Let’s see how deep the rabbit-hole goes
We live inside an elegantly-designed Matrix — and in this article I will walk you behind the scenes to reveal the hidden blueprint driving our reality.
This article describes an ontology concerning the root of all complex systems. An ontology is defined as the philosophical study of the nature of being, becoming, existence or reality as well as the basic categories of being and their relations.
This ontology will imply that all bodies and the laws that govern them arise from the geometry of the Circle, and even more specifically arise from its two fundamental constructors: the Line and Rotation. I will categorize these two fundamental concepts of a circle into a triangular tree-diagram (ie. hierarchy) that will reveal itself as the foundation for any living or non-living structure in the Universe.
As well, the two concepts will also be shown to be the ‘parents’ of the Rational, Irrational, and Imaginary numbers, which in turn give rise to all Mathematical Operators (addition, division, squares, roots, etc).
As Pythagoras once said — “All is Number”.
In this article, I will examine each node of a primordial hierarchy and describe dozens of unique symmetries that bind the nodes across a wide range of symbolic, philosophical, mathematical, and scientific associations. In the end, you will have been presented with a rudimentary explanation for the existence of a duality between a Physical and Abstract realm, forged by the Line and Rotation respectively.
It is in my belief that all Universal creations, tangible and intangible, should be naturally founded on a single hierarchical blueprint, including all religious, scientific, and dogmatic concepts such as Yin Yang, the Tetractys, Pascal’s Triangle, Euler’s Identity, Pythagorean Theorem, Fibonacci Numbers, Mandelbrot Set, Quantum Mechanics, and ultimately Plato’s, Theory of Forms.
In essence the following hierarchy will be presented as the base for all forms of nature, designed or otherwise scientifically discovered.
1. Introduction to the Pythagorean Tetractys
I will start by introducing to you an icon of old mathematics, that was once worshipped by Pythagoras and his followers around 500 BC. It has its roots in Vedic Mathematics and yet centuries later it would find itself to be the basic form for Pascal’s Triangle — the triangular array of the binomial coefficients.
The Tetractys is an exquisite example of how a simple visual pattern can have varied meanings and deeper levels of interpretation. The famous Pythagorean oath refers to the Tetractys, praising it as the source which contains the fount and root of eternal nature.
A profusion of insights can be derived from this symbol, concepts which are relevant to arithmetic (number relations), harmonics (musical tuning theory), and geometry (number relations in space).
Tetractys was conceived by one of the most brilliant mathematical minds of history — whom had spent decades travelling across the known world accumulating knowledge — and what he found in this simplistic form has influenced people, philosophers and scientists alike, since its inception.
In Ancient Greece, Pythagoras built an entire organization devoted to worship it, and this organization, referenced many times in works by Plato and Aristotle, is believed to have spread its ideas to the Kabbalah which then went on to heavily influence the teachings of Judaism, Catholicism, and Islam.
There is so much to be said about the Tetractys — even though it has been relegated by most scientists as little more than an ‘iconic image’ — that this article, which is ultimately a description of this form, only scratches the surface of the natural information it contains.
Throughout my book, Theory of Thought:Symbolism, I show that all symbols contain layers of meaningful information hidden beneath their surface waiting to be re-discovered and re-explained.
And according to my research — like Pythagoras once believed — I believe that the Tetractys should be held above all other symbols because it does appear to be the root (not at, but the).
2. Introduction to the Circle
Think of the circle as a seed that sprouts a hidden root that is embodied by the Tetractys.
Across most, if not all written accounts, our Creator has been referenced as a Circle, such as the Monad, the Sun, God, or the ‘Uni’-Verse (ie. Big Bang / singularity).
Throughout history, God has repeatedly been referenced to as the Geometer or Great Architect.
So God could be thought of as a reference to the circle along with the process which unfolds from it to reveal the subsequent creation of matter and life.
Thus the initial process of creation starts with:
1) a seed that sprouts a root;
2) a root capable of growing increasingly complex branches;
3) with the ultimate goal of manifesting intelligent forms of matter (ie. living beings).
Living beings are the only creations that can actively perceive the abstract realm — that’s why the ultimate goal of this hidden hierarchy (Tetractys) is to ‘create living beings’.
It should not come as a surprise to anyone that the circle represents the most basic shape and formulation of the Universe, considering such scientific concepts as the quanta of light (called photons), Atoms and Sub-atomic Particles, Spacetime Curvature, and the Fourier Transform (which can convert any waveform into a combination of ‘periodic’ Sine Waves).
All cyclical events, such as seasonal patterns or even a heartbeat, are manifestations of visible or invisible circles actively rotating around us. The concept of circles permeates our physical environments and our minds — and most if not all of our culturally iconic symbols are ultimately based upon the simple design of the circle, it’s rotation, and its periodicity.
And given enough cycles, the circle itself will be shown to have the necessary ingredients to sprout a complex living being.
To understand, start by considering the circle as the ultimate representation of One and Unity (ie. Monad), yet knowing that it can be conceptually divided into exactly two basic components — and in doing so, it can be reframed as a Duality (ie. Dyad) that naturally emerges from its singularity.
3. Deconstructing the Duality
Any circle should be described as a unit of two distinct concepts:
1) Line (thing / physical)
2) Rotation (action / abstract)
The concepts of the line and the rotation are each unique subsets of the circle.
1) The concept of the Line is a ‘thing’ of some linear distance.
2) On the other hand, rotation is an ‘action’ forcing a curvature.
So in a sense we have a duality between a linear thing and a rotating action, which is united into the concept of a circle.
You can simply recreate the two concepts and form a circle by
1) Placing a pencil on a table (Line)
2) Rotating it about its center (Rotation)
And yet another way of looking at the circle’s two primary components is by identifying the two features necessary in calculating PI:
1) Diameter (Line)
2) Circumference (Rotation)
For the sake of understanding this article you must now assume that every circle is forged from a precise DUALITY. This duality is ultimately regarded as the Line vs. Rotation and we will examine how these two low-level concepts extend into dozens of other interrelated, higher-level concepts.
4. Illustrating The Hierarchy
The circle is placed at the top of the hierarchy, with its two primary constructors branching off into separate child nodes. The two lower nodes represent the dyad that is encompassed within the monad. And all three nodes considered together is a Triad representing the complete notion of a system.
In Thought Theory, every object or idea can be considered as a system rooted by this precise hierarchy.
5. Re-Visualizing the System
The goal of Thought Theory is to deconstruct the concept of the circle into individual elements, apply deeper meanings to these elements, and re-visualize its components using a specific set of patterns.
Theory of Thought explores 5 different types of patterns. On the left is a system pattern — Notice how there are circles inside and outside of other circles.
In most regards, the circle is the ultimate container. It is the most mathematically efficient container with the highest area to perimeter ratio. As such the natural tendency for all cells, unless being used for a specialized purpose, is to take a somewhat spherical, round shape. Also, most of our ideas in group theory, relationship theory, and abstract categorization are almost always explained using circles (or circular nodes) — and whenever anyone draws a circle on paper it can be thought of as containing a meaningful system (ie. organization).
Circles are natural containers for some substance — which is part physical and part abstract.
By definition each system (ie. circle) must have exactly 3 necessary components to work as a container: It must have an inside and an outside that are separated by some border(ie. perimeter).
These 3 concepts are universally applied to all individual or collections of systems (ie. fractals).
In effect, the basic system pattern (BORDER-INSIDE-OUTSIDE) is just another way of viewing the three node hierarchical pattern (CIRCLE-LINE-ROTATION) shown earlier.
So the Hierarchical Pattern is the SAME as the System Pattern. They are just different perspectives applied with different, inherent meanings.
Both types of patterns explain the SAME system.
Can you understand, that we are able to take the same concepts and view them differently by applying some interrelated meanings to better grasp the underlying structure? This is the methodology used throughout Theory of Thought.
Each patterned diagram I provide is useful. However, the most challenging aspect is in finding the best methodology for illustrating and explaining this specific idea:
If Tetractys is like a prism, what do each of its facets look like?
6. A Basic Universal Blueprint
First and foremost, Tetractys is a hierarchy, and its two initial child branches should be thought of as mirror symmetries, forming left and right halves (ie. realms).
These halves are persistent across the entire hierarchy so I have added two boxes to encompass the halves and I included meanings in the upper corners of each box to help explain the influence each half has on the nodes within it.
You should think of the border that separates the two realms (ie. halves) as a type of mirror, with the purpose of:
1) Creating realms with symmetrical structures
2) Reflecting information (or energy) between the realms as a consequence of communication (or equilibrium)
The mirror is the border of the system that separates the inside versus the outside.
Note that any children which sit on the border will have strongly split meanings (as this node has contact with both the inside and the outside).
7. Various Meanings of the Duality
The Line vs. Rotation duality permeates all levels of complexity in the Universe and forms the base of everything. This duality comes to create our notions of Space vs. Time, Classical vs. Quantum, Gravity vs. Electromagnetism, Quanta vs. Waves, Rest vs. Motion, Positive vs. Negative, Men vs. Women, and spiritually, Yang vs. Yin (all respectively). Although the correlations are not always perfect, because these concepts are in themselves quite complex, it should be fairly intuitive to generally grasp how each concept is associated with one of the two sides of the duality.
You must understand that the concepts of the Line and Rotation, and the 5 relationship patterns that illustrate them, permeate all complex things including the construction of the brain, its 2-dimensional neurons in the neocortex, and the basic composition of thought itself.
8. Physical and Abstract Realms
In the same way that there cannot be an inside without an outside, there cannot be a Physical realm without an Abstract realm — and both realms exist and are created simultaneously, from the beginning moment of any system.
Note that thought is a manifestation that arose from the presence of an abstract realm in the Universe.
Socrates and Plato were among a handful of the most influential people in history and they referred to an abstract realm within their World of Forms. Plato himself was heavily influenced by the Pythagoreans!
And if you notice in the system pattern, a mirror separates the realms, but the mirror looks more like a wall (ie. cell membrane) — logically this does not make for an effective mirror.
So this is a perfect example of why multiple patterns are needed to visualize different perspectives. From the view of a hierarchical pattern shown here, the mirror appears to reflect some information between the physical and abstract realms (instead of blocking it like a wall).
Everyone tends to generally understand the Physical realm because we can see and touch it. But we have a hard time explaining how and where an abstract realm can exist, which is a big reason why the Theory of Forms has never gained real traction in the scientific community.
Generally, we see the physical world as:
- an infinite number of planes (ie. lines) that criss-cross each other in all directions (ie. Cartesian).
And now imagine that each of these planes:
- is continually rotating, and this planar rotation is the abstract realm.
So all around us are rotating planes, and although we can’t directly perceive their rotation, it is their perpetual rotation that is responsible for the unusual properties of Quantum Mechanics while also providing the building blocks for thought, thinking, and life!
And although the rotations are not physically seen — and have probabilistic effects rather than deterministic — they are capable of producing and maintaining interlocking circles everywhere around us!
And these abstract circles are capable of sustaining forms of information called symbols — hidden structures described by meanings, shapes, numbers, and letters.
All minds/brains can process symbols, consciously and unconsciously (perhaps even more so unconsciously).
The idea of interconnected symbolic structures that permeate time and space may be difficult to grasp at first.
On a scientific note, the rotation of the planes may be thought of as the underlying mechanism that gives rise to the Wavefunction in Quantum Mechanics.
The hidden rotation of the planes is an energy source for the mind and is necessary in giving rise to both the Physical and Abstract dualities simultaneously, for which the brain models while the mind inhabits (ie. Brain vs. Mind).
It is also my belief that as each plane is rotating it reveals an area filled with infinite discrete lines (which can later be explained as connections to other circles). This perpertual mirror symmetry creates real links between the abstract and the physical realms from which all minds draw energy.
And going a bit further with the concept — the entire rotational process appears to only require a 180° rotation of the plane (not 360°) — this is significant because the Triangle has 180° of angles.
Basically speaking — some sort of triangular shapes emerge from the rotating planes to become building blocks for the hidden structures that permeate a higher-dimensional space.
Note that the basic shape of the Tetractys and hierarchical pattern is triangular… the concept of triangles seem to emerge from circles naturally.
9. The Pythagorean Theorem
Before we get to the 3rd level in the hierarchy, it’s important to consider the special relationship between semi-circles and right triangles.
The Pythagorean theorem was first documented in Ancient Greece by Pythagoras (he was a very smart guy). His theorem is probably the most important equation discovered to date and is still used today to calculate any and all distances across a cartesian plane.
There are also derivatives of the Pythagorean theorem used to calculate angles and radians (curvature), which is directly related to Euler’s Formula. In computer science, the Pythagorean theorem can also be used to calculate ‘abstract’ distances — vectors across multi-dimensional maps of abstract values.
There’s a good reason that we only need a 180° rotation to create our circle, because that is the minimum angle necessary to produce a triangle, and more specifically, a right-triangle.
In Thought Theory, it is assumed that the Physical and Abstract spaces are connected together according to the Pythagorean Theorem, in a way that follows the relationship between Space and Time.
These right triangles are believed to hold the mirrors that balance (ie. equilibrate) the physical and abstract realms across all universal systems.
They equilibrate by dilating time — and in my opinion — time dilation may be a much more pervasive feature of SpaceTime then currently understood.
So as the concepts of the Line and Rotation give rise to the Physical and Abstract realms, a mathematical system must arise to create logical structures to inhabit them.
10. Three Fundamental Number Types
The next 3 nodes in the hierarchy, that descend from the Line and Rotation, are the Rational, Irrational, and Imaginary Numbers.
These three foundational types of numbers help forge a physical & abstract framework that goes on to support the construction of matter and information within our Universe.
You should have been introduced to these numbers very early in school, because they form the base of mathematics.
In another note to Pythagoras, legend has it that he was killed after teaching others about √2, a seemingly mystical number that was believed by his contemporaries to be an impossible number (hence irrational).
But we now know that these three types of numbers are crucial for scientifically describing all natural things. And that’s why they must be included as a low-level constructor within this blueprint.
The 3 number types is also a triad which has related meanings to the upper triad — these relationships are forged by symmetry.
When two or more things are related, they must contain symmetry (we are most familiar with mirror symmetry, however symmetry in Thought Theory goes beyond shapes and extends into meanings as well).
Currently, in number theory & set theory, the 3 fundamental types of numbers are generally visualized like so:
But I believe it’s best to illustrate them as distinct components of the system pattern.
Consider that the types of numbers are not human inventions — they are fundamental properties of the circle itself!
- Rational numbers are the Line.
- Imaginary numbers are Rotation.
- Irrational numbers are the Border.
So how can I determine if I am illustrating these concepts correctly? By analyzing symmetry. As long as symmetry is maintained, we should be on the right track.
So let’s take a closer look at the relationships between the 3rd layer nodes and the nodes above.
11. Rationals are Descendants of the Line
The relationship between rationals and the line is quite easy to grasp. It should be clear that Rational numbers exist on the number line, as ratios of discrete segments of lines, and are thus closely associated with physical (ie. planar) space.
A number line is a distance that starts from a single point and extends into a positive direction. In the Tetractys the concept of the line should NOT include negative numbers, as those are abstract concepts, but in applied mathematics negative numbers are part of the rational numbers because they are also represented as fractions.
But remember that the simplest idea of a line is one that goes from 0 to 1 (with 1 being any arbitrary distance).
Also note that base 10 rational numbers with infinitely repeating decimals (e.g. 1/7) only do so in certain bases (e.g. base 2, 3, 4, 5, 7, 10, 12, 16, etc.). So across all bases, all rationals can be expressed as integers, but in any specific base, all rationals are fractions, whose purpose is in comparing different line segments.
12. Imaginaries are Descendants of Rotation
The Imaginary number was conceived much later in history due to its mind-bending qualities. Firstly, it is associated with negative numbers (Square root of -1), which were also integrated much later in history than positive numbers. Over time, the imaginary number has come to be visualized as a rotation between axis in the complex plane.
Imaginary numbers require a higher level of abstract thinking, and are thus more complex. It’s quite fitting that the imaginary number is the descendent of rotation and exists squarely within the abstract half of the duality.
Also consider that the imaginary number is very practical and absolutely necessary when mathematically calculating quantum mechanics (QEM). They are heavily used in relation to quaternions, probability amplitudes, the Wavefunction, and particle physics in general.
13. Irrationals are Descendants of the Line and Rotation
“The then-current Pythagorean method would have claimed that there must be some sufficiently small, indivisible unit that could fit evenly into these lengths as well as the other. However, Hippasus, in the 5th century BC, was able to deduce that there was in fact no common unit of measure, and that the assertion of such an existence was in fact a contradiction.”
The Irrationals are not static numbers, like an integer or a simple fraction. In fact, all calculations of irrational constants, in any number base, can never be exact and will always produce approximations since they require calculating infinite numbers of fractions.
An interesting way to describe the infinite nature of an Irrational is by comparing it to the infinite diagonals that intersect the center of a circle. Think of each irrational as a process that produces a unique pattern of lines that eventually fills the circle one slice at a time.
On the right is Theodorus’ Spiral, which shows an elegant, shell-shaped symmetry between all surds (irrational numbers that are square roots of non-square integers). There is a theorem that proves that when expanding this spiral past the √17, none of the hypothenuses would ever perfectly overlap!
Irrationals can be thought of as ‘imaginary’ because:
1) they have a relation to the concept of ‘infinity’
2) they are lines with relationships to curvature
And in others ways irrationals are thought of as ‘rational’ because:
1) they sit upon the real number line
2) they can be represented using a series of fractions
Irrationals are best defined as infinite, recursive processes. Here are the specific ways to calculate PI, PHI, and e: (notice how the accuracy increases with with each step, yet never being exact).
You should begin to see that it’s the concept of Symmetry (ie. duality) that binds the nodes together as Tetractys grows in complexity.
14. Child Nodes Are Increasingly Complex
At the top of the hierarchy is the simplest of concepts — the circle. And all of its descendants are simply extensions of its unique traits. As more children are branched down the tree, increasingly specific traits are revealed. It also becomes increasingly difficult to define each child using exactly one meaning, because it inherits meanings from its parents and many more from its entire lineage.
There is a benefit in this complexity, for it allows each node in the hierarchy to be substituted with slightly different meanings (ie. concepts or traits), while remaining in-sync with its entire branch. From these substitutions we can learn more about how lineages intersect and complement one another.
Consider that this is the root of complexity and symmetry — it’s a necessary process that branches out into interconnected concepts that must originate from an initial symbolic seed!
15. 4th Layer and Basic Mathematical Operators
What’s a bunch of numbers without any way of performing calculations? And does the number 5 exist in a world where addition doesn’t yet exist?
It should make sense that the three fundamental types of numbers give rise to a layer of basic math constructs, called operators, that underpin arithmetic, geometry, algebra, and calculus. These basic constructs include addition, subtraction, multiplication, reciprocals, fractions, squares, roots, logarithms, and more.
These constructs are necessary for counting, grouping, scaling, growing, and changing individual systems or groups of systems. They may even be necessary to create the (infinite) abundance of Rationals and Irrationals that we find in nature. Perhaps, without these constructs, there are only a handful of rationals and irrationals that are actually present at the beginning… such as 0, 1, 2, 3, Phi, e, and Pi.
You can start to see that the purpose of this hierarchy is to establish a system (ie. an organization) with mathematical properties that can then logically interact with other outside or inside systems. The exact type of organization is insignificant — this is a blueprint for ALL organizations throughout the Universe (particles, cells, people, cities, galaxies, etc.), but should also include the Universe itself.
Now in order to determine the accuracy of my assumptions about this layer, let’s analyze the much larger assortment of symmetries that appear along with its nodes.
16. 4th Layer and Basic Mathematical Operators
Here we see that the Rationals give rise to positive numbers, integers, and the concept of addition. These are some of the simplest math constructs and as it were, the earliest discovered by humankind.
Note that all of these nodes exist in the physical realm (ie. line) of the duality. That makes sense, right?
17. Fractions, Reciprocals, and Ratios (Line)
From the Rational and Irrationals — being all Real numbers — rise the concepts of fractions, reciprocals and ratios. It is sensible that these concepts, that are particularly related to linear correlations, are located within the line’s side of the duality.
18. Squares, Roots, and Logs (Rotation)
From the combination of Irrationals and Imaginary, come the concepts of squares, roots, and logs. An entire class of Irrationals called Surds are created from square roots of non-square integers. These concepts lead to exponential correlations, curved trend lines, and the overall concept of growth. These concepts also imply motion and acceleration — as rotation can produce acceleration through angular momentum.
It’s fitting that all of this is located within the rotation (abstract) side of the hierarchy.
19. Negatives (Rotation)
Although negative numbers can be added and subtracted like Positive numbers, are described as integers, and included within the Rational numbers, they should not be considered to have a perfect mirror symmetry to the Positives: in the same way that a line and a curve are not mirror symmetries — yet they are practically opposites.
The concept of negative is of a different beast than the concept of positive, arising from different sub-concepts and leading to other unique concepts — particularly that of debt and finance.
Negative numbers were truly adopted thousands of years after positive numbers and they have become the instrument by which we have built entire global economies while re-defining the science of economics. They naturally open the door to advanced abstract thinking which has a tangible effect on our physical world. Hence, it makes a great deal of sense that the abstract concept of negatives, emerge from the Imaginary number, located on the rotation side of the duality.
Note that integers shows up twice in the hierarchy. I believe this is indicative of a rotational symmetry within the hierarchy itself! (Because all things, including this hierarchy are subject to the line and rotation).
20. Negatives Are More Complex Than Positives
Had the complexity of Negatives and Positives been equal — both would have been adopted simultaneously. That extra level of complexity took over a thousand years (or more), to truly understand.
If you notice, the circle rotates backwards (or perhaps forwards) from the implied direction of the line. And this is the right way of understanding negatives — it is not just going backwards on the same number line, it’s going backwards vis-a-vis a number line and doing so across a rotation (employing one of those abstract rotations mentioned earlier).
So you might say that subtraction does not remove complexity — it ADDS to the complexity (ie. appearing to remove some items from the physical while still increasing its entropy).
A negative number should not only be thought of as a left-handed direction vis-a-vis a number line, but also a curvature across an abstract realm.
So it’s really a combination of 2 concepts! (which have a lot to do with entropy and time)
On another note, this additional level of complexity may be the reason that a Neutron decays (higher complexity = abstract) while a Proton does not (lower complexity = physical)… more about this later
21. Opposing 4th Layer Symmetry
Let’s look at the elegant symmetry between the siblings in the 4th layer nodes.
The first symmetry is between the Positive and Negative on opposite sides,
The second symmetry is between the Fractions and Exponents within the middle.
Consider that both positive and negative nodes have connections to integers and the line and think how they might connect together if the hierarchy was reconnected via a cylinder or cone.
The second symmetry seems to differentiate between linear vs exponential growth.
The next step will be to understand these concepts further by substituting the 3rd and 4th rows with actual numbers…
22. Substitution With Integers
Substituting some meanings with integers yields many new symmetries while still preserving the previous ones.
Take a good look at the nodes I’ve replaced with numbers. They are all simple integers, that exist at the beginning of the number line, helping to define concepts such as even and odd, neutrality, singular vs. multiple, and perhaps some other basic concepts.
Let’s go over the symmetries that take shape from this fascinating node substitution.
23. Edge Symmetries
On the left side, which is governed by the Line, we have substituted the rational node with 0 since 0 is a rational number.
Conceptually, 0 also marks the start of any line. We then substitute 1 for the positive integers node and this action defines the line between the numbers of 0 to 1, defining a single unit of any length.
On the right side, which is governed by Rotation, we have substituted the imaginary number node for i. We have also substituted the negative numbers node for -1. Of course i and -1 are related since i is the √-1 (square roots are located in the abstract side).
Each horizontal layer also contains another interesting symmetry:
1) i is a rotation around the middle point of 0
2) -1 is the mirrored position of 1, with 0 in the middle
1 and -1 are thought of as mirror reflections or each other — for good reason — however as explained earlier, the concept of negative adds an extra value of complexity, which makes comparing positive and negative in some regard, like comparing apples and oranges.
Now if you locate the 1 and 2 in between of 1 and -1: they represent those levels of complexity I mentioned!
The hierarchy appears to set some ‘weights’ to certain concepts.
24. Substitution With 1 and 2
The [second] 1 symbolizes Fractions, Reciprocals, and Ratios — and it is the level of complexity of the line.
The 2 symbolizes Squares, Roots, and Logs — and it represents rotational complexity.
Two is the smallest positive integer which builds exponential results. 2¹= 2, yet 2²= 4.
2 is associated with growth, while 1 is associated with distance.
The number 2 is instrumental in understanding square roots, squares (ie. 2D shapes) and other concepts such as binomial expansion, exponential growth, and group theory.
Compared to the concepts inherent to 1, the concepts of 2 are more abstract (harder to understand) and inherently relate to higher complexity.
25. Substitution With 3
Out of all the symmetries found in this hierarchy, replacing the Irrationals node with the number 3 is possibly the least congruent. Perhaps the √3 would make more sense, right? (because square root of 3 is irrational, while 3 is not)
I’m going to provide several reasons for the substitution and try to explain how they are all valid.
They are not mutually exclusive.
They all work together to not only bind the number 3 with Irrationals, but to bind the number 3 to the center of this entire Tetrad.
1) Let’s first start with a simple explanation that speaks to its relationship with the Irrationals — but it’s hardly concrete on its own: I’ll state that there are exactly 3 Irrational constants that form the underpinning of the Universe — PI, PHI, and e. I will expand on this point later.
2) Next let’s think of 3 as the number that represents the combination of a Line and Rotation — perhaps as a number representing its complexity [1 + 2 = 3]. Therefore, the line and rotation, taken together must have a complexity of 3 — and perhaps this is what enables the 3D nature of space.
3) Next let’s think about 1, 2, and 3 as the first 3 positive integers in the universe. They are also the first three Prime Numbers, and all subsequent primes have a relationship with them, thru 6n+1.
They are also the only group of numbers that can be added or multiplied to produce the same value: 1 + 2 + 3 = 6 and 1 * 2 * 3 = 6.
4) And that brings us to what is perhaps the best reason why 3 is in the center — because of its relationship to the triangle and hexagon, which are both ultimately related to the 2-dimensional Kissing Number… which is 6:
Pictured above, the 3 appears to correspond to the 3 rows of 3 nodes (0–3-i, LINE-3–2, ROTATION-3–1).
It also seems to describe the relationship between 3 (1+2), 6(1x2x3), and 9 (3x3).
By the way, while only 7 circles can be neatly packed inside the larger circle (1/7), it turns out that there are exactly 9 circles worth of total area.
So what is the relationship between 3, 6, and 9? That’s beyond the scope of this paper, but you should take a good look at Vortex mathematics. It states that 3, 6, and 9 are conceptually different than 1, 2, 4, 5, 7, and 8.
When dividing 1 by 7 (the closest, simple approximation of PI’s irrational part), the answer [1/7 = .142857142857..] lacks the numbers 3, 6, and 9… but I digress…
Think about it — placing the number 3 at the center of a triangle makes perfect sense right?
Let’s just end this section by noting that the 3 is quite complex, which is expected given its position in the center of this hierarchy (ie. Tetractys is much more than just a simple, single triangle).
26. Equation Symmetries — Bottom Three Triads
The following two sections shows symmetry according to some basic arithmetic.
There seems to be an alternating symmetry between each group. These symmetries are similar to how Pascal’s Triangle is assembled (but not exactly).
27. Equation Symmetries — 4th Layer
Again, there seems to be a similar alternating symmetry between each group. These mathematical patterns should help validate the hierarchy’s structure.
28. Fibonacci Numbers
Although many scientists dismiss the idea that the Golden Ratio is a building block of the Universe since it is rarely found in fundamental physics, there should be no denying that the Golden Ratio is a pillar of mathematics and has been identified in dozens of basic scientific geometries and equations.
The Golden Ratio is also referred to as Phi, a mathematical constant of nature that emerges from the Fibonacci Series— a very simple recursive equation with a similar expression as Mandelbrot’s Set (Fractal).
Notice how each equation has one 1 and one 2.
In my opinion, these two equations are closely related — take a good look at how many differences and similarities there are. More about this later.
The Fibonacci Series creates a series of numbers starting with 0 and 1, and each subsequent number is the sum of the previous two.
0,1,1,2,3 are the first 5 Fibonacci numbers, and continually dividing the last pair of numbers together yields an increasingly accurate value for Phi.
Their appearance in the hierarchy (especially considering the fact that 1 appears twice) is interesting and may indicate that the idea behind the Fibonacci sequence (or it’s simple recursion) guides the creation of the hierarchy itself.
This may also be a hint that PHI is of some importance within the creation of the circle in some way that I have not yet explained.
Also note that the Fibonacci numbers are mostly grouped together on the Physical side, where addition, fractions, and ratios also reside (but does slightly extend into the abstract realm).
29. The Most Fundamental of Numbers
The numbers -1, 0, 1, 2, 3, i (along with PI, e, and Phi) are the most basic and widely used numbers in mathematics.
For example, 1, 2 and 3 are necessary constructs for determining Prime Numbers. They are the also the lowest factors in all higher numbers.
In Quantum mechanics, the Square Root of 2 and 3 are repeatedly used for calculating particle spin. The numbers are also closely identified with the four most basic shapes in geometry, Circles, Lines, Triangles, and Squares (2²), and are also found crossing the geometry of the Vesica Pisces (Union Pattern/Venn Diagram).
It’s only natural that a basic hierarchy would be structured atop these fundamental numbers.
30. Substitution with PI, e, and PHI
In this diagram:
3 is replaced with PI
2 with e
1 with PHI
Given that:
PI = 3.141…
e = 2.718…
PHI = 1.618…
The substitution is quite intuitive. It seems also natural to place the 3 most important irrational numbers in mathematics within the heart of the hierarchy.
There are some other implications here — an obvious one implies that PI is forged from both PHI and e, or vice versa. Of course on face value this is mathematically incorrect for a few reasons (Algebraic vs. Transcendental), however there may be some philosophical or other underlying implications to this symmetry. So this association may pave the way to some yet unknown equalities between these numbers or some of their derivatives (perhaps found in digit sums and vortex math).
If PI, PHI, and e, are not considered numbers, but in actuality are infinite processes of discreet steps, it may be reckoned that PI does in fact equal 3 at the first step of its process, inching towards its final value.
The same ideas may be applied to the values of e and PHI, as well as all other irrational numbers.
31. PI Substitution
Substituting 3 with PI is intuitive because PI = 3.1415… And as with any irrational, PI should be considered a recursive equation (ie. a process).
In one of its oldest known equations, called the Nilakantha Series, PI starts plainly as the number 3, and sequentially builds upwards by adding and subtracting small fractions, infinitely. So in essence, the irrational part of PI starts with the digits that come after 3, which are 0.141592…
The other famous equation is the Leibniz Series — which starts at 4. So perhaps PI is in essence a concept between 3 and 4, and its irrational part may be considered 0.141592… or 0.858407… (1-PI), depending on perspective - both decimals having possible relationships to this hierarchy.
Positioning PI on the border between Physical and Abstract, and on the same border as the concept of the circle itself is also meaningful — it implies that the border of the circle AND the border of the duality (which separates the physical and the abstract) are embodied within any equation that uses PI.
Pi is not exactly the circle or the circumference (border), it is the numerical representation of a process that divides (or unites) a duality.
This hierarchical framework seems to imply something about geometry and the structure of symbolism — a form created from a combination of meanings, shapes, and numbers.
Meaning and shape form the top of the hierarchy, while numbers emerge along the 3rd and 4th layers — therefore numbers are forged after shapes.
Meanings > Shapes > Numbers > Letters (pi, phi, e)
Lastly, notice that PI is the center of the hierarchy, which says a lot given that PI is found in all branches of maths and sciences and is undoubtedly the most important mathematical constant in the Universe.
32. e Substitution
Having e replace the number 2 is also sensible, since e is equal to 2.71828… But even more so, e is defined to be the rate of growth if we continually compound 100% return on smaller and smaller time periods. Therefore e is related to the concepts of growth and acceleration.
Without the infinite time periods (ie. only a single step/cycle) e would yield exactly 2. But when the idea of ‘infinite compounding’ is applied, growing 1 by 100% yields 2.71828… So e starts with 1, jumps to 2, and cycles towards 2.71828… one layer at a time.
Notice that e can be logically represented using a system diagram.
It’s thus fitting that the e node is associated with logs, powers, and exponents and is found on the abstract side of the duality, as a sibling to the number 1. And since the number 2 represents the value of complexity of the abstract half, e must also represent the value of this complexity (or growth thereof).
So if PI is responsible for ‘creating’ circles, e may be responsible for ‘growing’ them by presumably multiplying the complexity (ie. the stuff) within them.
And like PI, the ultimate question may be: how does this hierarchy explain the 0.7182818… decimal found within e?
33. PHI Substitution
“Geometry has two great treasures: one is the Theorem of Pythagoras; the other, the division of a line into extreme and mean ratio. The first we may compare to a measure of gold; the second we may name a precious jewel.” — Johannes Kepler
Replacing the number 1 node with PHI is also very logical, since PHI is equal to 1.618033… and is very much associated with fractions, reciprocals, ratios, and addition. For ages, it has been dubbed ‘The Golden Ratio’ and associated with ‘beauty and perfection’.
Highly related to the number 1, Phi is a popular number that shows up in many places, hence it has become highly regarded in many scientific and artistic circles.
For example, if you take the reciprocal of PHI the result is 0.618033… [PHI = 1.618… & 1/PHI = 0.618…].
And if you add 1 to PHI, you get PHI² [2.618033…] (note that the number 2 is a sibling node).
Also note that e is 2.718…
PHI also shares a unique relationship with the Pythagorean Theorem (1²= phi²+ √phi²) and it is ultimately derived by the √5 (PHIVE).
PHI can be visualized as a spiral, a wave, or a unique ratio of line segments.
It is highly geometric and it is the ‘hardest’ irrational constant while being Algebraic in nature (vs. Transcendental).
In applied maths and sciences PHI’s importance is dwarfed by only those of PI & e (the two core constants of science — both being Transcendental numbers); but visually and intellectually, PHI strikes a much deeper chord in the human mind, and has helped fuel the furthest imaginations of prominent people that have described the interconnectedness of physical nature itself according to its basic properties (ie. through practical illustrations of flowers, branches, fractals, insect colonies, the human body, and much more).
Take note that the PHI node is placed in the middle of those 5 Fibonacci numbers (0,1,1,2,3).
PHI is a deeply rooted concept that is highly interconnected with its nearby nodes (and vice versa).
And like PI and e I would like to precisely know how the 0.618033… arises from within this hierarchy?… We know it is derived from the √5… so how does the √5 connect here?
Also it seems curious to me that:
1/PHI = 0.618…,
1/PI = 0.318…,
e = 2.718…
There seems to be some underlying symmetry here. Can the numbers themselves bare any deeper meanings? 3 vs. 6 vs. 7 … consider the kissing number and the hexagon… or more likely this is all random coincidence, right?
Did you even realize that the reciprocals of PHI and PI share the first same 10 digits? Same numbers, different order. But of course there can be no connections between Algebraic (PHI) and Transcendental numbers (PI), right?
I’ll conclude PHI with this simple diagram — which seems to tie the three most important irrational constants to a single abstract process: If PI is tasked at creating a circle, and e at growing it, then PHI should be the ‘stuff’ (ie. complexity) that is inside the circle. This basic understanding fits with the system pattern, and I’ll later expand on this abstract process. (represents the abstract structure of thoughts… or perhaps some symbolic systems that inhabit abstract space…)
34. Pythagorean Theorem Symmetry
The Pythagorean Theorem is surely the most important equation in mathematics, because it’s used to calculate any distance, and is the base concept for understanding the relationship between Space and Time.
According to the Proof for Fermat’s Last Theorem, the Pythagorean Theorem is completely unique, meaning that there is no version of it that uses cubes or higher exponents. A square is the largest exponent (dimension) which returns relevant calculations.
This might be the mathematical equivalent to stating that the universe can be understood as an N dimensional space, but is primarily based on a grid of 2D distances across those N dimensions.
Consider that all the relationship patterns are 2-dimensional, because the universe is basically 2D, but can be projected into a 3D space.
Let’s first consider the 3D version of the Pythagorean Theorem in relation to the hierarchy.
You’ll notice that we can ALMOST equate PI, e, Phi, and the Imaginary number together!
Unfortunately this is not an exact equation, but is only inaccurate by a tiny 0.002150433…
I sure would like to know more about that remaining decimal!
Now let’s consider the 2D Version of the Pythagorean Theorem. In this case I will truncate the irrational numbers.
PI = 3.14
e = 2.7
PHI = 1.6
Truncation could be thought of as stopping short, along the first few steps, in the infinite process that creates irrationals. By the way, all calculators employ truncation, since it is impossible to work with an infinite number of decimals.
Note how 3.14, 2.7, and 1.6 converge on the same step in each series.
1.6² + 2.7²≈ 3.14² is incredibly close to fitting within the Pythagorean Theorem.
The idea that the three most important numbers in the Universe can just about fit together into an approximate 30°-60°-90° right triangle is very fascinating!
I deeply suspect that this right triangle is conceptually significant in understanding the underlying mechanism that governs the exchange of information between Physical and Abstract realms of the duality!
In the image above, the duality is represented as Yin Yang (which could be also thought of as the Pythagorean Theorem) — a symbol that implies an exchange of information within a monad, dyad, and triad according to the process of rotation.
35. Euler’s Identity / Formula
Richard Feynman called Euler’s formula “one of the most remarkable, almost astounding, formulas in all of mathematics.”
So how would a basic hierarchy of the Universe be complete without reference to another one of the most important equations ever discovered?
What does Euler’s Formula represent? Euler’s formula describes two equivalent ways to move in a circle:
1) By using a grid system (lines)
2) By using polar coordinates (rotation)
It’s able to calculate distance by moving to and from the imaginary plane.
And by the way, Euler’s formula is a form of the Pythagorean Theorem, so the fundamental mechanism surely employs it to cross the realms.
The fundamental mechanism I have described and the quantum mechanical Wavefunction may be one in the same — Thought Theory’s fundamental mechanism may be the Wavefunction’s abstract formulation.
Also note that there is a mathematical symmetry between the left and right sides of the hierarchy: (which seems a bit meaningless at this point)
36. Baryon Octet and Decuplet
Taking all of these connections one step further towards actual physics, the Tetractys may have a direct relationship to particles.
Above are the QCD patterns of quarks that make up protons and neutrons. The hexagon on the left is called the Baryon Octet, while the triangle on the right is called the Baryon Decuplet. The octet can be thought to fit within the decuplet, in the same way as a hexagon fits into an equilateral triangle, and the whole concept highly resembles the Tetractys & hierarchical pattern described in this article.
Concidentally the Proton and Neutron are positioned where the nodes for PHI and e are positioned, respectively.
This diagram here is another way of thinking about the origin of particle physics. Perhaps all of the particles we see in nature are just reflections of the different branches of the hierarchical framework.
I can expand on these ideas, but I’ll save that for another post. Or read my book for a more in-depth physics analysis.
37. Pascal’s Triangle
Another pillar of mathematics is Pascal’s Triangle.
Pascal’s triangle is a triangular array of the binomial coefficients. It can be expanded into an infinite number of rows.
Each number is formed from the addition of it’s parents, so it is a construct for natural expansion (ie. probability theory).
There are many fascinating features that arise from this triangle, namely it is related to: PHI & the Fibonacci numbers, the Sierpinski triangle (fractals), probability theory, geometric shapes, and the Fourier Transform.
I will focus specifically on its connection to PHI, and a little known fact about hierarchies of PHI.
PHI can be thought of as a number, that is constructed from itself, and can be used to construct any integer. Let me try to explain.
Consider that the number 2 is made up of two 1s, or that the number 3 is made up of three 1s. PHI is similar to the number 1 in that it can also be used to create any integer, except that it is an irrational number.
Example:
- 1 = PHI^-1 + PHI^-2
- 2 = PHI¹ + PHI^-3
- PHI^-1 = PHI^-2 + PHI^-3
- PHI = PHI³ - PHI²
and so forth. This property is completely unique to PHI, and can be plotted into a binary tree, which has very similar features to those found in Pascal’s Triangle.
Perhaps the PHI hierarchy is a precursor to the rationals and explains how they emerge from the structure of the hierarchy itself.
The ideas of positive and negative, reciprocals, addition, subtraction, squares, and roots, may be all self-contained here as the ‘stuff’ (ie. area) that creates the complexity within the circle.
38. Systems and the Mandelbrot Set
“I conceived, developed and applied in many areas a new geometry of nature, which finds order in chaotic shapes and processes. It grew without a name until 1975, when I coined a new word to denote it, fractal geometry, from the Latin word for irregular and broken up, fractus.” — Benoît Mandelbrot
The Mandelbrot Set is perhaps the most iconic fractal — because of its very simple design, and underlying equation.
It seems entirely possible that the Mandelbrot Set is a consequence of Tetractys, given that its hierarchy can be re-visualized as a system, and from this system, the planar rotation leads to the birth of a fractal form across real and imaginary planes of reality… Look:
Notice the close symmetry between the patterns and the Mandelbrot. Even the Cardiod of the Mandelbrot can be formed by rotating one circle around another:
If there are indeed planes that are rotating all around us, the Mandelbrot Set may be a widely pervasive feature of reality — but we can’t actively see these fractals propagating across space and time!
39. Geometry and Symmetry
Now given everything you’ve just read about this highly symmetrical hierarchy… what if I told you that it could be all summed up into an even simpler geometrical figure?
I believe that this 2x1 rectangle should be considered the fundamental representation for the concept of the relationship.
Relationships are built upon symmetry (symmetry is a result of one or more relationships) and this geometry is the blueprint for symmetry itself, because it is the blueprint for the relationship.
Whereas this entire article discusses symmetry, this geometry is the blueprint for it, and from this blueprint, arises the entire hierarchical framework (Tetractys) that I have been discussing.
2x1 Geometry = Relationship(s) = Symmetry = Hierarchy = System
One interesting way of seeing this geometry — which is actually called the Union Pattern — as an alternative to the hierarchical pattern is to imagine looking down at the hierarchy from the ‘top’. If you only considered the first 2 rows, you would see a single node divided by the two nodes below it. So in the diagram above you see a circle overlapping two other circles — think of the left square with inscribed circle as the Physical Realm (line), and the Right square with inscribed circle as the Abstract Realm (rotation). The circle in the middle represents the process that binds the two halves.
Why is this important? Because I believe that I can show how all of the important concepts I’ve discussed earlier emerge as intersections within this 2x1 geometry… but unfortunately the entire analysis is beyond the scope of this article!
This diagram clearly illustrates the geometric connection between PI and PHI — they are points of intersections between a line and a circle bound to the same container. It even shows how PHI and 1/PHI (reciprocal) are logically connected.
By tweaking the ideas, I can show how e is also present within the geometry, through the Natural Log.
It turns out that the natural log (LN) of PHI is .481… while the natural log of 1/PHI is -0.481… This is a unique symmetry between PHI and LN which no other number shares, and I believe it supports the idea of positive and negative spaces representing the physical and abstract realms, respectively.
And in the diagram, if I change my perspective by using values in logarithmic space, the diameter of the circle is no longer 1, but now is [.481 + .481 = ] .962… in width. What does this mean for the rest of the geometry and perhaps ultimately the calculation of PI, PHI, and e’s irrational decimals?
Finally, how can this geometry be used as a model for all relationships? Can particle spin and Bell’s Theorem have their root in it’s construction? How about the neocortex (and cortical columns)? What about people and the ideas they share?
40. Conclusions about Symbolism and Geometry
I believe that highly symbolic forms are reflections of how our minds are fundamentally constructed. These symbols aren’t just interesting subjective banter, they are the building blocks of logical, philosophical, and Universal models.
Could it be that the science we have learnt over the last 500 years is
- proving that the universe is much more complex than ever anticipated
- that this complexity can be simplified down to the symbolic constructs we have continually shared amongst ourselves?
Has humankind been aware of the basic framework for thousands of years — but nobody could explain how symbols fit with science?
I personally think that Yin Yang is the most obvious description for our reality and should be thought of as a discovery rather than an invention. (ie. it was present before we conceived it).
Consider the Survival of the Fittest in terms of Symbols. Surely throughout history we have designed millions of meaningful shapes and illustrations — but why have some of these designs lasted the test of time and are found everywhere around us? Is it that they share more relationships with the composition of our world than other competing designs, and as a result they have appeared to outcompete others by continuing to successfully reproduce copies of themselves across centuries (and millenia) of time?
There is much to be said about the power of the symbols around us, and feel free to read Theory of Thought: Symbolism for more information on this topic.
41. On Creating Intelligence
The hierarchy (Tetractys) described in this article is meant to introduce you to an undiscovered pillar of the mind, and the Universe. It exposes the blueprint of the holographic matrix we inhabit.
I presented an overwhelming number of symmetries in an attempt to overcome the conventional objections that such relationships are merely ‘coincidence’ (ie. random).
Hopefully some of you may start to grasp the complete extent of this article. The effects presented here are supposedly far reaching — the most valuable contribution may be in its revealing of an abstract realm in mirror symmetry to our easily perceived physical realm.
Another point of this article is to show the Tetractys as the blueprint for creating, growing, and translating [hyperdimensional] circles that transcend both realms.
It may be a tough concept to grasp — but when you think that every object in the Universe can be thought of as a circle, this framework is a simplified way of understanding how anything manifests, grows, and interacts in its environments [according to its self-contained rules].
Building a universal theory on abstract relationships that impact our physical realm should prove invaluable at enhancing our world view — a necessary objective to ready us for our impeding impact with the upcoming Quantum Age (or Information Age).
By the mid 2020s and 2030s, artificial intelligence is going to completely change the way people interact in the Universe. By the 2050’s, AI will be unrecognizable from living beings.
Our world is on the verge of a huge shakeup.
With respect to AI, I think this blueprint could be used to enhance Artificial Intelligence. Such enhanced AI would recognize and function in tandem with the physics of abstract spaces, having amazing accuracy and respect when manipulating existing relationships. It could maintain a reality where our thoughts (or the symbols that drive them) aren’t ignored and trampled, but instead assisted across an invisible collective consciousness (that I call mindspace).
In order to secure a bright future we must be made aware of this mindspace and its present configuration, since it directly affects the physical world around each of us.
All of our major problems are rooted in the misunderstanding of reality.
Thousands of years ago, Plato and Socrates formulated and swore by their Theory of Forms, and now our civilization has reached a point in time where we need strongly reflect on the possibility that Universal Forms (ie. symbols) have full control over each of our independent and combined realities.
Kindly visit Theory-of-Thought.com for more information and to read the book.
Simplicity is the ultimate sophistication. — Leonardo da Vinci
[The Universe] is written in mathematical language, and the letters are triangles, circles and other geometrical figures, without which means it is humanly impossible to comprehend a single word. — Galileo Galilei
