Nature’s Relationships : Fractals and Forests

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“Great fleas have little fleas upon their backs to bite ‘em,

And little fleas have lesser fleas, and so ad infinitum.

And the great fleas themselves, in turn, have greater fleas to go on;

While these again have greater still, and greater still, and so on.”

                                                                — Jonathan Swift:  1733

David George Haskell, in his book entitled “The Songs of Trees: Stories from Nature’s Great Connectors ” says:

“Virginia Woolf wrote that ‘real life’ was the common life, not the ‘little separate lives which we live as individuals.’ Her sketch of this reality included trees and the sky, alongside human sisters and brothers. What we now know of the nature of trees affirms her idea, not as metaphor but as incarnate reality . Like the union of leaf-cutter ants, fungi, and bacteria below the ceibo, a tree’s root/fungus/bacteria complex cannot be divided into little separate lives. In the forest, Woolf’s common life is the only life. Outside the laboratory, the complexity of the relationships among trees and other species increases by many orders. Decisions are made in these networks based on flows of information involving thousands of species. The chickadee’s culture looks simple in comparison. It is therefore not just the balsam fir tree that thinks but the forest. The common life has a mind. To claim that forests “think” is not an anthropomorphism. A forest’s thoughts emerge from a living network of relationships, not from a humanlike brain. These relationships are made from cells inside fir needles, bacteria clustered at root tips, insect antennae sniffing the air for plant chemicals, animals remembering their food caches, and fungi sensing their chemical milieu. The diverse nature of these relationships means that the tempo, texture, and mode of the forest’s thoughts are quite different from our own. The forest, though, also includes humans, chickadees, and other nerved creatures. A forest’s intelligence therefore emerges from many kinds of interlinked clusters of thought. Nerves and brains are one part, but only one, of the forest’s mind. “

Much has been written and said about fractals – those intriguing and highly irregular shapes that portray Nature’s objects and processes. Fractals are Nature’s geometric images. They are described as “self-similar” because they are endless inclusions of similar patterns within similar patterns, systems within systems.  If a system is self-similar, there is some feature that is constant at all scales of magnification. An object’s pattern looks the same close up as it does far away. This characterizes most natural systems such as trees, rivers, mountains, and the structure of mammalian lungs. Walk outside and look at a tree or sagebrush. There, you will see real-life representations of self-similarity.

The trees we see in Nature are self-similar. A magnified section of its branches looks about the same as the unmagnified tree. To illustrate, we can create a tree on paper or on a computer. We start with a vertical line that represents a single tree trunk with a length equal to 1. We grow the tree with the rule that, at each juncture, two branches are formed that are, perhaps, 60 degrees apart and may be half the length of the parent branch. We carry out this iteration multiple times. While we have created an idealized tree using an algorithm, keep in mind that a real tree contains algorithmic information within its genetic structure.

The term “fractal” was coined by Benoit Mandelbrot in 1975 and was derived from the Latin word “fractus” meaning “broken” or “fractured.” Fractal geometry is the geometry of irregular shapes that we find in Nature. It gives us the power to describe these natural shapes and to hint at unifying factors.

Self similarity is a very important aspect of Nature that provides unifying clues to the inter-connectivity that we see in Nature’s systems. Self-similarity means that as the magnification of an object changes, the shape or the geometry of the object does not change.

A fractal often has the following features:

• It has no characteristic size because it looks the same (at least approximately) at all magnifications. It is self-similar.

• A fractal is too irregular to be easily described in the traditional Euclidean geometric language that we learned in high school.

• Nature’s fractal structures are highly connected and hierarchical.

• While very complex, fractals can be defined by simple rules or recursive algorithms created by Nature’s genetic instructions.

Fractal theory provides a level of understanding about ecosystems because energy delivery systems within ecosystems are fractal. Energy transportation networks, like lungs, kidneys, and river systems, have fractal shapes because fractal shapes offer very efficient connectivity.

A good example is a mammalian lung. Alveoli are the tiny pockets in our lungs that store air for brief periods to allow time for oxygen to be absorbed into the blood-stream. In order to permit the absorption of sufficient oxygen into the bloodstream, the alveoli must have a very large total surface area. Human lungs contain 300 million alveoli with a surface area of 160 square meters — the size of a singles tennis court. The volume of a human lung contained in the chest cavity is only about 6 liters. So, this huge surface area is contained within this relatively small volume. This can happen only because the geometry of the lung structure is a system of convoluted fractal, self-similar surfaces. Much like crumpling up sheets of paper into balls and stuffing them into a bag. 

The efficiency of the lungs in diffusing oxygen from the inhaled air into the bloodstream is directly proportional to the available surface area. So, for a given volume of lung, it is highly advantageous to maximize the complexity of the structure so that the surface area is maximized. The highly complex fractal arrangement of the alveoli structures serves to maximize the surface area of the lung.

The same can be said about the ability of trees to absorb carbon dioxide and release oxygen. Think of a tree as an upside-down mammalian lung. Through the complex system of fractal branches and the leaves that are attached to these branches, trees and forests are highly efficient lungs of the earth.

Author David Haskell, in his quote at the beginning of this essay, describes Nature’s fractal, self-similar trees as “Nature’s Great Connectors”. His description is accurate and profound because self-similarity describes the strong relationship between the process of forming self-similar objects and the hierarchical connectivity of these objects. Self-similarity is connectivity. One segment of a self-similar system is created from a prior segment. A tree is composed of a hierarchical structure where a twig is grown from a branch, a branch is grown from a trunk, and so on. The result is a highly efficient system of energy flow conduits that bring energy from the sun and transport that energy throughout the plant. While this is going on, the tree transforms the sun’s energy into useful energy to be used by the plant and by the creatures who eat the plant. The fractal structure also serves as a lung that transforms carbon dioxide into oxygen for use by most life on Earth.

The fractal properties of these energy networks within any organism transcends all organisms. Without the fractal relationships that we have described, none of this could happen.

For Your Further Consideration

  • Our earth is a living system that transports and transforms the energy necessary for all life to exist. The key to an active group of ecoliterate humans that results in a healthy environment for all life on earth is the building of a systems view of life into the minds and hearts of humanity – particularly our youth. This worldview (the “Living Earth Story”) is supported  by the fact that all of Nature is interconnected and interdependent.
  • Environmental educators,  their students, scientists, and all stewards of Nature  are a powerful progressive force that, through their knowledge about Nature, through the legacies that they create for the future, and through their informed actions are capable of overseeing the well-being of our home —  Mother Earth
  • Environmental education is not simply offering facts. Environmental education must include the acts of passing a worldview of a Mother Earth on to Environmental education must be hands-on, and action-based if ideas, facts, and effective conservation strategies are to become a consciousness in the minds and hearts of all of our youth.
  • This website offers a free PDF book entitled “Empowering Stewards of Nature – Lessons From The Web of Life”. The book offers education methodology and content for creating Nature’s “Living Earth Story” within our youth and all stewards of Nature.. To download this book, follow the instructions on the right side of the web-site when you click the photograph of the book. 
  • If you are interested in working with me, other environmental educators, and other stewards of Nature to build a legacy of young people who will embrace and evangelize the worldview that “Everything on Earth is Connected and Interdependent”, please provide your questions and comments in the space provided below or by contacting me at my Twitter account @ballenamar.


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My name is Bill Graham. As a Marine Biologist who has worked in the US and Mexico for 30 years, I am a student of Nature, a teacher, a researcher, and a nature photographer. Through my work, I have acquired an ever growing passion for how everything in Nature is connected. Today, I travel extensively contemplating about, writing about, and photographing Nature’s connections. I also work with conservation projects in the USA and Mexico and mentor talented youth.

3 thoughts on “Nature’s Relationships : Fractals and Forests”

  1. Dear Bill, You posted this in August but I have just gotten around to reading it. This is the “magical” stuff that first drew me to your blob. I confirms what I and others have always maintained that everything is in Divine Order. Thank you.

    1. Thank you so much for your encouraging words. From your email address, I see that you are somehow connected to Harvard University. I did my master’s degree there in 1965. It was one of the great intellectual experiences of my life !!!!

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