What are fractals?

A fractal is a rough or fragmented geometric shape that can be subdivided in parts, each of which is a reduced-size copy of the whole.

Note: The word "fractal" comes from a Latin word that means "broken" or "fractured."

The concept of fractals is quite fascinating as it showcases the infinite complexity that can be produced from the simple and repetitive procedures. These are suitable for modeling various natural phenomena like terrains, plants, clouds, and feathers.

Classification of fractals

Fractals can be categorized based on their properties as follows.

Self-similar fractals

• Contains parts that are scaled-down version of the entire object.

• A tree with branches that resemble small versions of the tree itself is a good example of self-similar fractals.

Self-affine fractals

• Have parts that are formed with different scaling parameters in different coordinate directions.

• Examples include terrain, water, and clouds.

Invariant fractal sets

• Such fractals are formed with non-linear transformations.

• Examples include self-squaring Mandelbrot set.

Dimensions

Understanding dimensions in the context of fractals is essential as it helps to understand the complex nature of fractals. Considering dimensions help us to figure out the procedural generation, scaling, and the categorization of fractals.

In the context of fractals, two types of dimensions are important.

Topological dimension

It is the traditional concept of dimension that commonly refers to the geometric objects. It refers to the whole number dimension of an object based on its physical space. It is denoted by $D_t$.

• A line has a $D_t$of $1$.

• A square has a $D_t$of $2$.

• A cube has a $D_t$of $3$.

Similarity dimension

Similarity dimension ($D_s$) is a measure of how an object scales as we reduce the measuring length. $D_s$ is defined as follows.

Where,

• $N$ is the number of copies

• $\epsilon$ is the scaling factor

Example

Consider breaking down an object into $N$ smaller copies of itself, each of which are reduced by a scaling factor of $\epsilon$.

• For a line, if we reduce its size by a factor of 2 ($\epsilon = 0.5$), we get $2$ copies.

• For a squaure, if we reduce its size by a factor of 2 ($\epsilon = 0.5$), we get $4$ copies.

• For a cube, if we reduce its size by a factor of 2 ($\epsilon = 0.5$), we get $8$ copies.

Note: A fractal is any curve for which the similarity dimension is higher than the topological dimension.

Koch's curve

Koch's curveA fractal curve that can be constructed by taking a straight line segment and replacing it with a pattern of multiple line segments. is a fractal as the similarity dimension is higher than the topological dimension ($D_t = 1$).

The koch's curve is defined recursively as follows.

Here,

• Each line segment is replaced by $4$ smaller ones which are one third ($\frac{1}{3}$) of the size of the original line segment.

The next iteration is given below.

Similarity dimension for this curve is calculated as given.

Note:

The koch's curve is:

• Discontinuous everywhere

• Infinitely long

L-systems

Lindenmayer systems (or L-systems) are a parallel rewriting system that are mostly used to model the plant growth and natural phenomena.

L-systems have found extensive applications in computer graphics, art, and the generation of fractal patterns.

L-systems and fractals

• L-systems are linked to fractals by their ability to generate self-replicating patterns.

• When used, L-systems can generate the branching structure of trees, the arrangement of leaves, and the shapes of flowers.

• Such structures exhibit self-similarity in which smaller parts of the plant resemble the whole plant.

Example

Consider the following fractal.

• Start: $F$

• Variables: $F$

• Constants: $+ \;, \; -$

• Rules: ($F \rightarrow F+F-F-F+F$)

Where,

• $F$ means "draw forward"

• $+$ means "turn right $90^\circ$"

• $-$ means "turn left $90^\circ$"

Note: Visit Fractal lab to see interesting examples of fractals.

Demonstration

Overall, fractals, with their infinite complexity but self-similarity, offer an immersive representation of various geometric structures.