Patterns in Nature: How to Find Fractals
Our current feature exhibtion, A Mirror Maze: Numbers in Nature, takes a close look at the patterns that appear in the world around us. Did you know that mathematics is sometimes called the “Science of Pattern”? Think of a sequence of numbers like multiples of 10 or Fibonacci numbers—these sequences are patterns. In a way, whenever you observe a pattern in the world around you, you are doing math!
Patterns can be seen everywhere: in animals, vegetables and minerals. Have you ever observed the similarity between the shape of your lungs and the structure of a tree? Or maybe the pathways of lightning and the way a river breaks through the earth? These patterns are called fractals.
A fractal is a kind of pattern that we observe often in nature and in art. As, “whenever you observe a series of patterns repeating over and over again, at many different scales, and where any small part resembles the whole, that’s a fractal.”
Fractals are exciting, not only for their mathematical or conceptual representation, but also for the fact that you can visualise the math—and it’s beautiful!
The repetition that occurs in a fractal is called “self-similarity”. Another way to think of this is, when you zoom in on a small part of a fractal pattern, it looks just like the whole thing. One of the most famous fractals is the Mandelbrot Set.
The Mandelbrot Set refers to a fractal that a man named Benoit Mandelbrot generated from a simple mathematical equation with the help of computers. You may recognize the resulting images as a type of visual art that was particularly popular in the 1980’s when the Mandelbrot set was first identified. If you were able to zoom into the image below indefinitely, you would find that the pattern keeps on repeating infinitely.
The visualization of the Mandelbrot set shows that very complex, entirely unexpected structures can result from very simple mathematical rules. This understanding has applications that range from creating realistic computer graphics to modelling weather and financial markets.
A lung, lightning strike, or a branch are examples of a fractal that was studied even earlier than the Mandelbrot set, the Lichtenburg figure. These patterns were first studied by sending electrical currents through various materials and observing the resulting patterns. As voltage travels through a material, over time, the currents leak causing spreading, or branching into tree-like formations.
In fact, you can observe similar patterning in many natural phenomena, like roots, rivers, electrical currents and organs in the body.
Even though a fractal is, by definition, an infinite pattern and cannot be measured, the Koch snowflake lets us see that even though the perimeter of a fractal is infinite, the area is not. As you zoom into the edges of the snowflake, you would find that there are ever new emergences of the pattern, but the size of the snowflake itself doesn’t change.
This kind of fractal is commonly found in nature when we observe coastlines. You can’t really get an exact measurement of the land mass on Earth because the edges are not smooth, they are rough and variable, the Koch snowflake is a way of showing how the infinite irregularities can still be contained within an approximation of the whole.
What are some fractals that you have observed in nature? Have you ever seen fractals in art?
Here are some fractal references:
And a fractal activity: