Fractal Dimension

News excerpt:

Physicists use fractal geometry to study quantum systems in dimensions like 1.55 or 1.58 or anything between 1D and 2D.


  • Heisenberg’s Uncertainty principle:
    • The principle states that a particle’s position and momentum can’t be known simultaneously. This means whenever we measure an electron’s position, we won’t know its speed. When we measure its speed, we won’t know its position.
  • The dimensionality of a quantum system is a vital characteristic to bear in mind when physicists study its properties. 
    • For instance, electrons in a 1D system form a Luttinger liquid (not a liquid per se, but a model that describes the electrons’ liquid-like behaviour). 
    • In a 2D 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).

Behaviour of quantum system in fractal dimension:

  • Fractal dimensions refer to non-integer small dimensions.
  • A shape is fractal if it exhibits self-similarity, i.e., parts of it at a smaller scale resemble parts at a larger scale. 
    • Such shapes can be quickly produced by repeatedly modifying their 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 efforts, a fractal snowflake appears.
    • The Koch snowflake has a fractal dimension of around 1.26

What do fractals look like in nature?

  • Fractals are irregular, complex patterns at all scales and in all views. 
  • Some of the remarkable examples of such patterns include:
    • Design of human fingerprints, the stumps of trees
    • Shells of snails
    • System of human veins
    • Network of rivers as seen from high up
    • Edges of a snowflake
    • A bolt of lightning branching off in different directions
    • The shapes of clouds
    • The mixing of liquids of different viscosity
    • The way tumours grow in the body

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.
  • Researchers use 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. 
  • To study patterns in galaxies and planets
  • To study cell biology to make sense of some bacteria cultures. 
  • It is also used in chromatography and ion exchange processes.

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