The concepts of fractals and multi-fractal are scientific and/or mathematical terms with comprehensive explanations. By the end of this article, it is expected that the concepts of both are understood at least in theory.
Fractals
Consider an object such as a geometrical snowflake, a butterfly, or even an equilateral triangle. If you were to draw an imaginary line of division through any of these things, the result is that a mirror image exists on both sides of the dividing line. This indicates that the shape is symmetrical.
Even with the complex systems that function within a butterfly to keep it alive, its physiology can be perceived as two equal halves. Similarly, as complex as a snowflake may appear, it is possible to establish two completely identical pieces with such an imaginary line. This is a basic exhibition of symmetry.
Fractals are never ending patterns built on the concept of symmetry or asymmetry and, they are known for being scale invariant. This means that whenever a fractal pattern is generated repeatedly, there is no mutation of the initial shape regardless of the scale factor that is applied. Therefore, at varying levels of zoom, there is no discernable difference in the shape that is present. This repetitive generation without a visible change is possible because fractals have what is known as self-similarity. This type of recursive construct allows for the generation of shapes with an infinite perimeter, but a finite area.
Though the fractal pattern itself is endless, you can fit it within a defined physical space. This means that it is possible to place the entirety of the pattern within a regular shape such as a circle or a square. These shapes have a finite and calculatable area and, since they enclose the pattern, it too must have a finite area. However, since continuous zoom leads to the same pattern appearing into infinity and perimeter is measured using the outer lining of shapes, the perimeter of the pattern becomes infinite as the outer lining is continuously expanded.
While there are mathematical constructs such as the expression of fractal trees that illustrate the premise of fractals, there are also real-world phenomena such as the Romanesco broccoli vegetable, which illustrates a natural and self-similar fractal.
Note, that the usefulness of fractals is not restricted to shapes. There are also time related processes for which such self-similar fractal patterns can be established.
Multi-fractal
In short, the difference between fractals and multi-fractal lies in the level of complexity and the composition of each. As indicated before, they are a recursively generated, infinite, self-similar, scale invariant pattern. While they are quite useful and have many real-world applications, they are not necessarily able to capture the complexity of every system to which recursive pattern generation is applicable. This is where multi-fractals come into play. Whenever the fractal dimension is not enough as a single exponent to capture the dynamics of a system, a multifractal is used to then generalize the system to achieve the required complexity by using several exponents, which are referred to as the singularity spectrum.
Just as fractals are common and can be seen in a variety of real-world constructs and matter, multi-fractal systems are just as, if not even more common. There are quite a few complex operations such as human brain activity across neurons, the dynamics involved in the heartbeat, the way internet traffic flows across network nodes based on a routing protocols and tables, meteorology, and more that can only be captured using multi-fractals.
The reason for this is that unlike with fractals where the scale is adjusted at a constant rate yielding the capturing of constant pattern, multi-fractals can capture pattern variation. Therefore, multi-fractal analysis is used in tandem with lacunarity analysis (the study of the way patterns fill a space), along with various forms of fractal analysis to thoroughly investigate datasets as it allows for dataset variation capture.
Conclusion
While there are a lot of complexities involved in fractals and multi-fractals, the simple difference between the two is as follows:
- Fractals are an infinitely generated, self-similar pattern.
- Multi-fractals make use of fractals, in conjunction with other analysis techniques where there is variation present in the pattern complexity. At this point regular fractals are inefficient in representing the data.
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