Fractals in Nature
- Lokaditya Ryali
- Sep 25, 2020
- 4 min read
Have you ever found yourself doodling arbitrary repeating patterns (Fig. 1) while you were immersed in a deep thought or were bored? If yes, then let me tell you that these random patterns you were involuntarily sketching could very well be fractal geometries that have a whole separate branch of mathematics associated with them. The most basic definition of a fractal is that these are geometric patterns that repeat themselves at different scales. These are so basic that they can be generated using simple mathematical equations in an iterative fashion. Self-repeating geometries have been observed in nature for quite a while, but it was Benoît Mandelbrot in 1975 who had officially coined the name fractals for such patterns (derived from the Latin word "fractus," which meant broken or shattered glass).
Figure 1. Step by step generation of different example fractal geometries.
The concept of fractals has always fascinated me because of their simplistic origins yet complex final facets. Therefore in these series of blog articles, I plan to provide an overview of different aspects related to fractals such as fractals observed in nature, the math behind them, and their practical applications. In the current article, we will discuss various examples of fractals occurring in nature by classifying them into four major categories based on their scale. We will use a trickle-down approach by discussing examples from cosmic to the quantum scale, as shown in Fig. 2. I have intentionally tried to keep this article abstract in nature so that we get a sense of appreciation for the concept of fractals.
Figure 2. Different scales in nature.
Cosmic Scale:
Figure 3. Examples of fractals at the cosmic scale.
There is a separate division in the field of cosmology known as fractal cosmology, which investigates if the distribution of matter in the universe or the structure of the universe itself, is a fractal across a wide range of scales. Though the idea sounds very abstract and philosophical, there are several interesting examples which would make us rethink our understanding of the universe. Few of which are illustrated in Fig. 3.
Though not concretely proven, there have been studies that suggest that galaxies exhibit an explicitly fractal pattern up to a scale of about 100 million light-years. Furthermore, if published data is to be believed, even if the universe does become homogeneous at some point, it has to be on a scale larger than a staggering 300 million light-years across. The irony is that these scales are so huge that there hasn’t been enough time since the big bang nearly 14 billion years ago for gravity to build up such large structures. Fig.3a provides an illustration of a galaxy being represented by a Fibonacci spiral.
Researchers have shown that the complex and elegant rings of Saturn have a fractal structure, whose geometry can be explained using Cantor sets (Fig. 3b).
Human Scale:
Figure 4. Examples of fractals at the human scale.
There are many things in our observable daily life that have a fractal structure associated with them, few of these examples are shown in Fig. 4.
Each individual floret in cauliflower/broccoli tend to have a self-similar structure (Fig. 4a).
Snowflakes have a fractal structure (Fig. 4b). One of the common techniques that can be used to generate the complex looking geometry of the snow flake is the Von Koch’s snowflake curve shown in Fig. 5.
The human nervous system, bronchioles in the lungs (Fig. 4c), and blood vessels have a branching structure that repeats itself at different scales. Such a fractal-like structure allows them to be tightly packaged. For instance, it is estimated that the bronchioles in our rib cage constitute a surface area of a tennis court, while almost 60K miles of branching blood vessels and capillaries are packed into a human.
Similar to blood vessels and nerves, river bodies tend to have branching networks that sometimes demonstrate fractality (Fig. 4d).
Named after the British mathematician Michael Barnsley, the Barnsley fern is a fractal. Fig. 4e illustrates the four states of construction of the fractal fern. Highlighted triangles show how the half of one leaflet is transformed to half of one whole leaf or frond.
Large eddies formed in a turbulent flow break down into smaller ones, which in turn split into even smaller ones, in a fractal fashion (Fig. 3f).
The human heartbeat tends to have a statistically self-similar structure at different time scales (Fig. 3g).
Figure 5. Von Koch’s snowflake curve.
Micro/Nano Scale:
Figure 6. Examples of fractals at the micro/nano scale.
We all have heard about Brownian motion in our high school, which is basically the random motion of particles suspended in a medium. There have been statistical studies to find patterns even in these random motions and explore their fractality (Fig. 6a).
Fig. 6b shows a 3-D image of how a DNA packs itself tightly into a structure known as a fractal globule.
The micron-scale surface profile of any material has a certain degree of fractality associated with it (Fig. 6c).
Quantum Scale:
Figure 7. Example of fractals at the quantum scale.
Considering all the examples we have discussed so far, it would not be surprising if fractals are observed at the quantum level as well. In fact recently, a group of researchers from Princeton University have shown that this is indeed the case. Yazdani and his team were able to observe fractal patterns in the electron field interactions of a semiconductor material during its transition phase from metal to an insulator, using a scanning tunneling microscope (STM).
Gosh!! this article turned out to be longer than I expected it to be, but I hope that I was able to generate some level of curiosity in you about the concept of fractals. That being said, I am sure if you happen to be from an applied field you would be feeling, well all these patterns occurring in nature are fascinating and soothing to the eye, but what is their practical significance, like why do we even care if these things are self-repeating!!. Mandelbrot strongly believed that fractals are a much better way to model the natural world as nothing in the real world resembles perfect shapes, and there is a degree of roughness everywhere. In my upcoming blog articles, I will try to provide a brief context of the math behind fractals and their applications.
Clouds are not spheres, mountains are not cones, coastlines are not circles, and bark is not smooth, nor does lightning travel in a straight line — Mandelbrot
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