Tag Archives: Fractals

The Nature of Fractals – Part III: Our Chaotic Reality

April 8, 2019 By in Formal Science, Natural Science, Theology Tags: , , , , , , , , , , , , , , , , , 1 Comment
The Nature of Fractals – Part III: Our Chaotic Reality

This is Part III of a three part series on the Nature of Fractals. Readers can view the first two installments here: Part I and Part II.

In the previous part of this series, I introduced the thought that quantum mechanics are related to fractals. That combination further implies the idea that quantum objects represent the combination of spirit and matter, which themselves exhibit fractal properties.

The Hebrew Letter Kaf

A concept alluding to a transition akin to that seen in quantum mechanics can be found in mystic  interpretations of the Hebrew alphabet. The word “kaf” is made up of two symbols from the first letters of two other words – koach, meaning potential, and poel: “…suggesting that Kaf enables the latent power of the spiritual (the potential) to be made actual in the physical…”1  The symbol kaf represents a palm with a Yod in its middle where potential becomes reality and hearkens back to the concept of a quantum object collapsing into its matter form.

The combination of spirit and matter (see featured image) further suggests a direct link between Humanity and God. Our very thoughts may be thought of as quantum objects and underlie our reality, bringing into existence that which we interact with on a daily basis.

Imagine a being, a being so advanced and evolved, that he is One with the Word – he is indistinguishable from the Word. His thoughts are very special – they appear in his mind as quantum mechanical potential energy and take on a life of their own. Within the thought, there is Life – spirit and matter combine to form consciousness, one facet of the being as described in the dual-aspect theory.

In this manner, Sungsang and Hyungsang interact and form a human being. The thoughts or beings exist in the Universe of his mind, evolving themselves until they have each “had their due” and dissolve into nothingness. In this manner, each thought works to merge with the Word – thus further purifying the being. Thoughts beget thoughts of their own. which become fractal tessellations perpetuated throughout eternity. Fractals are left behind throughout the infinite hierarchy as evidence of the being’s work. At the End of Time, the being himself dissolves into nothing, a passing thought in a greater being’s mind.

Along similar lines, both the Bible and Mormonism support beliefs that are consistent with the previous paragraph. In the Bible we read in Romans (8:16-17):

“The Spirit itself beareth witness with our spirit, that we are the children of God and if children, then heirs; heirs of God, and joint-heirs with Christ; if so be that we suffer with him, that we may be also glorified together.”

In the Mormon faith, it is the belief that each individual can progress to the point to inheriting a universe of their own. 

“…Each one of you has it within the realm of his possibility to develop a kingdom over which you will preside as its king and god. You will  need to develop yourself and grow in ability and power and worthiness, to govern such a world with all of its people.”2

Lorentz Attractor

From a different perspective, fractal mathematics are a representation of chaos. “Fractals are related to chaos because they are complex systems that have definite properties.”3 It was discovered that there is indeed a pattern to chaotic systems. Using a number of different initial points for a given chaotic system and running the system for quite some time (a chore perfectly suited to automation), they all eventually resolve into a two-dimensional projection of a butterfly shape called the Lorentz Attractor. 4

When one thinks about the Earth and the moon revolving around each other due to gravitational attraction, the Earth is an attractor for the moon. In chaos theory, the “orbital” patterns seen in the butterfly shape seem to form orbits around what are called “strange attractors.” The butterfly shape is described in terms of fractal dimensions, which means that the shape’s dimensionality is not an integer (2-D, 3-D, etc.), but a fraction between two integers.

“…So, a fractal image is a visual representation of a strange attractor (or fractal space) that defines the orbit of a deterministic system that behaves chaotically…”5

So when we consider that fractals represent a form of order ubiquitous throughout Nature, it can be seen that they are indeed the embodiment of Ordo ab Chao – Order out of Chaos.

1 The Letter Kaf/Khaf, n.d.

2 “. . . the Matter of Marriage” [address delivered at University of Utah Institute of Religion, 22 Oct. 1976])  (Will Exalted Mormons Get Their Own Spirit Children and Worlds?, n.d.

3 Blumenthal, n.d.

4 Chaos VII : Strange Attractors, n.d.

5 Wrigley, 2017.

The Nature of Fractals – Part II: Of Spirit and Matter

March 11, 2019 By in Esotericism, Formal Science, Natural Science Tags: , , , , , , , , , , , , , , , , , , , , 1 Comment
The Nature of Fractals – Part II: Of Spirit and Matter

In the previous part of this series, I introduced the concept of a fractal. It is a construct which appears frequently throughout our daily lives and Nature herself.

So what could be driving this self-similarity at multiple scales? It is my belief that quantum mechanics is related to the phenomenon. Recently, scientists demonstrated that quantum mechanics has fractal properties.  (Dacey, 2010) Recalling some basics of quantum theory, sub-atomic quantum objects are thought to have both wave-like and particle-like (mass) properties. When a quantum object is in the wave state, it exists in a number of probabilities – a number of locations. When the wave function collapses, it becomes a definite particle of matter discernible by our instruments at a specific location. At this point, the particle not only takes on position and mass, it also takes on fractal properties.

As alluded to by Dr. Pincus in his quote in the previous article, forming a fractal mass particle is very efficient and effective in terms of energy conservation. I believe that matter retains both wave-like and matter-like quantum properties no matter its form, just to varying degrees between the two. It is the wave aspect of matter that allows it to interact with other object waves. When a given particle replicates to either a smaller or larger scale, the primary form is determined by the original object at the smaller or larger scale (thus the similarity), but that form is affected by wave interactions with adjacent objects.

Thus, the variability we find in Nature for naturally occurring fractals. In other words, if those interactions did not occur, then smaller and larger replications would be a perfect copy of the original. “Adjacent” particles do not necessarily correspond to normal concepts of space and time given the quantum mechanical property of entanglement, which states that a given particle can affect another particle at some physical distance. It is also my belief that this quantum object is the embodiment of Spirit and Matter – that the wave function seen in quantum particles is an indirect indication of the existence of Spirit. Our current instruments cannot detect or measure Spirit directly.

Sungsang and Hyungsang

Fractal mechanics in nature as described above suggest a dynamic relationship between the spiritual and the concrete – matter. A number of belief systems support this line of thought. An adjunct to Chinese philosophy’s Yin and Yang is the concept of Sungsang and Hyungsang. Where Yin and Yang describe attributes of an object, Sungsang and Hyungsang address the composition of the object – what it is made of. Sungsang refers to the non-visible aspects of an object while Hyungsang refers to the visible aspects of the same object such as mass, shape and the like. As an example, the Sungsang for a plant is Life. Both concepts are thought to stem from the Original Image, their origin.

Sungsang and Hyungsang exists in every physical object, from minerals to plants to animals to human beings in that order to form a kind of hierarchy. As one moves “up” through the hierarchy, the higher form introduces new attributes while taking on those lower down in the hierarchy. So plants, for example, introduce “Life” as it Sungsang, but also inherits physico-chemical aspects from minerals. The diagram to the right illustrates the concept. Sungsang and Hyungsang is one additional way to view and explain our inter-connectivity with Nature.

Note that the human level of the hierarchy adds the spirit mind and spirit body to the mix. Adherents to Unification Thought, those that believe in Sungsang and Hyungsang, believe that at the end of our lives, the material body – the portion of us that we inherit from the animal kingdom – falls away and our spirit, mind, and body continue to live on indefinitely.  (I. The Universal Image of the Individual Truth Body, n.d.)

Mosaic Pavement – Spirit and Matter

Another concept coming from the psychology of mind domain seems to exhibit aspects of  spirit and matter as well. Dual-aspect theory or dual monism addresses the mind-body problem – how the mind and body interact and exist alongside one another. As can be imagined, because mental processes are involved, some of the theories can get kind of exotic – and dual-aspect theory is no exception.

Under this Theory, the object under consideration, normally a person, is neither physical nor mental, but an inseparable combination of the both. A term used to refer to this combination in the readings I explored is “phental.”

The combination is not the simple combination of the mental and physical, nor is it reducible beyond its normal form. The mental and physical are then aspects of the phental.

A further refinement of the theory postulates that the phental is unknowable and that the mental and physical aspects are but a portion of the true self being exposed. This last part has some interesting implications and is a potential reference to what is stated above regarding the wave-like portion of a quantum object. As stated previously, I believe the wave-like portion to be an outward expression of the imbuing Spirit. This line of thought then makes one wonder what could be the underlying truth regarding the physical aspect of the phental.

The Nature of Fractals – Part I: The World Around Us

February 10, 2019 By in Esotericism, Formal Science, Natural Science Tags: , , , , , , , , , , , , , , , 1 Comment
The Nature of Fractals – Part I: The World Around Us

Man continually seeks deeper understanding of the world around him. From the deepest reaches of space, to the depths of our oceans, to the smallest particle, Humanity seeks to gain ever more profound insight into this world we all experience together. However, what if the clues to gaining some insight into our existence lie right before our eyes?

As I journey through my life, it continues to amaze me how complex and yet simple our existence really is. Humans have a remarkable ability to discern patterns. Repeating patterns are a phenomenon seen throughout nature, such as the fractal. Could our ability to discern those patterns and their existence be an indication of deeper truths for this reality?

Example of a Fractal

A fractal is defined as a “natural phenomenon or a mathematical set that exhibits a repeating pattern that displays at every scale.” No matter what magnification the observer uses, the same pattern is evident, just at a larger or smaller scale depending on the magnification used. The Mandelbrot Set is one such fractal and is illustrated to the left. Mandelbrot described the fractal as “…a rough or fragmented geometric shape that can be split into parts, each of which is (at least approximately) a reduced-size copy of the whole…” (New World Encyclopedia, n.d.)

One example of a fractal is seen in a hyperbolic fractal tessellation. A tessellation is a closed, countable set of tiles arranged so that they do not overlap with a repeating pattern. They essentially form a two-dimensional shape within the Euclidian Plane. A hyperbolic fractal tessellation combines the traits of a tessellation and a fractal in a manner similar to the illustration at the right.

Fractals can be seen in our daily lives. The manner in which this article was assembled has fractal patterns – start at the highest level, build a framework (outline), select one of the subsections and write to that, inserting a sub-framework around which the words are assembled, repeat until the depth of detail desired is reached. The antennas used in cell phones are fractal in design as well. This design was selected to solve an early problem with cell phones – the large number of different frequencies each phone had to receive. The length of an antenna must be a whole fraction of the wavelength of the signal for the signal to be received. Dr. Nathan Cohen discovered in 1988 that an antenna designed as a fractal could receive multiple signals because a fractal antenna realized antennas of multiple different lengths, either matching or a whole fraction of the wavelengths of the received signals.

Fractal Pattern in Nature

Fractals are ubiquitous throughout nature as well. From a certain perspective, the fractal antenna above was successful because it replicated the concept seen in nature. Some of the more commonly seen fractals include trees and ferns. For trees, think about how the trunk is the base for multiple large branches, which form the foundation for smaller branches, so and so forth to the leaves at the end of the smallest branches. Certain sea shells also exhibit a fractal pattern. You may wonder why natural systems behave in this manner. As quoted from Dr. David Pincus:

Essentially, fractal systems have many opportunities for growth, change and re-organization. Yet they also are very robust. They maintain their coherence; they hold together well, even under tough circumstances. They are balanced in this respect, between order and chaos. They are simple, yet also very complex. This balance is often referred to as “criticality.”

And the term “self-organized” is often added because systems tend to become fractal on their own, simply by putting a lot of system components together and allowing them to exchange information. Think of a party. All you need to do is come up with enough people at the same place and time and they will start to form complex patterns of connection with one another.”  (Z.McGee, n.d.) I like to think that fractals are so complex that they are simple.

Fractal Pattern in the Brain

It turns out that the brain is fractal, both in the way it is organized physically and functionally. On the physical level, at the smallest scales are the pyramidal neuron, which is the most common neuronal structure in the brain. These form into cortical columns, consisting of numerous pyramidal neurons. Finally, the Columnar Complex consists of a number of cortical columns. All of these structures exhibit branching both into and out of the arrangement.  (The Fractal Brain Theory, n.d.)

Indeed, illustrations of the neuron and its surroundings depict a fractal type of construction. Even the way the brain works is fractal in nature. Psychologists discovered in recent years that behavior patterns and social behavior adhere to those principles. So Humanity exhibits a fractal nature from the smallest to the most gross scale, which may explain our connectivity to Nature itself. One author describes this connectedness as “broadband connectivity” and explains how that may be related to our consciousness.  (Ph.D., 2009)