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.
About the Author
7 Sacred Geometry Symbols and Their Meanings
Sacred geometryis a discipline that studies the geometric shapes and patterns found in nature and in human-made structures. These figures are considered sacred due to their perfect symmetry and proportions, which are believed to reflect the harmony of the universe. In this article, we explore the meanings of seven sacred geometry symbols and how they can influence our spiritual and daily lives.
Table of Contents
- What is Sacred Geometry and What is its Purpose?
- 7 Sacred Geometry Symbols and Their Meanings
- History and Origin of Sacred Geometry
- How to Integrate Sacred Geometry into Daily Life
What is Sacred Geometry and What is its Purpose?
Sacred geometry is the study of geometric shapes that appear in nature and human constructions, considered sacred due to their perfect symmetry and proportions. These shapes include structures such as the cube, tetrahedron, and dodecahedron, among others. It is believed that sacred geometry connects with the order and harmony of the universe, providing a deeper understanding of the structure of reality.
These geometric figures are found in a variety of contexts, from ancient architecture to modern art. Temples and cathedrals often incorporate sacred geometry patterns in their design, using these shapes to create spaces that promote meditation and spiritual connection. Additionally, these figures are also found in nature, such as in the structure of crystals and the shapes of plants and animals.
Sacred geometry is also applied in meditation and spiritual practices. By meditating on these figures, one can harmonize internal energies with universal energies, promoting healing and spiritual growth. Geometric shapes act as a bridge between the physical and the spiritual, facilitating a deeper connection with the cosmos.
In the series Sacred Geometry: Spiritual Science, available on Gaia, it is explored how these geometric shapes are essential for understanding the structure of the universe and how they can be used in spiritual and meditative practices to enhance awareness and spiritual connection.
7 Sacred Geometry Symbols and Their Meanings
There are several sacred geometry symbols that are widely known and used in various spiritual traditions. These symbols include the Metatron’s Cube, the Merkaba, the Seed of Life, the Flower of Life, the Sri Yantra, the Torus Yantra, and the Celtic Labyrinth. Each of these symbols has a profound meaning and is used for different purposes in meditation and spiritual practices.
Before we explore each symbol in depth, it can be helpful to see a quick overview of what these geometric forms represent.
Quick Meanings of Each Sacred Geometry Symbol
- Metatron’s Cube: Represents the geometric blueprint of creation and is used for protection, balance, and energetic alignment.
- Merkaba: A star-shaped field symbolizing ascension, higher consciousness, and the union of body and spirit.
- Seed of Life: A seven-circle pattern representing the first stages of creation, new beginnings, and the foundation of all geometric forms.
- Flower of Life: A multi-circle design symbolizing unity, interconnectedness, and the underlying structure of life.
- Sri Yantra: A complex diagram representing cosmic order, divine balance, and the union of masculine and feminine energies.
- Torus / Toroid: A donut-shaped energy field symbolizing infinite flow, cyclical creation, and the movement of life force.
- Vesica Piscis: The intersection of two circles symbolizing the union of complementary forces, creation through duality, and the portal between the material and the spiritual realms.
Metatron’s Cube
Metatron's Cube is a sacred geometry figure that contains the five Platonic solids: tetrahedron, hexahedron, octahedron, dodecahedron, and icosahedron. These solids are arranged in such a way that they represent the fundamental structure of the universe. The cube is used in spiritual and meditative practices to connect with universal energy, providing balance and harmony to the user. Additionally, it is considered a powerful tool for protection and energy cleansing.
This symbol is not only visually impressive but also holds deep spiritual meanings. It is believed that Metatron’s Cube acts as a bridge between the material and spiritual worlds, allowing practitioners to access higher dimensions and sacred knowledge. Meditating with Metatron’s Cube can help align internal energies with cosmic forces, promoting healing and spiritual growth.
The Merkaba
The Merkaba is a geometric figure formed by two interlocking tetrahedrons that rotate in opposite directions, creating a three-dimensional star. This symbol represents the union of spirit and body and is used to facilitate spiritual ascension and the expansion of consciousness. By activating the Merkaba, one can balance and harmonize the body’s energies, facilitating connection with higher dimensions and access to elevated states of consciousness.
In spiritual practices, the Merkaba is known for its ability to protect and heal. Meditating with the Merkaba can help activate the energy field around the body, known as the Merkaba field, which acts as a protective shield against negative energies. Additionally, this energy field is believed to facilitate astral travel and mystical experiences, providing a powerful tool for spiritual exploration and personal development.
The Seed of Life
The Seed of Life is a sacred geometry symbol composed of seven interlocking circles that form a flower-like pattern. It represents the seven days of creation and is considered a symbol of fertility and creation. This basic pattern is the foundation of more complex figures like the Flower of Life and is used in meditation to promote healing and spiritual growth.
The spiritual meaning of the Seed of Life lies in its representation of the interconnectedness of all life in the universe. Meditating with this symbol can help connect with the energy of creation and foster a sense of unity and harmony. The Seed of Life is used in various spiritual practices to stimulate creativity, fertility, and the manifestation of new beginnings.
The Flower of Life
The Flower of Life is an expansion of the Seed of Life and is composed of multiple interlocking circles that create a more complex pattern. This symbol is considered one of the most sacred patterns and is believed to contain the secrets of the universe, including the shapes of the Platonic solids. The Flower of Life represents the interconnectedness of all life and is used to meditate on unity and harmony.
This symbol is not only visually appealing but also holds deep spiritual meanings. The Flower of Life is considered a visual representation of the life force energy that flows through all living things. Meditating with the Flower of Life can help activate this life force energy, promoting healing and spiritual growth. This symbol is also used to connect with universal energies and to explore the mysteries of creation.
The Sri Yantra
The Sri Yantra is a Hindu symbol composed of nine interlocking triangles that form 43 smaller triangles, all enclosed within an eight-petaled lotus. This complex geometric design represents the cosmos and the structure of the universe, symbolizing the union of the masculine and feminine. It is used in meditation and rituals to promote prosperity, peace, and harmony. The Sri Yantra is considered a powerful tool for manifesting desires and connecting with divine energy.
Meditating with the Sri Yantra helps align internal energies with cosmic forces, promoting healing and spiritual growth. This symbol is also used to balance the chakras and improve concentration and mental clarity. The intricate structure of the Sri Yantra acts as a map of the spiritual journey, guiding the practitioner toward a greater understanding and realization of their connection with the universe.
The Torus Yantra or Toroid
The Torus Yantra, or Toroid, is a geometric shape that represents a ring-shaped energy field, similar to a donut. This symbol reflects infinite energy and continuous creation, symbolizing the flow and balance of energies in the universe. The toroid is considered a model of the universe’s structure and is used in meditation to harmonize internal and external energies, facilitating a state of balance and well-being.
Meditating with the Torus Yantra helps activate the flow of energy in the body, promoting healing and spiritual growth. This symbol is also used to connect with universal energies, facilitating the expansion of consciousness and the integration of cosmic energies into daily life. The toroid is a visual representation of the human energy field, providing a powerful tool for transformation and spiritual elevation.
The Vesica Piscis
The Vesica Piscis is an essential figure of sacred geometry formed by the intersection of two circles of the same radius, whose centers lie on each other’s perimeter. This overlap creates an almond-shaped form at the center that symbolizes the union of two complementary principles. On a spiritual level, it represents the meeting point between the material and the divine, as well as the origin of creation emerging from duality. Due to its precise and symmetrical structure, it is considered a fundamental geometric matrix within many ancient symbolic systems.
This symbol has been used in spiritual and artistic traditions as a representation of birth, manifestation, and universal balance. Meditating with the Vesica Piscis allows one to work on the integration of inner opposites and to foster states of greater energetic coherence. It is also the structural foundation of more complex figures such as the Flower of Life, reinforcing its importance as a primordial pattern of creation and expansion of consciousness.
History and Origin of Sacred Geometry
Sacred geometry has its roots in ancient civilizations, where geometric patterns observed in nature were replicated. The Egyptians, for example, used geometry in the construction of the pyramids, believing that these shapes contained mystical and energetic properties. Egyptian structures were designed to align with cardinal points and certain astronomical events, reflecting a deep connection with the cosmos.
In ancient Greece, sacred geometry was formalized and deeply studied by philosophers and mathematicians such as Pythagoras and Plato. Pythagoras observed that certain geometric patterns and proportions repeated in nature and developed the concept of the Platonic solids, geometric shapes that represent the building blocks of the universe. Plato, in his work “Timaeus,” linked these solids to the fundamental elements of creation: earth, water, air, fire, and ether.
The cultures of India and China also incorporated sacred geometry into their spiritual and architectural practices. The mandala and yantra are examples of geometric symbols used in meditation and temple construction, aiming to facilitate connection with the divine and promote spiritual harmony. These patterns reflect the belief in an interconnected universe, where geometry is the foundation of all things.
During the Renaissance, interest in sacred geometry resurfaced in Europe, driven by artists and scientists such as Leonardo da Vinci and Johannes Kepler. Da Vinci studied and used the golden ratio in his art, while Kepler explored the relationship between the Platonic solids and planetary orbits. This period marked a fusion of art, science, and spirituality, highlighting the importance of sacred geometry in understanding the universe and human experience.
How to Integrate Sacred Geometry into Daily Life
Integrating sacred geometry into daily life can bring balance, harmony, and a deeper spiritual connection. Here are some practical ways to do so:
- Meditation with symbols: Use figures like Metatron’s Cube or the Flower of Life during your meditations. These shapes can help focus your mind and harmonize your internal energies.
- Home decoration: Incorporate sacred geometry patterns into your home decor. Placing paintings or objects with these symbols can create an atmosphere of peace and balance.
- Yoga and energy exercises: Use postures and movements inspired by geometric shapes during your yoga practice. This can help align your body and mind with universal energies.
- Art and creativity: Create your own designs based on sacred geometry. Drawing and painting these patterns can be a meditative and creative way to connect with the divine.
