Green Thumb to the Rescue - Science Matters

Science Matters

““To see a world in a grain of sand, or heaven in a wildflower.
Hold infinity in the palm of your hand, and eternity in an hour."

~ William Blake

Fractals in Nature

By Dr. Gordon A. Maclean

“Study the science of art. Study the art of science. Develop your senses – especially learn how to see. Realize that everything connects to everything else.”

― Leonardo da Vinci

For centuries observations of nature have revealed that there are certain patterns that nature uses seemingly repeatedly. In the 1970s, a new branch of mathematics, fractals, allowed us to describe what we see in nature in mathematical terms. Fractals are simultaneously very simple and very complex.

A fractal is defined as “a complicated pattern in mathematics built from simple repeated shapes that are reduced in size every time they are repeated” (https://dictionary.cambridge.org/us/dictionary/english/fractal)

If we start with a couple of very simple line drawings (a Von Koch curve and a Sierpiński triangle), it is possible to see how these patterns can develop.

Von Koch curve

From: https://www.sciencenewsforstudents.org/wp-content/uploads/2020/01/Von_Koch_curve.gif

Sierpiński triangle

From: http://fractalfoundation.org/resources/what-are-fractals/

I will not travel into the rabbit hole of fractal mathematics, which can become very complicated. Instead, I will concentrate on how fractals manifest themselves in our natural environment. At the end of this article, I have included several internet sites where you can see absolutely spectacular “fractal artwork” resulting from fractal mathematics called “Mandelbrot Sets” and allow you to create your own.

Mandelbrot Set. See the full (zoomable) image at https://upload.wikimedia.org/wikipedia/commons/2/21/Mandel_zoom_00_mandelbrot_set.jpg

If you read my previous article about Fibonacci sequences, you will find a bit of overlap between the two, with fractals being the broader context that Fibonacci’s work within. I will point these out as we go, but if you have not read that article, I encourage you to read that before continuing with this.

In nature, fractals are frequently on display if we know where to look for them. Trees, shorelines, major rivers, and mountains are all made up of smaller features indistinguishable from the larger if we have no idea of the scale of what we are seeing.

Take, for example, the following:

At first glance, given the structures and arrangement of materials, you could easily mistake this for a peak in the Alps or the Rockies. Take a step back, however, and the true picture reveals itself:

Both photos are courtesy of Paul Hepner, Marquette, Michigan.

In my Fibonacci article, I included a Nautilus Shell with the Fibonacci spiral overlain. If you look inside the Nautilus Shell, you will discover that it is constructed of a set of repeating chambers, with each chamber similar in structure to larger and smaller ones. The self-similarity between the chambers is the fractal aspect of the shell, and the size of each progression of spirals is the Fibonacci aspect of the shell.

Fibonacci aspect of the shell

https://dictionary.cambridge.org/us/dictionary/english/fractal

If you want to go looking for fractals yourself, and you are a gardener, plant some large leaf parsley, let it grow, and watch carefully over a period of a few weeks HOW its leaves and stems develop. From my experience, this plant is probably the easiest and fastest way to observe fractals, in action, in nature.

If you watch this parsley grow, you will see that it has a center (terminal) stem and two side (lateral) stems (see number 1 in the photo below.) If you let the parsley grow, you will see that the lobes of the leaves on both the terminal and lateral stems begin to deepen and become much more pronounced (see number 2 in the photo below.) Eventually, the lobe will deepen to the center vein of the leaf and “pinch-off,” forming new lateral stems.

If you look even closer at the leaves, you will see that the next generation of the lobe/stem process is already beginning (see number 3 in the photo.)

When you harvest the parsley, if you look closely, you will see many “generations” of this process on a single stem growing from the root, many laterals that turn into terminals in their own right, and many stages of the lobe development leading to another “generation” of the branching process.

Branching Process

Natural water drainage patterns are the best example of applied fractals in the larger landscape. Drainage patterns come in many basic patterns, all of which relate to the underlying geology with which they are associated.

Drainage Patterns

Howard, A.D. (1967) Drainage Analysis in Geologic Interpretation: A Summation. American Association of Petroleum Geologist Bulletin, 51, 2246-2259.

These drainage patterns are fractals, each with a specific pattern, and each branching pattern is unique from the others and follows self-specific “rules.”

The “dendritic” drainage pattern is perhaps the most dramatic and easily visible from space. In some areas of the following example, you can easily see four “generations” of the branching pattern.

Dendritic Drainage Pattern

https://www.britannica.com/science/dendritic-drainage-pattern#/media/1/157566/1258

Can you see other examples of fractals in the world around you? Look very carefully. Find the large, easily visible pattern, then look for it again and again at smaller and smaller scales on the same object. You will see it, and that beauty can be described mathematically.

How does the Fibonacci Sequence work into fractals? The Fibonacci Sequence is one mathematical expression of how the fractals work in nature. Romanesco broccoli is the most obvious example of fractals infused with the Fibonacci Sequence. This vegetable grows in a double spiral Fibonacci pattern. But this broccoli does not stop there. Each of the buds on the spirals (called meristems) comprises another “generation” of meristems with the same Fibonacci and fractal characteristics. The photo below makes it easy to see three “generations” of this pattern and maybe the start of a fourth.

Generations of Patterns

Because fractals are essentially mathematical, it is possible to convert these patterns into a set of “rules” and program those rules into a computer program. It is impossible to understate how these fractal rules of nature have been applied to our common culture, specifically in the field of graphic effects.

The snow image I presented above was a “real” example of how our brain, not knowing the scale of what we are looking at, can trick us into assuming what we are seeing. For movies and television shows, computer graphics can be used to trick us into believing that the actors are someplace VERY real looking that doesn’t exist except as a set of rules in a computer.

The first use of this technique was a one-minute segment in the 1982 movie “Star Trek 2: The Wrath of Khan.” The segment title “The Genesis Effect” was created using a series of fractals combined into a fly-over of a dead moon being transformed into a living planet may be seen here: https://www.youtube.com/watch?v=Tq_sSxDE32c

Since then, the use of fractals to create worlds, scenery, and background context far beyond what can be created physically has been incredible. In fact, you may no longer be able to identify “real” or “not real” if the background context is not obvious. And if it is obviously “not real,” enjoy and appreciate what you see with the knowledge that it comes from the foundational application of our real-life observations of the natural world.

Two fractal landscapes.

Fractal Landscape