, but does not uniquely describe nor provide enough information to reconstruct it.

For the same reasons of deceptive multiplicity that cause the illusion of motion, the human mind is accustomed to viewing physical objects enclosed in three spatial dimensions. Although we can feel one and two abstract dimensions, like surfaces and lines, we always conceive all objects as three-dimensional, because an abstract two-dimensional surface has no thickness, and this does not exist in the physical world that we normally conceive, neither does the abstract one-dimensional line which has not thickness or breadth. Actually, our feeling of dimensions is all deceptive, because they necessarily imply multiplicity which requires time iterations to be conceived in the mind, as we have already explained in Chapter VI of Volume II.

However, one simple way to imagine fractal dimensions is to consider every small rotation, between  and  for example, as a fraction of the whole dimension  plane. This means that they can be infinite number of dimensions between  and , so in reality fractals are as real as the whole integer dimensions, although both are some kind of intellectual reasoning to classify objects in space and time.

Therefore, a real three-dimensional space is an abstract mental construct that is symmetrical in all directions, while actual physical objects, since they have various imperfect symmetrical features, are less than three-dimensional, but they are also more than two-dimensional, because they are not completely flat. Likewise, a flat painting, for example, is more than one-dimensional curves, but it is less than two-dimensional, because a two-dimensional plane is completely flat and symmetrical or isotropic, i.e. it has no distinguishing features.

Koch snowflakes, as shown in Figure I.2, are a good example of fractals. Because these patterns repeat at various scales, the length of their circumference is infinite. In fact, this is also true for all kinds of shapes, even a simple square or circle, which are normally viewed in whole integer dimensions, because physical objects, no matter how smooth they look outwardly, are composed of discrete particles and even granular space geometry. So the whole concept of length or dimensions involves some approximation, which is usually performed through integration, or square-counting.

Although we do not realize it, but what we actually mean when we say that an object is three-dimensional is that it is enclosed in three dimensions, which actually means that its dimensions are less than three. Even a perfect symmetrical sphere is still less than three dimensional as far as it is finite, so it is in fractal dimensions. In reality, therefore, although the physical spatial dimensions are altogether not real, they are best treated as fractals rather than whole integers.

Figure I.2: In Koch Snowflake fractal, or Koch star, the progression for the area of the snowflake converges to 8/5 times the area of the original triangle, while the progression for the snowflake's perimeter diverges to infinity. Consequently, the snowflake has a finite area bounded by an infinitely long line.

In recent years, new and intriguing relationships have been discovered between symmetry and fractals, different from what is traditionally seen in symmetry considerations, and it is also referred to as scale-symmetry and self-similarity. In Euclidean geometry, shapes scale predictably according to intuitive and familiar ideas about the space they occupy. For example, when we scale a line segment by a factor of , its new length will be three times its original length. If we do the same with a square, its new area will be  times its original area, and that s because the square is two-dimensional. Yet if we apply the same scaling on a cube, its new volume will be  times its original volume, and that s because the cube is three-dimensional. In general, we can say that the ratio  of the sizes before and after scaling is equal to , where  is the scaling factor, and  is the dimension, thus: . By taking the logarithm of both sides we get:

 (1.5)

The same rule can now be applied to fractal geometry, and we usually do that by scaling down, by making , so that the the new size will be smaller and contained in the original size.

Figure I.3: Random Tree Fractal.