Quantum physics is too strange for many people to understand, and part of that strangeness is due to some of its counterintuitive features. For example, many quantum phenomena are subject to the Heisenberg uncertainty principle, making it impossible to know them with great certainty. According to this principle, we cannot obtain information about the position of a particle, say, until we actively check it.

This is different from, say, a soccer ball that has been kicked: we can calculate its position based on the information we get from Newton’s laws. In other words, obtaining information about a particle means collapsing its wave function. The wave function is a mathematical object that contains information about the particle, and ‘collapsing’ it means forcibly modifying it in a way that produces that information.

Before we obtain information about the location of a particle, however, it can be said that it is in more than one place and possibly in “contact” with other particles, even if they are physically very distant.

#### What are fractal dimensions?

Uncertainty is an inherent characteristic of all systems. It is independent of the precision or accuracy with which the system is measured. It simply exists, as an implicit element of the system’s existence. And it forced physicists to find a practical approach to studying quantum systems in order to get around the limitations it imposes.

One of the ways physicists responded was through so-called small non-integer dimensions, also known as fractal dimensions. The dimensionality of a quantum system is an important thing to keep in mind when physicists study its properties. For example, electrons in a one-dimensional system form a Luttinger liquid (not a liquid *by itself* but a model that describes the net behavior of electrons); In a two-dimensional system, particles exhibit the Hall effect (the conductor develops a lateral tension in the presence of a top-down electric field and a perpendicular magnetic field).

Obviously the question arises: how would a quantum system behave in non-integer or fractal dimensions?

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 omnipresent in nature, although it is also sometimes hidden from plain sight. A shape is a fractal if it presents self-similarity, that is, if parts of it on a smaller scale resemble parts on a larger scale. Such shapes can be easily produced by repeatedly modifying their edges using simple rules. Consider the Koch snowflake – a shape that starts as an equilateral triangle and, at each subsequent step, each side becomes the base of a new triangle. After many steps, a fractal snowflake appears.

The greater the “value” of a fractal’s dimension, the greater its ability to fill space as its shape evolves. For example, the Koch snowflake has a fractal dimension of about 1.26.

#### What are fractals like?

On the macroscopic scale, fractals can be seen as complex, irregular patterns at all scales and in all views, near or far. Some of the most notable examples of such patterns include the design of human fingerprints, tree stumps, snail shells, the human vein system, the network of rivers seen from above, the division of veins in a plant. leaf, the edges of a snowflake, lightning branching in different directions, the shapes of clouds, the mixing of liquids of different viscosities, the way tumors grow in the body, and so on.

There are fractals in the quantum realm too. In a study published in 2019, for example, researchers from Switzerland and the USA used X-rays to study the magnetic properties of a compound called neodymium-nickel oxide. They erased your magnetic order (the parts of your internal order imposed by magnetic fields) and then restored it. To their surprise, they discovered that parts of the interior of the material where the magnetization was in the same direction – called magnetic domains – had a fractal arrangement. They also found that the domains reappeared in almost the same positions they were in before they were erased, as if the material had a memory. All of these effects were due to the quantum physical properties of the material.

Another example of fractal behavior on a microscopic scale is available in graphene – a single-atom-thick sheet of carbon atoms bonded together. In this scenario, the surface density pattern of electrons has an almost fractal distribution.

#### What are the applications of fractality?

Historically, the first attempt to apply fractal analysis in physics was to Brownian motion – the rapid, random, zigzag movement of small particles suspended in a liquid medium, such as pollen in water. As such, the value of fractals is that they describe a new kind of order in systems that we would otherwise have overlooked. They open the way to possible new insights from otherwise familiar shapes, such as lines, planes and points, in the unfamiliar environment 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 antennas 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 bacterial 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 tree branches – they have far-reaching implications, even more so because they interact with different natural processes in a variety of environments. There are self-similar structures all around us that become increasingly complex over time. You just need to slow down and look closer, and you might glean some information that brings some quantum mystery into focus.

*Qudsia Gani is an Assistant Professor in the Department of Physics, Government College for Women, Srinagar.*

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