What are fractal dimensions?
It is independent of the precision or the accuracy with which the system is measured.
As an implicit element of the system’s existence.
And it has forced physicists to find a practical approach to study quantum systems in ways that can circumvent the limitations it imposes.
One of the ways in which physicists have responded is via the so-called non-integer small dimensions, a.k.a. fractal dimensions.
The dimensionality of a quantum system is an important thing to bear in mind when physicists study its properties.
For instance, electrons in a one-dimensional system form a Luttinger liquid (not a liquid per se but a model that describes the electrons’ liquid-like behaviour); in a two-dimensional system, the particles exhibit the Hall effect (the conductor develops a side-to-side voltage in the presence of a top-to-bottom electric field and a perpendicular magnetic field).
Physicists use the fractal geometry approach to study quantum systems in dimensions like 1.55 or 1.58, or in fact anything between one and two dimensions.
Fractality is ubiquitous in nature, if also sometimes hidden from plain sight.
A shape is fractal if it exhibits self-similarity, i.e. if parts of it at a smaller scale resemble parts at a larger scale.
Such shapes can be easily produced by repeatedly modifying its edges using simple rules.
Consider the Koch snowflake – a shape that begins as an equilateral triangle, and in each subsequent step, every side becomes the base for a new triangle. After many steps, a fractal snowflake appears.
The higher the ‘value’ of a fractal’s dimension, the greater is its ability to fill space as its shape evolves.
For example, the Koch snowflake has a fractal dimension of around 1.26.
Applications of fractality
The first attempt to apply fractal analysis in physics was for Brownian motion – the rapid, random, zigzagging motion of small particles suspended in a liquid medium, like pollen in water.
The value of fractals is that they describe a new kind of order in systems that we may have otherwise overlooked.
They pave the way to potential new insights from otherwise familiar shapes like lines, planes, and points, in the unfamiliar milieu of a space with non-integer dimensions.
Researchers have also used the concept of fractality in data compression, such as to reduce the size of an image when storing it, and to design more compact antennae without compromising their performance.
Some have also used fractality to study patterns in galaxies and planets and, in cell biology, to make sense of some bacteria cultures.
Fractal geometry has also found applications in chromatography and ion-exchange processes, among others.
Fractals are rooted in geometry but – like the fractal growth of branches on trees – they have far-reaching implications, more so as they interact with different natural processes in a variety of settings.
There are self-similar structures around us that become increasingly complex with time.
You just need to slow down and look closer, and you might just glean some information that brings some quantum mystery into focus.
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