What are Fractals?

What are Fractals?

If you look around you right now, depending on where you are, you’re likely to see to two distinct types of shapes: 1) blocky, linear and smooth if you’re in a manmade environment; or 2) branching, uneven and irregular shapes if you’re in a natural one. Why is there such a difference between the appearance of manmade and natural spaces? Why does one tend to look smooth, while the other looks rough? It comes down to one word: fractals.

A Brief History of Fractals

At the beginning of the 20th Century, mathematicians Pierre Fatou and Gaston Julia discovered fractal patterns while looking at complex mathematical systems. Back then, these objects defied linear analysis; they were considered aberrations or scary mathematical monsters, with infinite depth and complexity. They weren’t very popular and were forgotten until the late Belgium mathematician Benoit Mandelbrot discovered them again while working at IBM labs in Armonk, New York in 1980.

Fractals Contain Imaginary Numbers

To distinguish fractals from ordinary objects, you should know that fractal sets are created by algorithms that, in addition to ordinary integer numbers, also contain so-called “imaginary numbers”. This allows fractals to behave in much more complex ways, and describe more complex systems than ordinary numbers.

The Behavior of Fractals

Mandelbrot was the person who coined the word fractal. He used it to describe the behavior of financial markets and telephone line noise. The word fractal is derived from the word Greek “fractus,” meaning “fractured.” Mandelbrot noticed that telephone line noise is similar, whether you look at it over the course of an hour, a minute, or a second: you still see the same wave-form shape. In this sense, you can describe telephone line noise with a numerical dimension that applies at any time scale. The dimension defines the visual “roughness” of the signal; in other words, the dimension translates to how choppy it looks.

This is a very different type of geometrical logic than the one we were taught in school, where objects have a definite length and size. This is because, in school, we’re dealing with abstract objects that we imagine are perfectly linear and smooth. Nothing in the real world really looks like that!

If you take a look at almost anything natural under a microscope, you’ll see it’s full of fissures, pits and holes.

That’s because natural things are seldom perfectly flat beyond a certain scale. The closer you look, the more defects you’ll see.

Self-Similarity: Nature’s Preference

Nature seems to have a favorite type of shape: nature prefers that a structure looks similar whether you look at it close-up or from a distance. This is known as “self-similarity” and you can see it in almost any natural object or process you look at. You can see it in clouds, mountain ranges, coastlines, trees, your own hair, rivers, blood vessels, or the branching shape of your lungs.

In living things, fractal shapes allow them to maximize their surface area for exchanging energy or nutrients.

If all of your blood vessels were laid out end to end they would wrap around our Earth’s equator! Similarly, a fractal structure allows a tree to maximize the sun’s exposure to its leaves. There are many more leaves than branches and more branches than trunks.

Beyond Euclidian Geometry

Self-similar objects are those that look the same at any scale. They have lots of branches, are rough rather than smooth, and also tend to be very resilient to disturbances in their environments. The key characteristic of these fractal shapes is that, contrary to manmade things, they are not linear. Thus, they cannot be measured according to the principles of ordinary Euclidian geometry, which is the basis of modern geometry. You can’t get a definite measure of their length because their shapes do not resolve to a simple line.

The Challenge With Scale

A famous fractal example that Mandelbrot wrote about is the relation of fractals to the length of the coastline of England. While a standard automobile map would give a standard distance between two beaches on England’s coast, this is actually a fictitious number. The actual length, if you walked it, would be longer than the map shows because, in reality, you’re going to encounter rivers, inlets, eroded areas, and detours.

And if you were the size of a grain of sand walking along England’s coast, you would find even more spaces and detours, mainly between all the other sand grains that are larger than you. The smaller your ruler or yardstick, the greater the distance between two points on a self-similar, fractal object. This is because the coastline of England is not a straight but so-called “curvilinear.” (In case you are wondering, the fractal dimension of the West Coast of England is about 1.25: more dimensional than a straight line but less than a 2-D flat surface.)

Compare this with a Euclidean object like a flat sheet of glass. It’s smooth, with definite measurements of the distance between two points on the glass, whether measured in yards, feet, inches or even millimeters. It’s all the same distance at any real-world scale.

What is a Fractal Dimension?

Fractal objects are characterized as being “scale-free.” There’s no definite scale that defines them and their appearance looks the same no matter what the scale is. You can’t measure their absolute length but you can get a sense of their “roughness” and that’s what a fractal dimension is.

Clouds typically have a fractal dimension around 2.2 to 2.3, if you’re curious. That’s because clouds fill up more than a flat 2-dimensional surface like a tabletop, but are not entirely 3-dimensional because there’s so much space between the water drops. So they are actually objects that exist between dimensions! From the point of view of fractal geometry, clouds are truly interdimensional beings, as are many other natural things.

Take a tree, for example. The branching structure, as you move from the main trunk to the leaves or needles, become finer and smaller. You have one trunk but many branches and even more needles or leaves. The tree doesn’t completely fill it’s 3-dimensional space, as would a solid cube, but it’s much more than a flat surface. It’s an object between dimensions.

Fractal Flattening: The Hallmark of Modernity

Modern societies attempt to make things flat because they’re more manageable that way. Think of a highway, a dam, or an airplane propeller. Flatness adds to something’s predictability and controllability. I call this process “Fractal Flattening” and it’s a hallmark of modernity.

The author Dr. Edwin Abbott wrote a funny parody of this idea in 1884 in his book Flatland: A Romance of Many Dimensions. Abbott was making fun of Victorian England and its rigid attitudes. The story itself is about a society of flat shapes that don’t want to believe in or acknowledge the existence of complex shapes from Spaceland because they find them threatening to their social order.

Flattening Impacts Resiliency

The challenge and main drawback of flattening everything is that you reduce the amount of resiliency to sudden change and make things more vulnerable to catastrophes. For example, a fractal object like a tree can take a lot of wind as you see in a hurricane. Even if some branches snap off, you’ve still got all the other branches and the tree will grow back. It’s self-repairing, and so is the forest as a whole.

Now compare that to a manmade electrical grid with power lines going everywhere. If a few of those tree branches fall down on just a few powerlines during a hurricane, entire neighborhoods will lose power, sometimes for quite a long time. The system is not self-repairing and in fact, quite fragile.

Antifragile Systems Benefit From Stress

The author Nassim Taleb describes manmade systems that are vulnerable to stress as “fragile”. Conversely, those that benefit from disorder are “antifragile”. Antifragile systems are often characterized by fractal geometry, not linear structures. These anti-fragile systems benefit from chaos and grow stronger. (You can read more in Antifragile: Things That Gain from Disorder.)

The medical principle of hormesis states that some biological and living things can become stronger from stress. For example, small doses of a toxic agent can actually lead to immunity to that same agent. What is a poison at one dose can be a preventative medicine at another. Thus, fractal systems, because they are non-linear, can benefit from types of chaos that would topple manmade systems. In this way, stress can be beneficial to fractal systems.

Technology and Fractals

More recently fractals have been used to create the JPEG algorithm that is used to send pictures from one computer to another. The algorithm identifies self-similar patterns in pictures and allows them to be compressed. Similarly, fractals are used in satellite data compression. If you can identify repeating patterns in the data, you can summarize these as fractal equations which are faster to send wirelessly than every pixel in the raw picture.

In fact, modern cell phones use a fractal antenna shape to be able to work at different frequencies at the same time. Within the antenna are different shapes that correspond to the various radio bands that carry cell phone signals. These shapes are nested within one another so that even in a cell phone, many different frequencies are simultaneously at use. Research shows that fractal antennas perform at a higher efficiency than regularly shaped antennas like the old-fashioned whip antennas that used to be installed on cars.

Fractal Meditation

I often find it useful to meditate on fractal shapes when I’m walking or just relaxing. You’ll find if you do this it will help you feel connected to the larger natural forces that are all around you. It’s a great way to unwind and feel like your natural self.

Just spend a few minutes every day in the most natural environment you can find, or imagine one if you’re in a completely urban area. First, see and appreciate the branching structures you see around you whether they be trees, mountains, clouds, rivers, coastlines or entire galaxies.

Then visualize your own fractal structures like your hair, blood vessels, lungs, or beating heart which has its own unique fractal rhythm. Then imagine a connection between the fractal structures of your body and those in nature. Feel a sense of gratitude for this amazing geometry that nature has created for your body and your environment. Marvel at how well it works to keep you alive every moment of every day. And then see your boundless awareness stretching to the limits of the universe along with these fractal networks of matter and energy.

If you are struggling to call fractals around you to your mind’s eye, don’t distress. Watch this meditation series, featuring an enchanting flow of fractals.

Fractals and the Multiverse

One of my Taoist teachers, Mantak Chia, used to tell us that the structures that make up our body, like our hair, served as antennas to other worlds and dimensions.

He said we could pull in Chi energy from our environment and cautioned us not to cut our hair too short. In a sense, he was referring to how fractal shapes serve double duty as physical and resonant structures. In a sense, he was suggesting that we’re all fractal antennas!

We know that nature doesn’t waste anything. Wouldn’t it be amazing if all those fractal shapes we see in and around us not only contribute to physical resiliency but also serve as resonant antennas to parallel realities? In other words, the fractal shapes of trees allow them to collect a lot of energy from the sun. Perhaps it also lets them communicate with other tree worlds in parallel Earth realities. This is purely speculation on my part.

Take time to appreciate all the fractal structures in your life and how they support life in so many ways. And then the next time you’re designing something, consider whether it needs to be so straight, smooth or even. Perhaps your next project, task or work goal could benefit from some of nature’s fractal logic.



What Are the Platonic Solids, Pillars of Sacred Geometry?

Platonic solids are highly symmetrical three-dimensional geometric figures that have fascinated mathematicians, philosophers and mystics for centuries. They are considered the fundamental building blocks of creation in the universe, representing essential elements and profound spiritual concepts. In this article we explore what Platonic solids are, their origin, their spiritual significance and their connection to sacred geometry.

Table of Contents

What Are Platonic Solids?

Platonic solids are five geometric figures: the tetrahedron, the hexahedron (cube), the octahedron, the dodecahedron and the icosahedron. Each of them has equal faces, edges and angles, which gives them perfect symmetry. These geometric shapes were studied by the Greek philosopher Plato, who associated them with the fundamental elements of nature.

These solids are not only mathematically interesting, but also have deep significance in sacred geometry. They are believed to be the basic patterns that form the structure of the universe, from subatomic particles to galaxies. The symmetry and balance of Platonic solids reflect the inherent harmony of the cosmos.

In the series Sacred Geometry: Spiritual Science available on Gaia, Dr. Robert J. Gilbert explores how Platonic solids are essential to understanding the structure of the universe and our own inner energies. Through his teachings, he reveals how these geometric shapes can be used in spiritual and meditative practices to enhance spiritual awareness and connection.

Where Does the Name of Platonic Solids Come From?

The name “Platonic solids” comes from the Greek philosopher Plato, who described them in his work “Timaeus”. Plato linked each of these solids to the five classical elements: earth, air, fire, water and ether. According to Plato, these elements formed the basis of all matter in the universe, and the Platonic solids represented the perfect structure of each of these elements.

Plato was not the first to discover these forms, but he was the one who popularized them and gave them philosophical and cosmological significance. His association of solids with the fundamental elements influenced philosophy and science for centuries, highlighting the importance of geometry in understanding the physical and spiritual world.

One by One: The 5 Platonic Solids

The five Platonic solids are the tetrahedron, hexahedron (cube), octahedron, dodecahedron and icosahedron. Each of these solids has equal faces, edges and angles, which gives them perfect symmetry.

solidos platonicos

  • The Tetrahedron

The tetrahedron has four triangular faces, six edges and four vertices. It represents the element of fire and is associated with the solar plexus chakra. This solid symbolizes transformation and dynamic energy, and is used in meditative practices to manifest desires and promote mental clarity.

  • The Hexahedron (Cube)

The hexahedron, better known as a cube, has six square faces, twelve edges and eight vertices. It represents the earth element and is related to the root chakra. The cube symbolizes stability and solidity, providing a firm and secure foundation on both the physical and spiritual planes.

  • The Octahedron

The octahedron has eight triangular faces, twelve edges and six vertices. This solid is associated with the air element and the heart chakra. It represents balance and harmony, and is used to improve communication and promote mental and emotional clarity.

  • The Dodecahedron

The dodecahedron has twelve pentagonal faces, thirty edges and twenty vertices. It represents the ether or the universe and is associated with the crown chakra. This solid symbolizes connection with the cosmos and exploration of the mysteries of the universe, encouraging spiritual expansion and higher consciousness.

  • The Icosahedron

The icosahedron has twenty triangular faces, thirty edges and twelve vertices. It represents the water element and is related to the sacral chakra. This solid symbolizes fluidity and adaptability, promoting creativity and emotional expression in spiritual practices.

Platonic Solids in Sacred Geometry

In Sacred Geometry, the Platonic solids are considered the most basic and fundamental forms that make up the structure of the universe. These geometric forms, with their perfect symmetry and proportions, are seen as the building blocks of the cosmos, used at various levels of creation, from microscopic to macroscopic structures. Their symmetry reflects the harmony and order inherent in nature.

Each platonic solid has unique properties that make them powerful tools in meditation and energy practices. For example, the tetrahedron can be used to focus intention and manifest desires, while the cube helps to stabilize and ground energy. The octahedron promotes mental and emotional clarity, the icosahedron enhances fluidity and creativity, and the dodecahedron facilitates connection with the divine and expansion of consciousness.

The application of Platonic solids in sacred geometry is not only limited to spiritual practices; they are also found in art, architecture and science, demonstrating their relevance throughout the ages. These forms serve as bridges between the physical and the spiritual, providing a solid foundation for exploring and understanding the energies that surround us and the structures that make up our universe.

History and Origin of Platonic Solids

The history of Platonic solids dates back to ancient Greece, where they were first studied by the followers of Pythagoras and later described in detail by Plato in his dialogue “Timaeus”. Plato related each of these solids to the fundamental elements of the universe, attributing deep cosmological meanings to them and using them to explain the structure of matter and the cosmos.

In addition to Plato, other Greek mathematicians and philosophers, such as Euclid, also studied and formalized the geometric properties of these solids. Euclid, in his work “Elements”, defined and proved the properties of the Platonic solids, establishing a solid basis for geometry that would influence mathematics for centuries.

During the Renaissance, interest in Platonic solids resurfaced, especially through the work of the astronomer Johannes Kepler. Kepler used these solids to explain planetary orbits in his “Mysterium Cosmographicum”, proposing that the distances between the planets were related to the Platonic solids. Throughout history, these geometric shapes have continued to inspire scientists, artists and mystics, establishing themselves as symbols of harmony and perfection in the structure of the universe.

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