Fractal Dimensions : To integers and beyond
I’m sure anyone who is reading this blog is familiar with the term ‘dimension’, and more specifically, the usage of the term in mathematics.
Informally, dimension of a mathematical object is the minimum number of coordinates required to specify and point within it. For example, the dimension of a straight line is 1, since only one coordinate is sufficient to specify any point within it. Similarly, a plane has dimension 2, the inside of a sphere has dimension 3 and so on. We also naturally assume that dimensions are integers.
However, that is not always the case. In fact, there are subsets of the well known Euclidean space with dimensions which are not integers.
Fractals are such subsets, which have integer topological dimension but can have non-integer Hausdorff dimension.
Before delving into what makes fractals have such a unique property, let’s define what is a topological dimension and what is a Hausdorff dimension and how they’re different.
Topological Dimension
This is also called the Lebesgue covering dimension for a general topological space. However, since we’re just dealing with Euclidean spaces, it can be informally defined as the number of independent parameters or coordinates that can define the position of a point that is constrained to be in that space. By this definition it’s hard to imagine what a space with non-integer dimension would look like, since we don’t know how to define non-integer number of coordinates.
Hausdorff Dimension
For a set of points that define a smooth shape or a shape with a small number of corners (the shapes we usually come across in traditional geometry and science), the Hausdorff dimension is an integer which agrees with the sense of dimension defined in the previous section.
However, for less simple objects such as fractals, things change. Fractals have properties such as scaling and self-similarity. To assign a particular number to such shapes, integers are not sufficient.
Formally, the Hausdorff dimension is a further dimensional number associated with a metric space (a set where distances between all members are defined and follow certain properties). The more general sense of dimension is not associated with general metric spaces.
A fractal has an integer topological dimension, but when it comes to the amount of area it takes up, it behaves like a higher dimensional space. To get a more concrete idea about this Hausdorff or fractal dimension, we’ll try calculating the topological and fractal dimensions of a simple fractal.
An example — The Koch snowflake
In this section we’ll calculate the topological and Hausdorff dimensions of the Koch snowflake (shown below).
The Koch snowflake is iteratively constructed from an equilateral triangle. In each new iteration, the component line segments are divided into 3 parts of unit length and the newly created middle segment is used as the base of a new equilateral triangle which points outwards. The base is then deleted to arrive at a final object of unit length of 4.
Each line segment has been replaced with N=4, and each self-similar copy (the new lines) have length 1/S = ⅓ of the original.
In other words, we have taken an object with Euclidean dimension D and reduced its linear scale by ⅓ in each direction so that the length of the object increases to N=S^D
We solve for D by taking log each sides and get:
D = log(N)/log(S)
After the first iteration, N = 4 and S = 3,
Hence D = log(4)/log(3) = 1.26 (approx.)
This clearly shows that the original equilateral triangle (3 line segments) of dimension 1 has taken up a new non-integer dimension of 1.26 after the first iteration.
The Koch snowflake is a very simple example of a fractal, and its dimension (fractal/Hausdorff) of 1.26 essentially captures the complexity of how the detail changes with the scale. We can see that the structure becomes more complex and detailed in each successive iteration, and this is what the fractal dimension quantifies.
