What is a Fractal?
Fractal patterns, like many other concepts of mathematics, are full of paradoxes. A source of creativity and beauty, a fractal structure can also be a powerful tool to analyze and communicate complex ideas.
In mathematics, this property of an object is called self-similarity. Shapes that exhibit self-similarity are known as fractals. Fractal patterns are deemed as the most beautiful and exquisite structures produced by nature and are present all around us. We have an abundance of fractal geometry in nature like hurricanes, trees, mountains, rivers, seashells, coastlines, the edge of a snowflake, and many others.
These images could look very complex at first glance but if you take a closer look, both of these objects of nature follow a rather simple pattern. Each individual part of the plant looks exactly similar to one another and to the entire plant, just smaller in size. The same pattern is repeated all over at smaller scales.
There are many intricate plants and parts in nature like the ones shown below:
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Let us try to demystify fractal geometry in this article where we will understand what are fractals, learn about fractals math, and know types of fractals.
Fractal Definition
The fractal theory could be a little tricky to explain in a precise manner but most fractal patterns are linked by a set of common features:
Infinite intricacy
Zoom symmetry
Fractional dimension
Complexity from simplicity
We will now explain each of these features by considering the example of a fern. A fern is a great illustration of all the features of a fractal structure. A fern’s shape is intricately detailed where you would notice that the leaves are shaped like exact copies of branches but smaller in size. In other words, the whole fern is built by repeating the same basic shape over and over at ever smaller scales.
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Infinite intricacies - Polish mathematician Benoit Mandelbrot in his 1977 essay “The Fractal Geometry of Nature” wrote, “Nature exhibits not simply a higher degree but an altogether different level of complexity”. The first fractal was discovered in 1861 which created a massive stir in the mathematical community. A zigzag was constructed by German mathematician Karl Weierstrass which was extremely zagged. Even if you magnified the shape to the highest degree, you could only see a cascade of corners. Until modern computing, this zigzag remained just a glimpse of a completely new shape that was unravelled much later.
Zoom symmetry - Symmetry of an object is defined as an action, when applied to a shape, leaves it looking more or less the same. For example, a square shape has rotational symmetry since when we rotate it by 90 degrees there is no way to tell that the square had been spun. Fractal geometry blossomed into a new branch of mathematics largely due to Benoit Mandelbrot who worked for IBM in the 1960s. He had IBM’s immense computing power at his disposal and gave the most famous fractal structure called the Mandelbrot set. It had a completely new level of symmetry which is not present in ordinary shapes. If you zoom in to a small portion of a fractal, you would be amazed that you are looking at the same shape that you started with.
Complexity from simplicity - Despite the complexity of fractal structures, the shape arises from extremely simple core definitions. A small list of mathematical mappings can capture the shape of a fractal. These mappings tell us exactly how the small copies of a fractal are arranged to give the whole bigger fractal.
Fractals start with a simple geometrical object and a rule for how to change the object. This finally leads to objects which are very complex and their dimension is not an integer. British mathematician Michale Barnsley, in his 1988 book “Fractals Everywhere” says that “Fractal description of an object is a story about how the object grows” reminding us that science is not just about facts and objectivity. Science does have a narrative component also that we ignore quite often. Fractals remind humans that stories also are important. There is an algorithm in Barnsley’s book called the “Chaos Game” that lets computers generate a fractal quickly from its mappings. With its starting point in space, the chaos game tracks its motion while hopping around. Each of the hops is decided upon by selecting one of the mappings randomly.
Fractional Dimensions - The most striking feature of fractal patterns is that they are somewhere in between the normal one, two, or three-dimensional figures. We can define the dimensions of a shape as a measure of how rough the shape is or its size in one direction or how well the shape fills its surrounding space. To illustrate fractal maths imagine a piece of paper that has 2 dimensions (length and breadth). A solid sphere has 3 dimensions (length, breadth, and height) and it will fill more space than a piece of paper. Now if you crumble the paper into a ball, it will resemble a fractal shape which fills more space than the paper but not as much as the solid sphere. In the same way, the fractal geometry of our lungs allows them to pack a huge surface area (spanning a few tennis courts) into a small volume which is just about a few tennis balls. Since so much of the area is packed into such a small space, we are able to extract enough oxygen to keep us alive.
Any geometric pattern which occurs, again and again, makes a fractal structure. Fractals are everywhere around us in nature like ferns, trees, river networks, forks of lightning, etc. Fractal geometry could not be described by Euclidean geometry as it is an amalgamation of self-similar shapes that follow a simple and recursive definition.
FAQs on Fractal
1. What is a fractal in simple mathematical terms?
In mathematics, a fractal is a geometric shape containing a never-ending pattern that repeats at different scales. A key feature is self-similarity, which means that if you zoom in on any part of a fractal, you will see a smaller version of the whole shape. Unlike simple shapes like circles or squares, fractals describe complex and irregular objects found in nature.
2. Where can we see examples of fractals in nature and everyday life?
Fractals are surprisingly common and can be observed all around us. Some classic examples include:
Snowflakes: Each branch of a snowflake has smaller branches that mimic the larger structure.
Lightning Bolts: The main bolt branches into smaller, similar-looking bolts.
Ferns: Each frond of a fern is a miniature replica of the entire leaf.
Coastlines: The jagged pattern of a coastline looks similarly complex whether viewed from a satellite or up close.
Romanesco Broccoli: This vegetable grows in a natural, stunningly perfect fractal pattern.
3. How are fractals different from standard geometric shapes like squares or circles?
The primary difference lies in their complexity and dimension. Standard Euclidean shapes (like squares and circles) are smooth and have integer dimensions (e.g., a line is 1D, a square is 2D). In contrast, fractals are infinitely complex and rough. No matter how much you magnify a fractal, its complexity does not simplify. This property is described by a fractal dimension, which is often a non-integer value, indicating how it fills space.
4. What does the term 'self-similarity' mean in the context of fractals?
Self-similarity is the core property of a fractal where a part of the object is an exact or approximate copy of the whole. Imagine a large tree. It has big branches, which in turn have smaller branches, which have even smaller twigs. This repeating pattern at different scales is a perfect example of self-similarity. This allows immense complexity to be generated from a simple, repeating rule.
5. Why are fractals so important for computer graphics and image compression?
Fractals are vital in computer technology because they can generate highly complex and realistic-looking natural scenery (like mountains, clouds, or coastlines) from simple mathematical formulas. Instead of storing data for every single point of a complex image, a computer only needs to store the fractal equation and a few parameters. This makes fractal image coding (FTC) extremely data-efficient for creating detailed graphics and compressing images.
6. Can a shape have a finite area but an infinite perimeter? How do fractals make this possible?
Yes, and this is a classic paradox illustrated by fractals like the Koch Snowflake. You start with a simple shape (an equilateral triangle) and iteratively add smaller triangles to each side. At each step, the total area increases but remains contained within a finite boundary. However, the length of the perimeter increases with each iteration and mathematically approaches infinity. This demonstrates how fractals can possess counter-intuitive properties not seen in simple geometry.
7. Are there different types of fractals?
Yes, fractals can be categorized based on how they are generated. The main types are:
Escape-time Fractals: These are defined by a mathematical formula for each point in a space, such as the famous Mandelbrot Set and Julia Set.
Iterated Function Systems (IFS): These have a fixed geometric replacement rule, like the Sierpinski triangle or the Koch snowflake, where a shape is repeatedly replaced by smaller copies of itself.
Random Fractals: These are generated using stochastic (random) processes, which are useful for modelling natural phenomena like landscapes or coastlines.
8. How does the 'three-quarter power law' in biology demonstrate fractal principles?
The three-quarter power law relates an organism's metabolic rate to its body mass. This relationship is a direct consequence of the fractal geometry of circulatory and respiratory systems. Organs like lungs and kidneys use fractal branching networks to maximize their surface area for efficient exchange of oxygen and nutrients within the limited volume of the body. This space-filling, efficient design allows a small percentage of body volume (like the 3% for arteries) to service every single cell, a key principle of fractal efficiency.
