Dimensions

We live in a three-dimensional world: That is, we need three coordinates to specify the location of any point. The coordinate system we use locally might be the orthogonal (right-angular) Cartesian grid with x-y-z axes, or more practically for a global description, a location of a point in terms of latitude, longitude and altitude. When we restrict our attention to a subset of the world we can often make do with a description in terms of fewer coordinates. For example, a point on the surface of a table can be described in terms of two coordinates. Sometimes a description in terms of fewer coordinates is a useful and economical approximation. Consider a ball of string lying on top of a table. From a very large distance it might appear to someone as a point object, its location in space given by three coordinates: Far away it is a point and so zero dimensional. When one approaches closer it will look disc-like and two-dimensional. Even closer up the three dimensional structure of the ball will become apparent. Finally one realizes that the ball is made of string, a one-dimensional object, and so any point on the ball of string can be located by tracing out a single coordinate along the string! In the example above we have used two definitions of dimensions. Firstly there is the Euclidean dimension ($D_e$): The number of coordinates required to specify an object. Secondly there is the Topological dimension ($D_t$), which, roughly speaking, is a measure of the intrinsic dimension of the object. For example, a thin string has topological dimension one but when it is spread out in space, as in the ball, it has a Euclidean dimension of three. Topology is often called "rubber" geometry because it deals only with the qualitative shape of an object. If the object is imagined to be made of rubber then by stretching (but without tearing) it can be deformed into another topologically equivalent object. Thus, a curve of any shape is topologically equivalent to a straight line, and has a topological dimension of one. The Euclidean and topological dimensions are always integral. For characterizing fractals it is useful to introduce another definition called the similarity dimension which is often fractional. To motivate this definition, consider first a unit Euclidean line, square and cube, each divided into N equal self-similar parts of linear dimension $s$. For the line, since $N s=1$, each smaller part has a length $s=1/N$. For the square, $N s^2=1$, so each smaller square has length $s= 1/N^{0.5}$, while since $N s^3=1$ for the cube, each smaller cube has length $s=1/N^{1/3}$. The values of $s$ above are called scale factors.
Definition of Similarity Dimension: If an object of unit size contains $N$ self-similar copies of itself of size $s$, then its similarity dimension $D_s$ is determined by the equation

$$N s^{D_s} = 1 .$$ (2.1)

For the Euclidean figures above, $D_s=1$ for the line, $D_s=2$ for the square and $D_s=3$ for the cube. These numbers are identical to the topological and Euclidean Dimensions for these figures. Let us rewrite the above equation in the form

$$D_s = { \log (N) \over \log (1/s) } .$$ (2.2)

Now we can find the similarity dimension of the Koch curve. At each observation scale, the curve contains $4$ self-similar copies of itself of size $s=1/3$, so

$$D_s = { \log (4) \over \log (3) } = 1.2618...$$ (2.3)

Thus the similarity dimension of a Koch curve is larger than its topological dimension which is one, but smaller than its Euclidean dimension of two. Since $D_s$ for a Koch curve is larger than that for a line but smaller than that for area, one can roughly say that the Koch curve is more than a line but not quite a plane. We are now in a position to appreciate the following formal definition of a fractal:
Definition of Fractal (formal): A fractal is an object whose similarity dimension is larger than its topological dimension.
An equivalent way of thinking about equation (4.18) and fractal dimensions is as follows. Imagine measuring everthing relative to the length of a measuring stick. So if a smooth curve can be covered using $N$ units of that stick of measure $s$, the length of the curve can be estimated as $N s$. If a stick half in length, $s/2$, is used instead, one would need $2N$ units to cover the same smooth curve, and the final length of the curve will come out to be the same. For a fractal curve the above is no longer true. As the size of the measuring stick is changed, the total length of the curve changes, as we saw for the Koch curve. Historically this fact was observed by Richardson: He found that the length of some borders between countries seemed to increase when the length of the measuring instrument was reduced. Can you explain in physical terms what is happening in this case ? Here is another example of a fractal. Mandelbrot discovered that noise in telephone lines is clustered and can be modeled as a Cantor set. The Cantor set is generated as follows: The initiator is a unit line element. The generator involves removing the middle one-third of the unit line. After this first step the figure consists of two line segments, each one-third in length. The procedure is repeated endlessly, each time removing the middle one-third of the remaining line segments.





The Cantor set has an infinite number of points but it is of width zero (technically one says it is of measure zero). The Cantor set is in some sense the opposite of the Koch curve: In the generation of the Cantor set, at each step (prefractal) the line segment was made shorter by one-third while in the Koch curve it was made longer by one-third. Therefore one would expect that the Cantor set is in fact 'less than a line', just as the Koch curve was considered 'more than a line'. This is confirmed by computing its similarity dimension. What are the topological, Euclidean and similarity dimensions of the Cantor set?