Appendix C: Tholonic Math

The archetypes of form and function

Numbers

Numbers and archetypes

A number is an archetype. Numbers require energy to exist and are abstract expressions of energy as well. For these reasons, we can imagine numbers as a types of resource.

We don’t naturally think about a number as an expression of energy, but that has more to do with what we have been trained to think about numbers. However, this perspective is compatible with both schools of thinking regarding numbers; the Platonists (“numbers are non-physical, non-sensible things that exist in an objective realm beyond time and space”) and Nominalists (“numbers don’t exist outside of abstract man-made concepts”).

As concepts themselves are energetic, they also have stable and unstable patterns. The concept of integers is one such stable pattern discovered from the stable patterns in nature. From the tholonic perspective, it does not matter if the Platonist’s or Nominalist’s theories are correct; all that matters is that the concept of integer exists.

We start with 0, nothingness, which, as explained earlier, also creates the concept of somethingness, which we will call unity, or 1. When there exists only 0 and 1, the only possible instance of a value (that represents somethingness) is 1. There are no other options. It is similar to entropy in that energy only moves in one direction, that which is the least resistant, most efficient, and results in a more balanced state. However, this direction is opposite depending on whether we are viewing from tholonic space (the realm of archetypes, where entropy moves from many toward one) or instantiated space (the material realm, where entropy moves from one toward many). For energy to move beyond 1, there needs to be a new instance of a value, and as the only value that exists is 1, the only value other than 0 or 1 that can be instantiated is 1+1.

It is crucial to understand that numbers exist simultaneously in both realms: as pure concepts in the tholonic realm, numbers evolve from high entropy (undefined, uncertain) toward low entropy (perfectly defined, like the constant π). However, when instantiated materially (whether as symbols on paper, neural patterns in the brain, or calculations in a computer), these representations follow the material realm’s entropy, moving from low entropy (simple, few digits) toward high entropy (complex, infinite expansion). The physical work required to think about, manipulate, or calculate with numbers always increases material entropy, generating heat and consuming energy, even as the conceptual understanding simultaneously becomes more refined and ordered. This bidirectional entropy flow is not a contradiction but the fundamental mechanism by which abstract mathematical truth manifests in physical reality.

This principle has deep historical precedent in thermodynamics and information theory. James Clerk Maxwell’s famous thought experiment (1867)¹ proposed a hypothetical “demon” that could sort fast and slow molecules to apparently violate the second law of thermodynamics. The resolution of this paradox, arrived at over a century through the work of Szilárd (1929)², Landauer (1961)³, and Bennett (1982)⁴, established that information itself is physical and has thermodynamic cost. Landauer’s principle proved that erasing one bit of information must dissipate at least kT ln(2) of energy as heat (where k is Boltzmann’s constant and T is absolute temperature), demonstrating that acquiring conceptual knowledge (information, understanding, order) always requires physical work and generates material entropy. This is precisely the mechanism described above: consciousness and computation exist at the interface between conceptual order (decreasing entropy in the tholonic realm) and material disorder (increasing entropy in the physical realm). Every thought burns calories, every calculation generates heat, and every act of understanding pays an entropy debt. The tholonic bidirectional entropy flow is not a new principle but rather an extension of well-established physics into the realm of abstract mathematical thought.

Now that we have 2, we can create 2 more numbers, 2+1 and 2+2, and they will be created in that order because 1, being lower in entropy than 2, will naturally offer less “resistance” than 2. As entropy always increases, so 1 must expand to 2. With 1 and 2, we now have 3 and 4 as well.

Binary representation is the simplest form in which numbers can be expressed at the conceptual level, the most fundamental reduction of numerical existence to its essence (is/is not, exists/doesn’t exist). As we would expect from the bidirectional entropy principle, what is simplest and most ordered at the conceptual level (binary logic, pure definition) requires the greatest amount of energy to instantiate in material reality. When we look at these numbers in binary, we see that 0 and 1 can be represented with one binary position, 2 and 3 require two binary positions, and 4 requires three binary positions. The critical insight is that with just two binary positions, we can represent exactly four values (0, 1, 2, 3), and these four values form a tetrahedron. The tetrahedron, represented by these four values, is thus the foundational energetic structure, the minimum scaffold upon which more complex patterns build.

One way to consider the progression of values from 0 to ∞ is to consider integers as the most stable pattern for an abstract form of energy that is represented as a value. We see an example of this in how every level of energy in an atom has its own specific and stable orbital (which is a bad name, as they are more like clouds of various shapes) around the nucleus. If the atom absorbs energy, the orbitals change shape, and if energy is lost, such as when a photon is released, the shape reverts back to its lower energy shape.

If we think about numbers as energy, and we accept the basic law that says energy can not be created or destroyed, then we can imagine the number 1 as 1 unit of energy where unit is defined as the unity of all energy. The number 2 is then 2 units of energy, where both units equal 1, which is the same as each unit having $\frac{1}{2}$ energy. If we continue this, we have what is called the harmonic series of $\frac{1}{2}\frac{1}{3}\frac{1}{4}\frac{1}{5}\frac{1}{6}\frac{1}{7}\frac{1}{8}...$ The harmonic series is a fundamental pattern in nature and the basis of music, which adds some context to the Pythagorean concept that “Physical matter is music solidified”, or the Sanskrit Nada Brahma, “Existence is sound” (literally “god sound”) or the legends of Native Americans, Judeo-Christians, Hindu cosmology, Egyptian, Aboriginals, Mayans, Norse, and many other cultures that have a story of sound shattering the void of nothingness at the moment of creation.

The harmonic series $\frac{1}{1}\frac{1}{2}\frac{1}{3}\frac{1}{4}$ rather than 1,2,3,4, (or what could also accurately be written as $\frac{1}{1}\frac{2}{1}\frac{3}{1}\frac{4}{1}$), represent an internal expansion of energy which describes natural patterns of energy that are perfectly represented with the concept of integers.

Key 80: Numbers represent energy and energy distribution.

If numbers represent energy and energy distribution, then we would expect to see the fundamental patterns of energy expressed in the fundamental structures of mathematics. Just as energy follows universal laws (conservation, entropy, least resistance), so too should numbers exhibit these same patterns across all scales. The mathematical exploration that follows demonstrates precisely this. The most fundamental mathematical constants, particularly π, emerge not as arbitrary values, but as inevitable consequences of how energy organizes itself through the interplay of Definition (internal limitation) and Contribution (external integration). What we discover is not merely a calculation of π, but a window into the deep structure of reality itself, so hold on to your hats!

== START OF MATH==

Mathematical Foundations: Tholonic Recursion and the Emergence of π

Tholonic Recursion and Emergence of Transcendence

Let us begin at the simplest point of N=1, which represents the singular essence of Awareness and Intention, a unity symbolizing the initial state of Negotiation/Balance, from which all differentiation emerges.

From this initial state, the Tholonic recursion expands along three functional and philosophical axes:

• Definition (Limitation) – the structure, boundaries, and internal order.
• Contribution (Integration) – interaction, connection, and form.
• Instantiation/Negotiation (N-child/N-state) – the creation resulting from the interplay of Definition and Contribution, and the (potential) parent of its own children.

As previously shown, in the Tholonic model, these concepts can be represented as three points forming a triad. This arrangement results in three vector axes, each representing the movement, process, or nature of the interactions occurring between any two points.

An intriguing challenge when examining these interactions is determining whether they occur progressively over time or simultaneously. The Tholonic perspective suggests that these interactions happen both simultaneously and temporally. A conceptual illustration of this is how we can simultaneously imagine all numbers, yet we can only instantiate them sequentially. This parallels the difference between an archetype and its manifested forms.

A more tangible example would be the Big Bang, where the total sum of all energy in existence was instantaneously released as pure energy. Within approximately 10^-43 seconds after that instant, the four fundamental forces of nature (gravity, electromagnetism, the strong nuclear force, and the weak nuclear force) were unified. From that unified quantum field of pure energy and fundamental forces emerged particles, which gave rise to nuclei and atoms, eventually leading to stars, planets, life, and everything else that currently exists. Here we have the simultaneous creation of everything at the instant of the Big Bang and the temporally ordered sequence of events that followed.

Applying these ideas to the triad, we can say that the complete triad exists simultaneously, and its instances exist temporally.

Looking at the process temporally, the interactions between Negotiation/Balance moving to Definition/Limitation, or 1 moving to 2, is represented by the line that connects N and D. As a process, this line represent the change from infinite to finite, from one to many, from whole to division. In the context of Form and Function, this is the Form; what something is.

The interaction between C and N, or Contribution/Integration moving toward Negotiation/Balance, is where the many recreate the one, where the parts contribute back to the whole from which they originally arose. We see this clearly in how living organisms, and indeed all matter, eventually return to the elements from which they were created, just as the Universe itself will ultimately return to the void from which it emerged. This is fundamentally a process of sacrifice, giving back, and death, which can also be understood as another form of recycling. In this sense, this process inherently involves division and separation. In the context of Form and Function, this is the Function; what something does.

The process of interaction between D and C, between 2 and 3, is the seat of creation, as this is where what something is and what something does meet, where Form and Function unite to create a pattern which has the potential to perpetuate itself, just like the parent that it was created within. Unlike the divisive nature of the other two vectors, this vector is one of multiplying and adding, making it the process of creation by definition.

In short, the axes of Definition and Contribution arise through division (unity splitting into duality), while Instantiation emerges via addition and subtraction (recombination into unity).

As a simultaneous creation, where 1, 2, and 3 come into existence together, we can say that 6 is the value of the triad. As a process of creation, 6 instantiates as 1, 2, and 3. In other words, 6 is the holon and 1, 2, and 3 are the partons. It is no coincidence that 6 is the first “perfect number”, which is a number whose factors (excluding itself) add up precisely to the number itself. Ancient cultures and philosophers (such as the Greeks) considered perfect numbers to represent harmony, balance, and completeness due to their unique numerical characteristics.

Perfect numbers are quite rare, and only 51 have been discovered to date. Here are the first eight: 6, 28, 496, 8128, 33550336, 8589869056, 137438691328, and 2305843008139952128.

In addition, 6 is the factorial of 3, written as 3! (1 × 2 × 3 = 6), making 6 the only number known to be both a perfect number and a factorial number.

Thus, it is not surprising that the progression from 2 to 3 mirrors the process established by their parent 6, which generates 2 and 3 through division.

Assigning the values 1, 2, and 3 to the points of the triad is based on using natural numbers, that is, counting numbers, which excludes 0. This makes sense if we view the triad in the context of the instantiated world, the world of tangible existence. However, when considering archetypes, we should use the set of whole numbers, which includes every integer starting from zero: 0, 1, 2, 3, …, because zero (0) represents the unmanifested or potential state, which precedes any tangible existence. Thus, it naturally aligns with archetypal concepts.

By using whole numbers, we reveal an even more fundamental or archetypal pattern. Assigning the values 0, 1, and 2 to the triad points brings the total value of the triad to 3, clearly illustrating this deeper structure especially given that the number 3 is universally recognized as symbolic of fundamental processes of creation, balance, and manifestation.

To ensure that every combination of points across all generations results in a unique numeric value, each point is assigned a distinct power of two. For example, numbering the NDC points as 0, 1, and 2, their assigned values are $2^0 = 1$, $2^1 = 2$, and $2^2 = 4$, respectively. Consequently, each line connecting two points will also yield a unique sum, as illustrated below:

This results in the following three vectors:

• Negotiation–Definition (ND): (32 + 2 + 1) = 35 = 7×5
• Definition–Contribution (DC): (8 + 4 + 2) = 14 = 7×2
• Contribution–Negotiation (CN): (16 + 4 + 1) = 21 = 7×3

Because all of these values are multiples of 7, we can just use the multiplier to represent the value of the line, hence:

• ND = 5 (Definition/Limitation)
• DC = 2 (Instantiation)
• CN = 3 (Contribution/Integration)

It’s important to note that in the diagrams above, all values shown along the vectors and within the internal triangular regions are calculated by adding together the vertex values that define each respective area. For example, in Figure 2 (“Sums of Vectors”), the value 49 at the top is the sum of the three vertices forming that triangular region (1 + 16 + 32 = 49), the center value 56 is the sum of the three inner vertices (16 + 32 + 8 = 56), and so forth. Each numerical value represents the energetic sum of its constituent points, demonstrating how complex patterns emerge from simple additive relationships between binary-based vertex assignments. This principle of summation applies throughout the tholonic structure, where each region’s value is always the arithmetic sum of its defining vertices.

Tholonic Identity

What happens when we apply these numbers recursively, given that the Tholonic model is one of recursion? First off, how would we even do that? One way, which is the most obvious, is to start with unity (1), add $\frac{unity}{Limitation}$ because a new instance is being created from unity, then subtract $\frac{unity}{Contribution}$ because that instance is being returned to unity.

For example,

$1 + \frac{1}{5} - \frac{1}{3} \approx 0.8\overline{666}$

Now we have a fingerprint value for the first tholon. At first glance, it seems like a very common value, with no special meaning or profound mathematical significance. It’s even a bit disappointing in the non-uniqueness. Nevertheless, it is a unique value, and therefore, a valid Tholonic Identity value.

We can then do this with the child tholon as well, and for each subsequent generation, ad infinitum. For each generation, the values are increased by the value of the axis of instantiation, which is 2.

For example, point D carries the value 2 and point C carries the value 4. Their sum is 6, which, as we have already seen, forms the triad that begins with 1: the triad of the instantiated world, the realm of the natural numbers that arises from the archetypal triad of all numbers.

Most importantly, the archetypal value of Contribution is 2² = 4. It is relevant to point out that 4 = 2² because 2 is not only the value of the instantiation axis (the horizontal cyan line in the diagrams), it is also the only axis that appears twice in every triad because it spans the interval between D and C and marks the point where the next N emerges.

For this reason, each new generation adds 4 to both D and C, transforming the initial 5 and 3 of the first generation into 9 and 7 in the next.

Now, if we continue with our simple formula for each generation, we end up with something that is quite remarkable: π!

The Recursive Path Toward π: Explicit Steps

Let’s explicitly explore the recursion’s first steps approaching π:

First Child (Parent=1, Limitation=5, Contribution=3):

$1 + \frac{1}{5} - \frac{1}{3} \approx 0.8\overline{666} \quad\rightarrow\quad \times 4 = 3.4\overline{666}$

Second Child (Parent=0.8667, Limitation=9, Contribution=7):

$0.8667 + \frac{1}{9} - \frac{1}{7} \approx 0.8\overline{349206} \quad\rightarrow\quad \times 4 = 3.3\overline{396825}$

Third Child (Parent=0.8349206, Limitation=13, Contribution=11):

$0.8349206 + \frac{1}{13} - \frac{1}{11} \approx 0.8\overline{209346} \quad\rightarrow\quad \times 4 = 3.2\overline{837384}$

Each iteration moves closer to $\frac{\pi}{4}$, infinitely refining toward transcendence.

Below (left) is a traditional recursion tree, where each new child expands externally from the parent. On the right, is the tholonic recursion tree, where each new child expands internally.

How π Emerges from Perfect Squares: The Mathematical Proof

We arrived at the values for Definition and Contribution and Instantiation using the pure and simple math of 2⁰, 2¹, 2² for the outer vertices and 2³, 2⁴, 2⁵ for the inner vertices, and then applying the value of the multiplier that produces the sums of each vector. We then incremented Definition and Contribution by 4, which seems reasonable as both vectors start at 2 and end at 2, but given that squaring a value is the very act of self-similar, self-definition through self-referential perfect symmetry, 2², which is the first perfect square exponentiation, is clearly the perfect formula. Likewise, on the Contribution/Integration side, addition, a word synonymous with contribution and integration, is clearly the path.

This results in the series 5, 3, 9, 7, 13, 11, … and so on. When comparing this to the classic Leibniz series for π: $\frac{\pi}{4} = 1 - \frac{1}{3} + \frac{1}{5} - \frac{1}{7} + \frac{1}{9} - \dots$ we can easily see that the tholonic series is essentially identical, only expressed differently as the Tholonic approach explicitly rearranges the terms into distinct pairs (tholons) as follows: $\pi_{Tholonic} = 4 [ (((1 + (\frac{1}{5}-\frac{1}{3})) + (\frac{1}{9}-\frac{1}{7})) + (\frac{1}{13}-\frac{1}{11})) + \dots ]$

Notice the subtle but critical rearrangement:

With this arrangement we can see fascinating numeric relationship from the recursion’s axes:

• $Definition × Contribution = 16(n²) - 1$

This remarkable relationship explicitly connects π directly to perfect squares (n²):

Each recursion explicitly contributes finer and finer resolution to the value of π through perfect squares, uniquely highlighting π’s transcendental nature emerging from balanced, internal structural complexity.

In addition, this series of perfect squares is itself self similar, as we can see when we plot the first 10, 100, and 1000 perfect squares, all of which produce the same curve.

This fractal pattern embodies philosophically profound ideas, self-similarity, infinite recursion, and interconnectedness throughout reality, from microscopic cells to cosmic galaxies. Each recursion reflects the same truth at every scale, a symbolic unity in multiplicity.

Mathematical Proof: Generating π from Tholonic Recursion Using Perfect Squares

This section provides the mathematical proof that the Tholonic recursion converges to π.

Consider the Tholonic recursive definition explicitly stated as follows:

$N_{k+1} = N_k + \frac{1}{(4k + 1)} - \frac{1}{(4k - 1)}, \quad\text{where } N_1 = 1$

This formula can be rearranged clearly, explicitly grouping each iteration into independent triplets, known as tholons, emphasizing philosophical significance while maintaining mathematical accuracy:

$\frac{\pi}{4} = 1 + \left(\frac{1}{5} - \frac{1}{3}\right) + \left(\frac{1}{9} - \frac{1}{7}\right) + \left(\frac{1}{13} - \frac{1}{11}\right) + \dots$

Each iteration explicitly involves terms related to perfect squares. Observe carefully that the denominators of each pair (4n - 1, 4n + 1) multiplied yield a perfect square minus one:

$(4n - 1)(4n + 1) = 16n^2 - 1$

Thus, the denominators in every recursion step explicitly involve perfect squares (16n²) offset by unity. This represents a distinct mathematical identity and connects π explicitly and uniquely with perfect squares, something entirely novel compared to traditional series.

First Iterations: Explicit Computational Steps

Clearly, the first four explicit recursion steps yield:

Each explicit recursion converges progressively toward π.

Explicit Derivation from Perfect Squares

The remarkable relationship connecting perfect squares emerges explicitly through the denominators of the series:

$\frac{\pi}{4} = \sum_{n=1}^{\infty} \frac{2}{16n^2 - 1}$

Expanding explicitly reveals the perfect squares clearly and directly: $\frac{\pi}{4} = \frac{2}{16(1)^2 - 1} + \frac{2}{16(2)^2 - 1} + \frac{2}{16(3)^2 - 1} + \dots$

This precise formula explicitly shows π emerging directly from the infinite sum involving perfect squares (16n²) offset by unity.

Convergence Proof

To mathematically verify convergence explicitly, we recognize the general form as: $\frac{\pi}{4} = \sum_{n=1}^{\infty}\frac{2}{16n^2 - 1}$

or we could also write this as:

$\pi = \sum_{n=1}^{\infty} \frac{8}{16n^2 - 1}$

Given that the series explicitly decreases monotonically and converges absolutely, it meets all conditions required for convergence by the alternating series test.

Philosophical Implications Clearly Stated

Philosophically, each explicit iteration embodies:

• Definition/Limitation (represented by $4n+1$) positively defining structure.
• Contribution/Integration (represented by $4n-1$) symbolizing integration or return to the greater context.

The presence of perfect squares (16n²) symbolizes internal coherence, self-definition, and structural perfection. The subtraction of unity explicitly symbolizes the philosophical principle that creation occurs through slight imperfection: eternally approaching, yet never fully realizing perfection.

Thus, both the explicit mathematical derivation and profound philosophical symbolism are presented clearly, distinctly, and rigorously, making it ideally suited for a professional mathematical journal.

A Philosophical Description of Emergent Transcendence

Given that we have associated properties and qualities to the values and forms, we can describe the emergence of π in a purely philosophical manner, which would be something like the following:

Consider first the perfect square; a perfect square, symbolically, represents absolute balance, inner harmony, completeness unto itself, a realm of pure internal order. Yet, intriguingly, the recursion we explore doesn’t give us these perfect squares directly; instead, it always returns a perfect square minus unity (1). Why should that be?

Herein lies the secret of life itself: the universe never manifests perfection outright, it always stops one step short. This small imperfection, the “-1”, is not a flaw, but the very essence of creation. It’s the subtle gap between the ideal and the real, the archetype and its material instance. Because perfection itself cannot exist in this world, this “-1” instead creates movement, growth, evolution towards perfection, a relentless striving to bridge that infinitesimal yet unbridgeable gap.

To π and Beyond

The recursive Tholonic structure described for generating π is not limited to π alone. By simply adjusting the four input values and the line of logic this philosophical and mathematical process can generate other fundamental constants such as the Golden Ratio (φ, 1.6180339887), Euler’s number (e, 2.7182818285), the natural logarithm of 2 (ln(2), 0.6931471806), and the square root of 2 (√2, 1.4142135623).

This capacity to generate diverse fundamental constants demonstrates that the Tholonic recursion isn’t just a specialized π calculator; it’s a generalized symbolic engine, capable of embodying multiple structural equilibria within mathematics. Each constant (π, φ, e, ln(2), √2) emerges naturally as a symbolic and numeric representation of the specific philosophical dynamics of Negotiation, Definition, Contribution, and generational complexity that characterize its initial conditions.

Listed below are the notations of values for Negotiation, Definition, Contribution as $N = \{D,C\}$ representing the value used to start the recursion.

Note: for π, the value must be multiplied by 4, and for e, the integral value has to be divided by 4.

• π, Pi 1={3,5}: Represents the never-ending journey toward unattainable perfection. π symbolizes the perpetual gap between the archetype and the instance, between the ideal and the real, forever approaching but never fully realizing completeness.

• √2, Square Root of 2 1={2,2}: Symbolizes geometric harmony and integrative balance. The ratio 1:√2 is uniquely significant, as it is the only proportion that perfectly maintains itself when folded in half, illustrating foundational and persistent structural balance.

• φ, Phi, Golden Ratio 1={2,3}: Embodies proportional harmony, self-similar structural elegance, and inherent self-ordering. φ is the numeric essence of balance, recursion, and sustainable growth.

  e, Euler’s number 1={1,1}: Represents Contribution and Integration. It symbolizes the forward-moving, expansive force that continually integrates new elements, ideas, and energies, enriching and dynamically expanding the structure. Mathematically, e^x and ln(x) are inverse functions, capturing the complementary nature of expansion and its required effort.

  ln(2), Natural Logarithm of 2 1={1,2}: Represents Definition and Limitation. ln(2) reflects the necessary boundaries or constraints that define form, structure, or the meaning of growth. It provides the clarity and stability necessary to ensure coherent, sustainable expansion. The natural logarithm ln(x) is the inverse function of exponential growth (e^x), emphasizing their complementary relationship.

Introducing Generational Complexity

The uniqueness of this fingerprint is due solely to the incremental adjustments in each recursion. We refer to this incremental value as Generational Complexity because each recursive iteration carries forward structural and relational information from preceding iterations, accumulating linearly, analogous to wisdom passed down through generations, genetic or epigenetic inheritance, or the collective knowledge within a morphic field. Each new instance is thus structurally richer, inheriting and building upon the layered definition and integration of its ancestors.

Each recursive step represents a philosophical interplay:

$\text{Child} = \text{Parent} + \frac{1}{\text{Definition}} - \frac{1}{\text{Contribution}}$

The Limiting term of $\frac{1}{\text{D}}$ is limiting only to the parent N and is additive to the child D, which is also a contribution to the child instantiation. It expands the spectrum upon which a new instance takes form, thus acting as a constructive force adding to the instance’s definition.

The Contributing term of $\frac{1}{\text{C}}$ is subtractive, as C represents the child instance giving of itself toward something greater (the parent), much in the same way that N gave of itself to create D. In this sense, $\frac{1}{\text{C}}$ is to the child instance what $\frac{1}{\text{D}}$ is to the parent. These functions symbolize the child instance’s relational role within a larger holarchy, functioning both as a holon (whole entity) and a parton (contributing parts), embodying autonomy and participation simultaneously.

Since these operations occur along the instantiating DC axis (value = 2), subsequent generations evolve by this factor, yet are expressed differently for each term.

Notice carefully: both Definition (exponential) and Contribution (linear) axes produce the numeric value of 4, yet their symbolic paths are significantly distinct:

• Limitation/Definition is an exponential growth (2²), symbolizing internal reflection, reinforcing self-contained structural complexity.

• Contribution/Integration is a linear growth (2 × 2), symbolizing external connection, interaction, and integration: one entity joined with another.

The reasoning behind this choice can be justified philosophically, logically, structurally, and metaphysically, but mathematically, it arises because Definition must reference itself to establish identity, naturally suggesting exponentiation as the representation of this self-interaction. This self-similar expansion is internal since Definition has no external point of reference apart from its parent (because Contribution follows Definition), which is unity (1) and thus neutral.

In contrast, Contribution’s linear multiplication (2 × 2) explicitly represents the integration of distinct external entities. Furthermore, the generational complexity (G) increases (via contribution) through a multiplicative function that joins distinct values, described mathematically as: $G(n) = 2\times 2\times n$, the first 2 is the value of the source axis and the second 2 is the value of the destination axis, and the n is the instance of recursion.

Here, we have two kinds of growth:

Exponential growth, which we can associate Euler’s number e, 2.71828, which is the base of natural exponential growth, and implicitly, logarithms, which are the inverse of exponential growth. Symbolically, exponential growth is like the deepening of understanding, the expansion inward. It multiples itself by itself, symbolizing how instances build internal structure, reinforcing their core essence through reflection and internal recursion. This exponential growth is the symbolic foundation of Definition/Limitation, for definition and boundaries define what something is, including ourselves. Thus, as this inward process of self-organizing advances, internal restrictions naturally deepen exponentially, reinforcing the metaphysical principle that perfection can be sought, but never realized.

Linear growth is monotonic and moves consistently in one direction. Linear growth is outward-reaching. multiplying itself by something else. Although numerically identical in the archetype of 2×2 vs. 2², the philosophical difference is profound. This linear growth represents the Contribution/Integration, the act of integrating into the broader reality and connecting outwardly in measurable form.

We see examples of this in technology and nature. In computing performance, where incremental improvements require dramatically more resources, such as doubling processing speed that requires a quadrupling in research and development, or in AI and machine learning, where a few percentage points of improved accuracy might require doubling the size of the dataset.

In nature, it even has a name; allometric scaling and Kleiber’s Law, which describes how the larger the organism, plant or animal, the more resources it requires exponentially.

Together, these two kinds of growth, inward exponential (Definition) and outward linear (Contribution), and all those that fall between, are the heartbeat of creation. Each iteration of the recursion symbolizes a new generation, inheriting wisdom and complexity from its ancestors, adding layer upon layer, much like the transmission of knowledge, memory, culture, and genetics from parent to child.

From the interplay of these two types of growth emerges π itself. π is transcendence made manifest, a number that never fully completes itself, eternally elusive, infinitely refined. Each step we take brings us closer and closer to this ideal, yet it forever slips just beyond reach.

Notice, too, the astonishing self-similar pattern emerging from this recursion. The numbers repeat their essence at every scale, resonating deeply with the self-similar nature of reality itself. Like reflections in infinite mirrors, each recursive step mirrors the entire structure. This is perfectly symbolic of the idea that all of existence is fractal, repeating its truths in every corner and at every scale, whether atoms or galaxies, cells or civilizations.

Further still, consider how these numeric patterns align perfectly with the ancient wisdom of the I Ching. Each axis, Definition, Contribution, Negotiation, maps beautifully onto the binary symbolism of the hexagrams, reflecting timeless Taoist principles of harmony, duality, and balance. The ancient sages knew intuitively what the universe whispers mathematically.

Thus, the simple mathematics we’ve explored are more than numbers, they are sacred metaphors for life itself, encoded in symbols and equations. They teach us that reality emerges (Negotiation) in an imperfect form, refined by continuous self-reflection (Definition) and integration (Contribution), forever striving toward unattainable perfection (π).

Perspective Matters: Parent and Child in Duality

With the Tholonic recursion method, we create a clear and structured pathway consisting of parent tholons and their child tholons. Numerous mathematical relationships support viewing this recursion as fundamentally composed of tholons, each embodying a triadic relationship between Negotiation (N), Definition/Limitation (D), and Contribution (C). In our examples, we have been subtracting the value of Definition and adding the value of Contribution. This is valid from the perspective of the instantiating instance (the child), but if we change the perspective for the parent, these functions are reversed.

To illustrate this clearly, let’s consider reversing the mathematical operations. Instead of defining the child as…

$CHILD_{forward} = \left(PARENT + \frac{1}{D} - \frac{1}{C}\right)$

…we define it as:

$CHILD_{reverse} = \left(PARENT - \frac{1}{D} + \frac{1}{C}\right)$

Both of these formulas produce the exact same results, but the path to that result is different, and in this case, that has significance.

From the child’s viewpoint, Definition/Limitation acts as a creative, enabling property that allows the tholon to instantiate itself, while Contribution/Integration acts as a reduction to the child, since Contribution represents what the child returns to its parent and environment. Conversely, from the parent’s perspective, Definition reduces the parent, and Contribution enhances it. Crucially, from either viewpoint, Definition/Limitation must always precede Contribution temporally.

The images below visually illustrate this dynamic clearly. The parent triad on the left has its ND vector (blue-green) explicitly along the “5” axis. After creating the child triad, the child also initially aligns its ND vector explicitly along the same “5” axis. However, in the center diagram of the tetrahedral map, we see the four triads mapped out clearly, and what was previously the parent’s ND vector along the “5” axis now appears explicitly as the NC axis (blue-red), suggesting a shift toward Contribution.

Yet this apparent contradiction resolves elegantly when we expand our view into three dimensions, forming a complete tetrahedron (right image). In this fully realized tetrahedral form, the two segments of the “5” axis merge into a single coherent vector. Simultaneously, the red (C), green (D), and blue (N) points merge into a unified fourth point, represented by the central white dot. What was originally the parent’s blue N-state thus transforms into a new white N-state. This new white point inherently contains all RGB components and therefore simultaneously functions as Negotiation (N-blue) relative to the red and green corners, as Contribution (C-red) relative to the blue and green corners, and as Definition (D-green) relative to the blue and red corners.

When we combine both perspectives of forward and reverse, their sum converges exactly to $\frac{\pi}{2}$. Mathematically, this happens because the sum of the forward and reverse child values is always twice the parent value:

$CHILD_{\text{forward}} + CHILD_{\text{reverse}} = 2 \times PARENT$

Philosophically, this happens because the child is a child to its parent, but it is also a parent itself, resulting in two parents.

Conceptually, this result is ideal in Tholonic terms. A tholon, being inherently tetrahedral, requires the interaction of exactly one parent and one child to form a stable tetrahedral mapping. Each complete tetrahedral interaction thus naturally corresponds to a value of $\frac{\pi}{2}$ (which equates to 90° in radians). Consequently, a fully realized tholon, which is comprised of both a real tholon and its virtual counterpart, produces as complete a value of π (which equates to 180° in radians) as its context and limits allow.

Interestingly, two tetrahedrons converge to π, yet also create a perfect cube while π defines the shape of a sphere. At first glance, the relationship between the tetrahedron, the cube, and the sphere seems ironic. A cube and a sphere are eternally separated by the transcendental π, proof that a perfect cube can never precisely transform into a perfect sphere. This impossibility arises because π, the constant defining the relationship between linear and curved geometries, is transcendental and irrational. Yet intriguingly, two tetrahedrons, each Tholonically valued at $\frac{\pi}{2}$, combine perfectly to form a cube, explicitly assigning to the cube a Tholonic value of π. Thus, even as two perfectly rational tetrahedrons merge seamlessly into a cube, their Tholonic identity encodes the inherent impossibility of its perfect transformation.

However, beyond irony, this relationship offers profound philosophical insight. The value of a Tholonic tetrahedron, as revealed through this recursion process, symbolizes an infinite pursuit of Perfection, always approaching but never fully attaining it. This reflects precisely how a cube may infinitely approximate a perfect sphere but never fully become one, constrained eternally by π’s transcendental nature. Thus, the impossibility of perfection is intrinsically embedded within the Tholonic tetrahedron itself, reflecting the fundamental truth that perfection can be eternally approached but never fully realized.

Here, again, we see the same interplay of Perfection, Refinement, and Growth that was shown to us previously using the binary values of the vectors and vertices, as described above.

Ultimately, this recursive mathematical structure as well as the associated binary values reveals both mathematical elegance and deep philosophical wisdom, unifying ancient knowledge and modern insight into a coherent vision of existence which exposes some truths on life and reality:

• Perfection is eternally approached but never reached.
• Imperfection creates movement, growth, and infinite refinement.
• Existence can be described as the interplay between internal reflection (Limitation) and external interaction (Contribution).
• Reality manifests through fractal, recursive self-similarity.
• Wisdom lies not in attaining perfection, but in the endless pursuit of refining oneself toward it.

Binary Symbolism and the I Ching Connection

Note: A much deeper exploration into the I Ching and the Thologram is covered in the book “Introduction to the Tholonic I Ching”. Additionally, the I Ching itself has been reinterpreted through the Tholonic lens in the book “The Tholonic I Ching”.

The symbolism inherent in this recursive process, which is based explicitly on exponents of 2, naturally corresponds to a binary system, making it deeply connected to the symbolism found in the I Ching, itself fundamentally binary in nature. This connection between the ancient wisdom of the I Ching and mathematical symbolism has a significant historical precedent. Rather than viewing the Tholonic I Ching as an entirely new concept, it should be seen as a direct continuation and expansion of earlier explorations, specifically the groundbreaking work of Gottfried Wilhelm Leibniz, who himself studied the I Ching extensively.

Leibniz was first introduced to the I Ching in 1697 through correspondence with Father Joachim Bouvet, a French Jesuit missionary stationed in China. Bouvet’s letters included detailed diagrams of the 64 hexagrams and explanations of their meanings. This correspondence allowed Leibniz to recognize immediately the remarkable alignment between the system of broken and unbroken lines used in the I Ching and the binary arithmetic system that he himself had independently developed several years earlier.

Leibniz was deeply fascinated by this discovery and elaborated on it in his influential essay, Explication de l’Arithmétique Binaire (1703). He regarded this discovery as profound evidence of a universal truth that transcended cultural boundaries. For Leibniz, it validated both his mathematical explorations and his broader philosophical and theological beliefs.

Indeed, Leibniz’s binary system was fundamentally rooted in philosophical and theological ideas rather than purely practical mathematical applications. Leibniz perceived binary arithmetic as symbolic of divine creation itself. In his interpretation, the number “1” represented unity or God, and the number “0” represented nothingness. This binary opposition mirrored his philosophical conviction that God created everything out of nothing (creatio ex nihilo). Thus, Leibniz saw his binary arithmetic as the perfect mathematical representation of creation itself. This philosophical and theological dimension was central to Leibniz’s thinking about binary arithmetic, which, at the time, had not yet found practical applications. It was only centuries later, through the works of figures like George Boole and Claude Shannon, that binary arithmetic became central to communications and computing technology. For Leibniz himself, the binary system was primarily a philosophical and theological tool and a symbolic framework for understanding creation and illustrating the harmony between mathematics, philosophy, and religion.

In a letter to Duke Rudolf August of Brunswick in 1697, Leibniz explained clearly the spiritual essence of his binary arithmetic:

“The essence of my binary arithmetic is that I use no characters other than 0 and 1, and when reaching 2, I start again… This resembles creation ex nihilo, because all numbers are created from 1 and 0, just as all beings are created by God (the Unity) from nothing.”

The Tholonic I Ching presented here aims explicitly to formalize and further develop the insights that Leibniz initially realized. It is profoundly significant, and not coincidental, that the Tholonic recursion method presented here uses Leibniz’s series to calculate π, simultaneously reflecting deep symbolic relationships with the I Ching. This remarkable alignment represents a significant step in bridging the gap between metaphysics and physics, philosophy and mathematics, intuition and rigorous computation.

Below is one such clear illustration of this profound connection.

The Connection to the I Ching Hexagrams

Remarkably, the numeric values of the three axes, 35, 14, and 21, along with the value of the resultant center triad (56), perfectly align with the binary symbolic language of the I Ching. Not only does each vertex and its opposing vertex sum to the total number of hexagrams (64), but each numeric axis value corresponds philosophically to hexagram pairs representing the universal dualities of balance, definition, and integration, encoded millennia ago.

The table presented shows the 64 I Ching hexagrams arranged in binary order as 32 distinct pairs, each pair embodying opposing yet complementary aspects of a single conceptual theme. Significantly, the first two pairs of Primordial Duality and Cycle of Influence (or scope), represent the fundamental axis bridging the two primary classes of hexagrams (ascending and descending). These two initial pairs function collectively as one profound foundational concept, setting the stage for the entire recursive symbolic structure.

Highlighted clearly within the table are the specific pairs that correspond directly to the axes explicitly identified in the Tholonic map: Growth (Increase–Duration), Perfection (Before Completion–After Completion), and Refinement (Decrease–Influence). These three pairs precisely define the starting points of the Tholonic recursion process. Even more striking than their symbolic alignment with the recursion is their remarkably uniform distribution, perfectly dividing the hexagrams into symmetrical segments of seven stages each, reflecting deep numerical, philosophical, and metaphysical coherence.

We can clearly see the relationship between the Parent and Definition axes symbolically represented by the hexagram pair 41䷨₃₅ Decrease (reduction, sacrifice) and 31䷞₂₈ Influence (harmonious joining) which, as previously noted, aligns perfectly with the underlying mathematics. This vector corresponds precisely to the path of Refinement that these two hexagrams embody, allowing us to state explicitly that refinement itself is fundamentally an act of definition and limitation.

The vector connecting Parent to Contribution aligns with the path of Perfection, represented by the archetypal hexagrams 63䷾₂₁ After Completion (balance, order, and harmony that has been successfully achieved), symbolizing the ideal of perfection, and its counterpart, 64䷿₄₂ Before Completion (transitioning to balance, order, and harmony), symbolizing the continual striving toward that ideal. These two hexagrams thus represent the complete spectrum of perfection, from aspiration to attainment.

Further, the vector linking Definition and Contribution, which, according to the Tholonic model, symbolizes the spectrum across which new instances of creation are instantiated, thus aptly named the axis of instantiation, corresponds directly to the path of Growth. This path is specifically defined by the hexagram pair 32䷟₁₄ Duration (stability, sustainability, endurance) and 42䷩₄₉ Increase (growth, expansion), providing as explicit and clear a definition of growth as can be found within the symbolic language of the I Ching.

At the heart of these interactions lies the central triad, embodying the path of Equilibrium, which we can also interpret as the Balanced Path.

This idea that the “way” or the “passage” is one of balance is reflected in the ancient truths of Buddhism’s Middle Way, Confucianism’s Doctrine of the Mean, Ancient Greek’s Golden Mean, Taoism’s wu-wei, Islam’s wasatiyyah, Judaism’s shvil hazahav, and in many indigenous traditions. Here, we’ll refer to the Path of Equilibrium as simply The Passage.

The Passage symbolizes the authentic and balanced way forward, represented powerfully by the cardinal pair of hexagrams 11䷊₇ Peace (balance, integration, flourishing) and 12䷋₅₆ Standstill (obstruction, blockage, disharmony).

The Passage

The message of the Path of Equilibrium is clear: when Heaven and Earth unite harmoniously, peace arises; when they separate, stagnation inevitably follows.

In Tholonic terms, this profound insight reveals that when consciousness and intention focus on integrating the perfect archetype with its imperfect instances, in other words, seeing the perfection inherent within imperfection, Awareness and Intention manifest as growth via peace and harmony. Conversely, when consciousness and intention focus instead on separating the perfect archetype from its imperfect manifestations, that is, dwelling upon imperfection or lack, Awareness and Intention manifest as stagnation via disharmony and disintegration.

Application: Ohm’s Law and the Tholonic Structure

Within the tholons are numerous relationships, and some of these relationships can be visually mapped to the 2D tholon map. For example, we could map the 12 formulas of Ohm’s law to get the following:

==END OF MATH==

This shows the 4 trigrams that make up a tholon, however, depending on the perspective of view of the tholon, the formulas will be different. There are many ways to view this mapping, but this is perhaps the simplest. Besides the basic 12 formulas, we also see that the trigrams have two “directions”. One direction (the inner circle) is the direction of the formula, but this only applies where division is involved because but . The other direction is an arrow that starts at the initiating value, the point of the N-source, and terminates at the child N-source.

For example, using P=I×R, we can see how the mapping assigns P as the blue N-source, R as the green dot of Definition, and I as the red dot of Contribution. This then creates a yellow (additive color of red and green) spectrum between I and R, which in the case of these formulas, is where the math operators of ×, ÷, √ and the function = exist. This yellow spectrum is that of Cooperation or Conflict, so it makes sense that it would also be where we you find × and ÷, and suggests that, in this case at least, cooperation is literally twice as efficient as division, given that the relationships in multiplication-based trigram works in both directions.

You will notice that there are 12 yellow spectrums. These are the only the locations capable of creating a new child N-source, Of those 12, half are the function “=”, and half are operators. Off those 6 operators, 3 are constructive (×), 2 are destructive (÷), and 1 is reductive (√, which is a special case of destructive). If we look at the yellow spectrum of only the trigrams where V is the blue N-source, as this is the archetype, we see that each of the 3 sets, or each side, always has a pair of function and a single instance of another.

Side 1, Side of Contribution as multiplication

Side 2: side of Definition as division.

Side 3: side of Negotiation as equalization.

This beautifully shows how the primal attributes of Negotiation, Definition, and Contribution continue to reappear in every child and instance. In this case, we can see N, D, and C attributes within each N, D, and C as C_NCC, D_DND, N_DNN. Notice how × and ÷ never share a side, which we would expect to see given that they can only interact via =.

Looking at all the functions of a side, we see that we have 3 multiplications and 6 divides, which makes sense when you consider that multiplication works the same in both directions, so we need twice as many divisions to represent with division what multiplication can represent with 3.

New Data?

Applying Ohm’s law to the tetrahedron shows some new relationships. Readers who enjoy numbers may find this next (short) section interesting.

We can satisfy all 12 formulas of Ohm’s law as they appear on a tetrahedron, but we can also find some new data now which may appear as useless, or may be meaningful if we know how to use it, as we have claimed that “There is no useless data, only data we have yet to understand how to use” and “All data is valid”.

Previously (and in the left-most image above) we saw how the contribution of red divides the green on the right side but constrains the green on the left side . Can we apply these same concepts to other scopes, such as electricity? Does current (I) “divide” resistance (R) and is resistance (R) “constrained” by current (I)? Yes, but first we have to recognize that while “constrain” and “divide” have different meanings in their usual context, they have the same mathematical meaning. To describe the function as X being constrained by Y, because X is being defined by, or measured by Y, is not inaccurate, as X is indeed being “kept within certain limits” in as far has how it is being measured. We can then say “resistance is defined by current” and “current is limited by resistance”, both of which are technically correct statements.

In the right-most image above we have the 2D tetrahedron map with only V, I, and R values mapped to each trigram, along with the operators (÷, ×). The gray operators are associated with the inner trigrams, and the black operators are associated with the outer trigram.

Ohm’s law states that , so in the image above, all lines that connect V to R are assigned the ÷ function. The same goes for and I×R=V. Note how only N-sources have the functions of ×.

Let’s apply values to these points to make it clearer. Above is a tholon map where R=2, I=3, V=6 (and the implied P=18). These numbers satisfy the formulas of Ohm’s law as you can see in the image above. These are not arbitrary numbers. They are the smallest numbers that represent the properties of each point, mathematically speaking. Why do we not start with the number 1 then? Because the properties of 1 are the antithesis of creation. 1 represents perfect unity and balance, whereas every number greater that 1 represents separation and imbalance. For example, if we applied the values of 1 to I and R, then the values of V and P will also equal 1.

Every positive integer greater than one can be written uniquely as a product of primes ~ Euclid, 4th century B.C. “The Fundamental Theorem of Arithmetic”

The number 1 is unlike every other number and stands in a class all by itself:

• It is neither a composite number nor a prime number, though it used to be considered a prime number.
• Any number multiplied by 1 remains that number.
• 1 is its own factorial.
• 1 is its own square and square root, its own cube and cube root, etc.
• 1 is the result of the empty product, as any number multiplied by one is itself.

In fact, from the philosophical perspective of the Neoplatonists, the number 1 is not even considered a number in the traditional sense, but rather the source of all numbers.

So, why not start counting with 2 rather than 6? Because we assume that the children of N, when recombined, must equal N.

We can test these numbers by applying the formulas of Newton’s 2^nd and see that the bases of the trigrams apply multiplication, and the sides apply division, and consequently create perfect inverted replicas of themselves in the children.

There are some interesting relationships and values below. I can’t tell you what they mean or represent, but it would be irrational to assume they have no significance without knowing anything more about them. Everything that leads up to these new values is quite reasonable and provable. It is just a matter of discovering where these new values and formulas apply. To explain these new values here would border on sadistic given what the reader has had to suffer through already, and as it qualifies as a book unto itself. However, there are just a couple I would like to share.

If we look at current (I, red, 3) “dividing” resistance (R, green, 2), on the right , and apply the operator of division associated with the 6-3 vector, or side, we have 2²÷3, which is $1\frac{1}{3}$ (which we’ll call R_i). On the left we have resistance (R) constrained by current (I) , which is 3²÷2, which is $4\frac{1}{2}$ (which we’ll call I_r). Neither number seems particularly significant, but if we multiply these numbers together, which we would because they are connected via the “base” of the trigram, the 3-2 vector, which has the operator of ×, we get R_i×I_r=6, which is the value for V, so Ohm’s law has not been broken. Using this same reasoning we can apply this to the three 6s, but as they are on a vector of multiplication, we would say 6³, which yields 216 (which we’ll call V₃).

We now have new properties of R_i, I_r, V₃. We also know that R_i×I_r=V. This means we can also say $\frac{V}{R_i}=I_r$ and $\frac{V}{I_r}=R_i$, all in accordance to Ohm’s law. We can also calculate a new value for P as V×I_r=P, but this gives us the value 27, which is a different value from P=18, so we will call this P₁. Let’s test these values by applying the other formulas and see if we get the correct results. We can apply the formula I_r²×R_i=P₁ where we’d expect P₁ to equal 27, and it does. Likewise $\frac{P_1}{I_r}=V$.

We have exposed at least 4 new values within the tholon map. Now all we have to do is find out what they mean and where they are applied in the real world. Here are all the values listed.

• R = 2
• I = 3
• V = 6
• P = 18
• R_i = $1\frac{1}{3}$
• I_r = 4.5
• V₃ = 216
• P₁ = 27

Regarding V₃, 216, this is a fascinating number that has a history that goes back to at least Plato, because we know (generally accepted by experts) that 216 is called Plato’s Number and was described to Plato by Socrates who said (when explaining how to properly breed humans):

For the number of the human creature is the first in which root and square multiplications (comprising three dimensions and four limits) of basic numbers which make like and unlike, and which increase and decrease, produce a final result in completely commensurate terms.

(The word “cubed” was not in use in those days, but 216 is the first (smallest) cube which is the sum of three cubes. It also makes more sense how this might apply to human breeding when you know that back in those days, odd numbers were considered male and even numbers were considered female. Math class back then must have been a lot more interesting.)

Plato elaborated on this in his “Republic”:

Now for divine begettings there is a period comprehended by a perfect number, and for mortal by the first in which augmentations dominating and dominated when they have attained to three distances and four limits of the assimilating and the dissimilating, the waxing and the waning, render all things conversable and commensurable with one another, whereof a basal four-thirds wedded to the pempad yields two harmonies at the third augmentation, the one the product of equal factors taken one hundred times, the other of equal length one way but oblong,-one dimension of a hundred numbers determined by the rational diameters of the pempad lacking one in each case, or of the irrational lacking two; the other dimension of a hundred cubes of the triad. And this entire geometrical number is determinative of this thing, of better and inferior births.

The above has been interpreted to mean 6³, which Plato alludes to elsewhere, as 6 (considered the number of marriage back then) is the product of female (2) and male (3), and 2³ × 3³ = 216. While 216 is the most accepted answer, great thinkers spanning over 2,000 years, such as Aristotle, Proclus, Marsilio Ficino, Gerolamo Cardano, Eduard Zeller, Friedrich Schleiermacher, Paul Tannery, Friedrich Hultsch, and more, have all tried to decipher Plato’s enigmatic text. 216’s fame may even go back as far as Pythagoras, as it is the sum of cubes of the Pythagorean triple (3,4,5, hence 3³ + 4³ + 5³), or even Ancient Babylon.⁵

Later in history, this pedigree number fell slightly into infamy when John the Apostle declared in Revelations that 666 was the number of the Beast (the beast that comes out of the sea, not the one that comes out of the abyss of the Earth), but also referenced this this number as the the number of man. 216 might have escaped scot-free if John had not specifically said that the number 666 was to be calculated (or counted depending on translation) to learn its true number.

Here is a call for wisdom: Let the one who has insight calculate the number of the beast, for it is the number of a man (ed: or “humanities number”), and that number is 666 . ~John, Revelations 13:18 (NIV)

Mathematicians even classify any prime number that contains the sequence 666 as Belphegor’s Primes, named after one of the seven princes of Hell who helps people make discoveries and seduces them in to thinking they will be become rich as a result. Believe it or not, treasure hunting with demons was a very popular career choice until until the Industrial Age.

Because 216 = 6×6×6, it gained a dubious reputation by association with the Beast of Armageddon and started popping up in some strange places, thanks to humanities generally suspicious nature and love of doom. Fans of the arcane ignore the most likely suggestion that John may have been telling the reader to add the numbers together, because in the original Greek it was penned as 600+60+6, and “Six hundred threescore and six” in the most popular King James Bible. However, the odd coincidence that John’s number of man (666), which needed to be “calculated”, has a strong relationship with Socrates’ “number of the human creature” being 6×6×6=216, which also needed to be calculated, is curious… even more curious is why these old timer’s preferred to shroud their message in incomprehensible riddles rather than just tell us the damn number! Ug… philosophers. Regardless, 216’s clout in the religious, mystical, and tin-foil communities holds up today. Just do an Internet search for “216” to see for yourself. Coincidentally, 216 × 10 happens to be the years the procession of the equinox moves through each of the 12 constellations… just sayin’.

The above example is based on only 1 of the 3 sides of a 1^st generation tholon, and there are countless generations!

Squares and Roots

What is the significance of a square or square root? We know what it means mathematically, but how would that meaning be understood in a different context, such as society or law? The challenge with applying one relationship from one context to another is that without well-understood metrics, attributing foreign attributes to a relationship runs the risk of being very subjective, biased, and filtered through a cultural or moral lens that itself may be very biased. This is the reasoning behind many beliefs and superstitions. Nevertheless, we will try our best to avoid those pitfalls.

If you measure the long end of any A or B series piece of paper, such as the standard A4 size of paper, and divide that length by the short edge, you get the square root of 2. This is not a coincidence, but rather the very clever work of the Germans in 1922. The reason for this is so that a piece of paper of the dimensions $\frac{height}{width}=\sqrt 2$, when folded in half (along the long edge), will remain $\frac{height}{width}=\sqrt 2$. So, no matter how many times you evenly fold it, it will always remain the same relative size. This can only occur when the ratio is $\sqrt 2$ , making $\sqrt 2$ a very self-similar or fractal concept.

The number 2 is especially unique in the way it is the only number where 2² = 2×2 = 2+2 = 4. No other number has the same value when squared, added or multiplied (except 0). Tholonically, a single point is nothing but a concept that has no dimension or form, but 2 points is the first instance of a duality and dimension, making it the most fundamental number of creation, or the transition from concept to form, so we would expect to see some property of self-similarity given that self-similarity is a fundamental property of existence. Perhaps that is why the number 2 is not only the first prime number, but the only even prime number as well. It may also be the only number that inspired a murder (allegedly). We already told the story of Hippasus, but what wasn’t told was that along with his other mathematical crimes, he broke the sacred belief that all numbers must be expressible as fractions by proving that $\sqrt 2$ was an irrational number. His discovery was declared “Fake News” and he was promptly dumped at sea (or abandoned on a deserted island).

In Chapter 4: Laws, we showed how $1-\frac{1}{3} +\frac{1}{5} -\frac{1}{7} +\frac{1}{9} -\frac{1}{11} +\dotsc =\frac{\pi }{4}$, but if we use all numbers rather than just prime numbers, we get a very different answer that adds to the significance of the number 2:

$1-\frac{1}{2} +\frac{1}{3} -\frac{1}{4} +\frac{1}{5} -\frac{1}{6} +\dotsc =ln( 2) =\ 0.69314718056$

In other words, the sum of 1 divided by all odd numbers (which is ∞) minus the sum of 1 divided by all even numbers (which is ∞) equals the natural Log of 2.

We also see the significance of 2 in the Riemann Zeta Function (which we explain and look at in Appendix F, “An Unexpected Pattern”: ζ(2)= and ζ(2^-1)=0, or ζ() = 0.

is also the basis of the Silver Ratio. This is a less famous Metallic Ratios, which include gold, silver, bronze, copper, etc. While the Golden Ratio’s mean is between 1 and 2 (1.618), the Silver Ratio’s mean is between 2 and 3 (2.414), the Bronze Ratio between 3 and 4, and so on. Here are all the metal ratios:

The Golden Ratio is often credited for Silver Ratio accomplishments, such as the spiral of the Milky Way, facial proportions, architecture (especially in Japan), bird flight patterns, and more. Notice how the Silver Ratio is the only ratio (other than 1:1) that can be determined by only using the number 2.

Here are some thoughts as to how squares and square roots might apply to non-math contexts.

When we square something, the units-of-measure and the unit-count are the same. For example, 6² , or 6×6, says we have a quantity of 6 counts of a unit-of-measure that equals 6. The thing we are measuring, and the units-of-measure are the same. We are using 6 as an objective unit-of-measure to measure something that is essentially subjective.

Why is it subjective? Because when we say a×b, we are implicitly saying concept × count. These concepts are archetypes, such as “apple”, “cost”, “voltage”, ”mile”, etc., which only exist within the context of the intelligences that created them. They are a collectively created and accepted archetypes that have no meaning outside of the context they exist within. An archetype does not exist in the material world, but instances of archetypes only exist in the materials world. In this way, archetypes and concepts are subjective as they only exist in the mind of the conceiver.

We can see this clearly if we look at the history of any concept based on material existence. For example, the concept of “apple” once referred to bananas (14th c.), and generically referred to all fruits, including nuts (17th c.).

Another example is the “mile”. The Romans, who invented the mile as 1,000 paces, knew that a mile was much shorter on rainy days over harsh terrain that is was on dry, flat fields. As the concept of the “mile” was adopted around the world, each culture adapted its definition, resulting in wildly different definitions. The Hungarian Mile example, was 8.9374 km! It wasn’t until 1959, over 2,000 years after the concept was created, that the mile was internationally settled to be 1.609344 km. Even the metric system is susceptible to contextual vagaries as it is based on the naturally observable data, which is contextual to humans on Earth. It’s worth nothing that the original concept of the modern metric systems was created simultaneously, by a French Roman Catholic Abbot, Gabriel Mouton, in 1670, who proposed using the Earth’s circumference as the basis of measure, and an Anglican clergyman, John Wilkins, who, in 1668, proposed using the length of a swinging pendulum with a half-beat of 1 second. Even though Wilkins’ book was published 2 years before Mouton’s, he specifically criticizes the idea of using Earth’s circumference as a standard, suggesting Mouton introduced his idea prior to his publication.

As for count, this is a strictly abstract mathematical concept that has no material existence, therefor no instantiation. Regardless of the context, 2 apples, 2 miles, 2 dimensions, 2 galaxies, etc., 2 retains the exact same meaning. However, things get fuzzy in extreme contexts, such as distance and time related to things moving at or near the speed of light but even then, the limitations is because of the thing, not the concept of 2.

For these reasons, we say that archetypes derived from instances, such as “apple” and “mile”, are subjective, and instances that are derived from archetypes, such as numbers, are objective. However, objective and subjective are like any polarity; everything exists on the spectrum between these poles, and nothing actually exists as a pure state of either.

So, when we say a×b, or concept × count, we are saying subjective × objective.

This may make sense when we say apple × 2 = 2 apples, but what if we say current × resistance = voltage, or I×R=V? Obviously, both I and R are subjective, collectively agreed upon measures. Wouldn’t this be like saying apples × oranges? No, because apples and oranges are defined as the fruits of specific types of plant, while current (I) is already the results of numerous functions and operators, such as , and the same is true for resistance (R) and voltage (V). Apple and orange as concepts can’t be reduced any further and remain apple or orange. When we do break down apple into its components, its partons, we find ourselves in the world of biology and geneticists, which have their own non-apple centric concepts, and apple is just a particular collection of 57,000 genes (twice that of humans!). In the same manner, we could reduce the macroscopic concepts of current, voltage or resistance to their microscopic components, but then we and up in the world of quantum physics, chemistry, etc.

This subjective/objective duality is effectively the same as the quantitative/qualitative duality. Here are those definitions again:

• Quantitative data can be counted, measured, and expressed using numbers. Quantitative data has an objective agreed upon value.
• Qualitative data that is descriptive and conceptual. Qualitative data can be categorized based on traits and characteristics.

Of course, when we are speaking metaphorically, we are referring to the qualitative properties of math, not the quantitative, so none of these claims apply to math that can be done on a calculator (probably).

However, this metaphorical relationship appears in real-world instances, such as the relationship between gravity and consciousness. For example, in an earlier section we showed how the gravitational field g was a product of G×M₁×M₂ where:

G = universal gravitational constant. M₁ = mass of the observing physical system (the observer). M₂ = mass of the observed physical system (the observed).

These definitions are verbatim from the research paper, and depend on the very same concepts of subjective/objective to describe the most fundamental force in reality.

Bringing this back to squares and square roots, squaring a number is metaphorically equivalent to something using its own properties to evaluate its own existence, not unlike the 1^st dot that became aware of itself through the 2^nd dot, creating the 2 states of objective and subjective.

Can we say that when something is squared, it is the objective measure of a subjective existence? Can we see these two conditions, subjective and objective, as 2 dimensions of something’s existence? We know that 2 dimensions defines an area, and we also know that when two 1-dimensional (lines) are at their two most contrasting position (90°) they automatically define a 3^rd dimension that is a multiple of .

We can see a fascinating example of this if we look back to Chapter 7.5 The Thologram and revisit the CHSH experiment which shows that $\frac{2\sqrt 2}{2}$ is the difference between relativistic and quantum realities.

The relevant part is how, when looking as the diagram of right angles and equilateral trigrams above, we see that in the RA trigram, which represents the maximum separation of the lines, we always have a connecting like (hypotenuse) that is a multiple of $\sqrt 2$, but in the equilateral trigram, where all lines are equally distributed, all lines are equal.

Does this suggests that the quantum model is related to a RA triangle with sides of length 2, and the relativistic model is related to an equilateral triangle of with sides of length 2? Mathematically, they do seem related, as $\frac{quantum-probabilities}{relative-probabilities}$ and $\frac{hypotenuse}{side-length}$ both have a $\frac{n\sqrt 2}{n}$ relationship.

From the ancient Greeks to modern philosophers, geometry and math has been fundamental in describing and understanding the world not just from a technical perspective, but even a moral, ethical, legal and social perspective. What follows is similar in that it uses math and geometry to support a philosophical premise. It needs to be stressed, again, that this exploration is not intended to shed new light on math or geometry, but rather, uses math and geometry as guideposts of reason. With that said, here is another view of the extra-mathematical significance of the number 2.

Tholonically, 2 is the result when the 1^st instance of 1 created a new instance of 1 and is equivalent to 1+1. However, the resulting refers to the total number of 1s, not the components of a singular 1. The components of 1 is just 1, as 1²=1, but the components of 2 are the pair of values that equal $\sqrt{2}$ (1.414). Here, we have multiple true statements that are different yet have the same result:

• 1+1=2. This is how 2 came into existence. The objective instance of 2.
• 1.414² = 2. This is how 2 defines itself, like a type of numerical DNA. This is the subjective instance of 2 which has qualitative or subjective value of 1.414 that is measured by its quantitative or objective value of 1.414 is also 2.

This is another example of the fractal nature of numbers.

What is the reasoning behind this claim? We can say that the original source of 2 is 1+1 because we can only know the square root of 2 after we have created 2, so 1 + 1 = 2 = $\sqrt{2}^2$, in that order.

The difference between 1 and 1.414 is the difference between creating something (1+1) and the creation itself (1.414²), or perhaps the difference between the objective, or the quantitative and the qualitative. And what is that difference? It appears to be 0.414 because 1.414-1=0.414, which also happens to be the inverse of the Silver ratio, $\frac{1}{2.414}=0.414$.

What does 0.414 represent then? If 1+1 created 2, and 1.414² describes what was created, can we say that 0.414 is the difference between the creator and the created? Let’s go with that for now and see if it holds up.

What is the difference between creator (1) and created ($\sqrt{2}$)? Is it similar to the difference between parent and child? man and god? flower and seed? Tholonically speaking, there are 2 types of tholons, that which can create new tholons, and that which cannot. When a tholon cannot create, it is the end of the line for that form of expression. The difference between creator and created is simply creation, which is an act on intention. Creators create, creations cannot create, that is, not until they become a creator, if they can. This suggests that measuring the number 2 by creation would tell us about the its ability, or perhaps potential, to create.

How would we define the concept of 2 by the difference between creator and created? We can measure 2 by the unit of creation, or put it another way, .

What is $\frac{2}{0.414}$ ? It is 4.828. And what is 4.828?

• Silver Ratio × 2.
  
• $2+2\sqrt{2}$. This is interesting as 2 and $2\sqrt{2}$ were the limits of relativistic and quantum probability in the previous test.
• The area of an octagon (having sides of length 1). Likewise, $s=\sqrt{\frac{A}{4.828}}$ can be used to approximate the side length $s$ of a regular octagon with area A. In addition, an octagon with an area of $\frac{2}{0.414}$ has a perimeter length of 2³, or 8. The octagon is of special interest as it is a 2D version of the 3D octahedron, the shape of nothingness. Related this this, the surface area of an octahedron is $\sqrt{2}$ when edge length = 0.414.
• The tangent of $\frac{1}{2^{2^2}}$ of 3 circles (67.5° out of 720°) plus the cotangent if $\frac{1}{2^{2^2}}$ of a 1 circle (22.5° out of 360°). This is interesting as it shows a relationship between 1 and 3. In this case, with circles, but we see this in the tholon as well.

…and a bunch of other stuff as well, for example, see image below. In that image, notice how, when the height = 1, the red area is $\frac{0.414}{2}$, the green area is $1+\frac{0.414}{2}$, and the base length is $\frac{2}{0.414}-2=4.828-2=2\sqrt{2}=2.828$.

This is also interesting because a circle can be easily divided into 4 identical quadrants. All 4 quadrant are the partons to a circles holon, but the archetype of a quadrant is itself a holon and it, too, has partons. These partons can be 4 equal divisions of 22.5°, or 2 partons only, one of 22.5° and one of 67.5° (or there can be 4 and 2 at the same time). Why do we say this? Because the patterns these partons create are based on the most stable patterns of creations, and specifically, 0.414, $2\sqrt{2}$, and the Silver Ratio, 2.414.

For example, 22.5° is $\frac{1}{16}$ (or more significantly $\frac{1}{2^{2^2}}$) of a complete circle of 360°, and the tangent of 22.5° is 0.414. The remaining 3 segments of 22.5° together creates a segment that is 67.5, whose tangent is 2.414. These 2 segments together create a 90° quadrant with total tangent value of 2.828 (interestingly, tan(90)=∞). Unfortunately, $\frac{360}{67.5}$ give us a rather clumsy $5\frac{1}{3}$, but 720, or 360 × 3, 3 circles, divided by 67.5 gives us 16, which is that same $\frac{1}{2^{2^2}}$ relationship that 22.5 has with 360. In this way, each of these 2 segments which make up the the quadrant, which is the archetype of a the most fundamental shape in existence, will naturally tend to create a single a full circle and 3 full circles. In other words, where there is 1 there is 3, and where there are 3 there is 1.

So far, this falls more on the supportive side of the claim that 0.414 is the value of creation, or should we say a fundamental value of creation, as there are many more, such as the Silver Ratio, which is based on the same numbers:

This unique value of 0.414, which we are suggesting is the difference between the creator and the created, a number of creation and intention, would probably have some interesting relations with the number 3, itself a number of creation, at least in the realm of tholonic archetypes and their instances. Let’s see.

= 7.242. This number also happens to be 2.414 × 3, or $1+3\sqrt{2}$. Again, the Silver Ratio appears, but this time as a multiple of 3. So far, so good, but there’s more,

This is fantastically elegant in so many ways, and the recurring principle seems to be the Silver Ratio, which is based on 2.

For simplicity, let’s assign 0.414 to a variable Λ (upper case lambda).

We could also then look for the qualities of creativity within the original source with 1÷Λ, which shows us that Λ is the inverse of Silver Ratio, or 2.414. It also shows us several other interesting relationships especially when we map these values to the tholon. In the image below, green lines represent division, and red lines represent multiplication, in accordance with how we mapped Newton’s 2^nd law to the tholon.

Below is the 2D map for this above image to make the relationships more obvious, for example, how the lower-left trigram has the values Λ, +1, +2, +3; how lower left and lower right are naturally inverted, and when combined, equals the baseline of the top trigram which also has the values of Silver Ratio +Λ, +1, ×2, or how the central trigram is ×1, ×2, ×, which naturally match the 3 corner values.

The arrangement of these values on the tetrahedron do not conform to Newton’s 2^nd, however, they seem to have a similar pattern, because if we apply the formulas of Ohm’s law to these values, we get some very interesting values, all of which are the result of the numbers 1 and 2 only, and in one case () we see how it relates to π:

$1.848=\sqrt{Silver\ ratio\times2}$ and $2\cos{(\frac{\pi}{8}})$

Additionally, if we multiply all the values of each type (the equivalent to V, I, R and P), we get the a total value for each group where R=1/8, P=8, I=2.230, and V=6.309. With these values, the formula for volts produces =1, and the formula for resistance produces = 2.828. These two values represent the N-source (1), and the N-child’s spectrum (2.828), so, clearly, there is an order, as can be seen in the upper-right image that shows the formulas, but not exactly the same as Newton’s 2nd. For example, the value for that vector which is the spectrum of instantiation between D and C has a value of the Silver Ratio+Λ (2.414+0.414), but if we replace the values with the Newtonian values of 6, 2, and 3, that same vector has the value of 6, and as we pointed out, 6 is the value of the initial creation, given it must be the value of the N-state. In one case, the value is arrived at by adding Silver Ratio+Λ, and in the other, it is arrived at by multiplying a duality (2) by the trinity (3), which is itself a number of instantiation if we accept “from 3 come all things”. Qualitatively, these are the same expressions with 2 different contexts and scopes.

This value of Λ (lambda, 0.414) might be a good example of a tholonic qualitative value in that it was derived by the qualitative “formula” of creation or intention = created - creator. As we would expect, this number appears in nature, specifically, it is the ratio of the radius of a sphere that can fit in the empty space of a tetrahedron (tetrahedral void) created by 4 spheres. This may sound very obscure, but it is quite critical in the way atoms pack themselves together. This tetrahedral void is also a octahedron (the shape of nothingness), but made of spheres instead of tetrahedrons.

Another interesting point about 0.414 is that if we calculate the axis of symmetry between the values 1, 2, and 3, using the standard quadratic equation of ax²+bx+c=0, we get the two values of and , or and . Notice how these 2 results match the 2 values in the above 2D tholon map where 1 connects to Λ, the difference being the addition of i.

Black Holes and Electricity

While we are on the topic of the number 2, there is another interesting pattern related to black holes.

The formula to calculate the radius of a black hole is quite simple: $radius=\frac{2G\times mass}{c^{2}}$, where G is the gravitational constant. Because we know that $mass=E/c^2$, we can write this formula as $\frac{2G\times(E/c^2)}{c^{2}}$. If we apply the rules described in Chapter 5: Gravity where we claim that G is equivalent to V (voltage), and in Chapter 5: The Laws, where we claim that c (speed of light) is equivalent to I (current) and equate E with P (power), we can write this formula as: $\ \frac{2V\times(P/I^{2})}{I^{2}}$ . Because $\frac{P}{I^2}=R\ (resistance)$, we can then say $\ \frac{2V\times R}{I^{2}}$. Using the smallest integer values that can be applied to I, V, and R, as described in Chapter 5: The Laws, we can write this formula several ways:

$\frac{(2\times 6)\times 2}{3^{2}}$ = $\frac{12\times 2}{9}$ = $\frac{24}{9}$ = $\frac{8}{3}$ = $\frac{2^3}{3}$ = $2\frac{2}{3}$ = 2.666…

$\frac{2^2\times 6}{3^{2}}$ = $\frac{R^2\times V}{I^{2}}$ (using all properties of R, V, I)

$\frac{2^3\times 3}{3^{2}}$ = $\frac{R^3\times I}{I^{2}}$ (using factorials)

$\frac{64/3}{8}$ = $\frac{8^2/3}{8}$ = $\frac{2^6/3}{2^3}$ = $\frac{R^V/I}{R^I}$ (based on 8)

There are many more ways to express this, each pattern exposing a different aspect of the same thing. Whether this “means” anything I can’t say. Perhaps this is just data waiting to be turned into information, but I find it interesting how in describing the virtual radius of a black hole (being it’s in the context of electricity), we end up with a cube of length 2 and a volume of 8 (i.e., $2^3$), and how $\frac{volume\ of\ cube}{dimensions}$ = radius of black hole in a different context. Given that the cube is also the archetypal geometry of Contribution/form/integration, and that black holes not only create galaxies, but may well create universes, this is an especially coincidental relationship. The cube is also associated with acceleration in Newton’s 2^nd law, effect in the cause/effect chain, and the archetype of movement, according to the association made in Chapter 5: The Laws.

It’s worth pointing out that when we use a trigram with side lengths of 2, we get the ratios of $\frac{quantum-probabilities}{relative-probabilities}$ as described above.

Prime Numbers

In the thologram, we consider prime numbers part of a more extensive set of prime numbers and their products (a number that can only be created by multiplying prime numbers greater than 3 together). This was not done by design. As we’ll demonstrate, this set of numbers naturally occurs in the thologram in the green dot of Limitation/definition.

Likewise, the blue dot of Negotiation/balance and the red dot of Contribution/integration each have their own sets of values, and each set contains $\frac{1}{3}$ of all numbers. Given that primes occur less often as the numbers grow in size, then for primes and products to maintain a $\frac{1}{3}$ balance of all numbers, their products must increase proportionally… and they do. Below is a chart that shows this.

This primes and products set of numbers can be seen as either the remainder of all numbers after removing all the ÷3 and ÷2only sets, each of which is $\frac{1}{3}$ of all numbers, leaving the remaining set which must, therefore, be $\frac{1}{3}$. This remainder is also all the numbers that are prime or a multiple of primes greater than 3, which just happens to be $\frac{1}{3}$ of all numbers. The former reason is consequential, and the latter is causal, which we would expect if the tholon of primes and products is both a holon and a parton.

Prime numbers are defined as a number that is “divisible by exactly 2 natural numbers; itself and 1”, natural numbers being any positive integer. As previously mentioned, the “exactly by 2” stipulation is recent and was only added for convenience. While this definition describes a prime number, it doesn’t say anything about the nature of prime numbers. A mathematician or cryptographer could explain the nature of prime numbers much better than whatever is said here. Still, we can at least say that prime numbers are the keystone building blocks of numbers, as all numbers can be reduced to a product of prime numbers, and prime numbers can’t be reduced at all.

From the tholonic perspective, which considers A&I to be the source of all energy, concepts are an expression or instance of energy. While this energy may not be in the form of electricity or heat (although a nominal amount of both is required to maintain a concept), it is energy nevertheless and will contextually conform to fundamental laws of energy.

As numbers are concepts, how can we model them as forms of energy? As we have already seen, energy oscillates. We can look at numbers as different oscillations. This is similar to how we look at harmonic series, and we can apply those ideas here. We can imagine the most basic number, 1, as a singular instance of oscillation or as a single cycle, if we use the traditional vocabulary of harmonics. The number 2 would then be 2 instances of 1, 3 would be 3 instances of 1, 4 would be 2 instances of 2, etc.

This idea creates simple waveforms at first, but that simplicity is due to 1, 2, and 3 being prime numbers (here, we take the Euler approach and recognize 1 as a prime number). If you look at #4 in the image above, it looks exactly like 2. Why not just show a waveform that had 4 cycles? If we look at numbers like instances of energy, w-e have to apply the laws of energy, the primary of which is “energy travels the path of least resistance” or the law of entropy. Because 4 is two 2s, the energy signature, so to speak, is that if 2, as 2 is what creates 4. Reducing a number to its most basic components, or prime factorization, will always reveal the smallest prime numbers that can create a particular number. Think of the number value as a value of resistance. 1 has the lowest resistance, so energy will always prefer to travel down that path, and we see that as 1 is the only number by which every number can be divided (and remain a natural number).

As 1×1 = 1, we can’t create 2 from 1, so a unique concept of 2 has to be created. The same is true for 3. 4 is a more efficient way to express 2×2 (shown as 2 thin blue lines which appear as one as they are overlapped), but as 2 requires less “energy” or has less “resistance” than 4, a 4 will always be expressed energetically as two 2s. 5, being a prime number, has no factorization other than 1×5, so it instantiates as a wave of 5 cycles. 6’s prime factorization is 2×3, and together they create a new unique waveform (black line).

We can see from this little sample that the prime numbers (1,2,3,5) and their squares (1,4,9,25) appear to be pure, perfect, or monotone waveforms until we get to 6, the first number that has a complex waveform.

Here is what the first 60 natural number looks like using this technique (even numbers = red, odd numbers = blue):

We first notice that prime number waveforms get denser and denser, which we would expect as they can not be reduced to any smaller number, but they also evolve in unexpected ways as we go high and higher.

Here are the first 60 prime numbers (on a slightly expanded scale):

Another pattern we see is how exponential values are also “perfect”:

While these patterns may appear identical, they are not. The chart is too small to see the detail, but while the form is identical, the amplitude is not. 32 is 16 times larger than 2, but they both have an identical waveform. The #2 waveform is the simplest waveform that remains unchanged and the most common of all the “pure” waveforms.

Regarding the 3 columns of the tholon, we have the following bird’s-eye view.

Some of the patterns are surprising. In the higher numbers, we can see the more extreme offsets from a perfect cycle of the prime number forms. This is because they come near to a perfect circle, but not exactly, so each cycle’s offset increases a small amount each time. In practical terms, were prime numbers a sound wave, it would be a perfect monotone that never changes tone but with varying volume. #1994 is the only instance (from the limited samples) where the resulting waveform is that of 0.

The waveform of #1 rarely appears, and when it does, it’s unexpected. For example, it first appears (after #1, of course) in #1997, and its reverse form in #1999.

Note: It may be that the correct waveform for #1 is not , but should instead be half of that, , which is how a cycle of 1 is shown in classical harmonics. Assigning #1 to a full cycle has more to do with keeping to tholonic patterns. In either case, the results are quite similar.

If we look at the “Right” columns of the tholonic values, we see both the numerical values and the functional values are identical, and they are only prime numbers and products of prime numbers (excluding 2 and 3). The “Left” column shows all even numbers for the numerical values and flipping between even numbers and prime-based numbers on the functional side. The “Center” columns show a perfect progression by 3 for the numericals and a perfect progression by 1 for the functionals.

Each column represents one of the 3 types of numbers; ÷2only, ÷3, and divisible by primes greater than 3. It also shows us that even numbers divisible by 2 and 3, such as 6, are actually in the class of ÷3, or center values but just happen to be divisible by 2.

In the thologram, 3 is considered the first number from which all numbers emerge, as it is the first point of creation of a trigram, and its functional value, or archetype, is 0, the nothingness to the somethingness of all numbers. 0, 1, 2, and 3 express the first archetypes of numbers, those being nothingness, somethingness, even, odd, and prime and create the 4 classes of number that every number is a member of. All numbers that follow have properties based on these archetypes.

For this reason, 4 is considered a generation of 1. However, as each generation is reversed from the parent, 7 is actually the 1^st generation of 1, and 4 is the inverted transition of 1 to 8. An interesting observation is how the top 2 trigrams contain the longest set of contiguous set of numbers, which happens to be the set of 0-9. In this diagram (right) we use all 6 colors to show the difference between each generations, as we showed previously for the trigrams, even though we only used 3 colors for simplicity sake.

Getting back to prime numbers, the following image of the waveforms of values created by the first 16 prime numbers demonstrates exactly how this fractal expansion works as we can see how iteratively embedded waveforms build one top of each other.

1 1×2 = 2 2×3 = 6 2×3×5 = 30 2×3×5×7 = 210 2×3×5×7×11 = 2,310 2×3×5×7×11×13 = 30,030 2×3×5×7×11×13×17 = 510,510 2×3×5×7×11×13×17×19 = 9,699,690 2×3×5×7×11×13×17×19×23 = 223,092,870 2×3×5×7×11×13×17×19×23×29 = 6,469,693,230 2×3×5×7×11×13×17×19×23×29×31 = 200,560,490,130 2×3×5×7×11×13×17×19×23×29×31×37 = 7,420,738,134,810 2×3×5×7×11×13×17×19×23×29×31×37×41 = 304,250,263,527,210 2×3×5×7×11×13×17×19×23×29×31×37×41×43 = 13,082,761,331,670,030 2×3×5×7×11×13×17×19×23×29×31×37×41×43×47 = 614,889,782,588,491,410

Or, perhaps better displayed in the following format with the cross-sum values, which is interesting as they are all 6 or 3 (after 1 and 2, of course).

1 → 1 1 × 2 → 2 2 × 3 = 6 → 6 6 × 5 = 30 → 3 30 × 7 = 210 → 3 210 × 11 = 2,310 → 6 2,310 × 13 = 30,030 → 6 30,030 × 17 = 510,510 → 3 510,510 × 19 = 9,699,690 → 48 → 12 → 3 9,699,690 × 23 = 223,092,870 → 33 → 6 223,092,870 × 29 = 6,469,693,230 → 48 → 12 → 3 6,469,693,230 × 31 = 200,560,490,130 → 30 → 3 200,560,490,130 × 37 = 7,420,738,134,810 → 48 → 12 → 3 7,420,738,134,810 × 41 = 304,250,263,527,210 → 42 → 6 304,250,263,527,210 × 43 = 13,082,761,331,670,030 → 51 → 6 13,082,761,331,670,030 × 47 = 614,889,782,588,491,410 → 93 → 12 → 3

Thologram and Prime Numbers

The embedded nature of the thologram naturally creates sequences of many different patterns and values. The simplest sequence is that of incremental counting starting from 0, as in 0, 1, 2, 3, 4, 5, etc., and being the simplest sequence, we would expect to see some very fundamental patterns emerge. Below are some diagrams and tables that illustrate this simple sequence.

We end up with the “Sequential values” table of 3 columns of values; center, right.

• The center column, which is where the counting starts, has the sequence 0, 3, 6, 9, etc. All the numbers are divisible by 3 and are colored blue.

• The right column has the sequence 1, 5, 7, 11, etc. All the numbers are prime numbers or products of prime numbers that are not 2 or 3 and are colored green. We’ll refer to these values a rPrimes.

• The left column has the sequence 2, 4, 6, 10, etc. All the values are divisible by 2 and are colored red.

Because of this commonality by column, we can assign the right, center, and left columns the properties or functions of ×1, ×2 and ×3.

The table on the left, “Prime factorization”, shows this multiplying function for each column and the values that, when multiplied by the columns function, result in the sequential value.

In this table, the factorials are the same coloring as Sequential values.

• rPrimes are green cells.
• divisible-by-2 based products are red cells.
• divisible-by-3 products are blue cells.
• divisible by 2 and rPrimes are yellow cells.

In both tables, the numbers 0 and 1 are considered special cases as they are “special” numbers.

Here’s an example of how to read the “Prime factorization” table values:

The sequential value of 28 has the prime factors of 2×2×7. As 28 is in the left (×2) column, we show only 2×7, which when ×2 is 28. These truncated factorializations are called the trigram values. The function of the column (i.e., ×1, ×2, ×3) is applied to the trigram value to produce the sequential value. In this case, 2×7, being in the left column, is multiplied by 2, resulting in 28. So, a trigram value of 7 produces the sequential value of 7 when in the center (×1) column, 14 when in the left (×2) column, and 21 when in the center (×3) column.

The 1^st value of the center column is the 1^st instance, where the counting, or creation, begins from. The 1^st instance of a value is then 0 in the center (×3) column. In the “Sequential values” table, all values in the the center column are always, and only, products of 3, so we know that the center value will always be 3 × the level of the embedded trigram. i.e., the 4^th level of an embedded trigram will have a center value of 3×4. The exception to this appears to be the initial value of 0, but as this initial value is a product of 0×3, 0 is still a product of 3.

The “Pattern of variance” charts plots the difference between sequential values in each column of “Sequential values”. For example, the right column has the values 1,5,7,11,13, etc. The difference between these values is 5-1, 7-2, 11-7, 13,11, or 4,2,4,2,etc. The center difference between center column values is always 3, and the difference between left columns values is 4-2, 8-4, 10-8, 14-10, or 2,4,2,4, etc., which is exactly the same as the right column difference, but offset by one place.

This variance is also seen the numbering in the values along a the left and right axis of the trigram.

We now have different models of progressions depending on our perspective, because, if you remember how the sequence of the thologram was generated, each child thologram was the inverse of the parent, which creates a different sequence than that created by the column functions.

This difference in sequence only effects the attributions of the polar children of Definition and Contribution, and not the attribution of Negotiation, but the difference is that in the column functions, they alternate position every other generation. So, either the polar attributes oscillate and all the increments remain the same, or the polar attributes stay the same, and their increments oscillate. This is similar to sitting in a train at a train station, and the train next to you starts to move. For a moment, you are not sure which train is moving. If all we are interested in is the relationship between the 2 trains, it doesn’t matter which train is moving.

You can see how the right column also holds all the primary numbers, except the numbers 2 and 3, and we know that prime numbers are perfectly unpredictable because we know that the pattern of prime numbers are unpredictable, and if the primes are unpredictable, then the products of primes are also unpredictable. So, while we can’t predict where a prime number will be, we can predict where either a prime number or a product of prime numbers that are not column functions of 2 or 3 will exist. The rule is, any number that when divided by 2 and by 3 result in two fractions is a prime number or a product of a prime number. This holds true for any number after 4. Interestingly, by this rule, the number 1 is a prime number and 2 and 3 are not. For the record, up until the early 20th century, 1 was a prime number, but was demoted from prime status simply due to convenience. Euler, perhaps the greatest mathematician who ever lived, took a more fluid approach; sometimes he used 1 as a prime, and sometimes he adopted the Greek idea that 1 is not even a number, but rather the progenitor of numbers. This seems the most sensible approach because unlike all other numbers, the number 1 has many faces.

The thologram has the concept of a tholonic prime, which is a unique value that can’t be a product of any smaller numbers. This is more or less the same definition as a regular prime, but it includes 1, and even 0. In this way, the first order of tholonic primes are 0,1,2,3, (which are also the values of the first tetrahedron) and all numbers that follow are secondary tholonic primes (which are the same as modern traditional primes).

Another interesting pattern we can see in the numbers of these 3 columns is their crosssum values, or the value of the all the numerals in a row added up, i.e., 123 as 1+2+3=6 (see Appendix F, “An Unexpected Pattern” for more information on crosssums). These cross-summed values, while loosing their quantitative value, expose their qualitative values. If we look at how the values in the “Sequential values” table change, we see 3 distinct patterns. The ÷3 (center) column follows the simplest pattern 3, 6, 9, and is the only pattern that is a single “frequency”. The ÷2only (left) is 2 “frequencies”, or rather “waveforms”, that of binary growth, [2,4,8] (or [2¹, 2², 2³]) and that of prime number growth, [1,5,7]. Again, notice how 2 and 3 seem to be outside the set even though they are prime numbers. We can say that [1,5,7] are the first 3 prime tholonic prime numbers excluding 2 and 3.

The crosssums of the center and the left columns together use every number 1-9. The Primes & Products (right) column starts with 1,2,3, and then continues with the same frequencies as the left column, but in opposite, or reverse order.

In total, we have 3 unique ‘frequencies’, the 3-based [3,6,9], the 2-based [2,4,8], and the prime-based [1,5,7]. If cross-sums show the qualitative properties, then we can say that the [3,6,9] is the pattern of Negotiation, balance, and stability. The right column, that of Limitation and definition, is an unending stream of the numbers [2,4,8,1,5,7], and the left column, that of Contribution and integration, is the stream of [1,5,7,2,4,8].

The difference between these infinite series of numbers, if we add the right column and subtract the left, is +1, -1, +1, -1, +1, -1, and it is this oscillating series that defines the spectrum across with new N-states can appear. You’ll notice this series of primes and products does not start out with this pattern of [1,5,7,2,4,8], but rather with [1,2,3,5,7,2,4,8], but the [1,2,3] is then replaced with [1] for the remainder of the series.

We argued previously that the first tholon, which contains the numbers [0,1,2,3], should be considered in a class by itself, as it (and its numbers) is the archetype of all the trigrams (and numbers) that follow. In this number series, we are literally being told that the number 1, the archetype, or tholon, of all numbers, contains 3 partons, and those partons are themselves tholons that exist as the 3 columns of the thologram, ÷3, ÷2only and primes and products of which the tholon of 1 is also a parton. This is another example of the fractal and self-similar nature of numbers.

This anomaly in the first instance of the pattern is significant in another way as well as it appears in, and only in, the domain of Limitation and definition, as it is the archetype, or tholon of [0,1,2,3] that not only forms the 1^st tholon of creation, but limits and defines every number that exists.

If the concept of 1 is a tholon, then what are the 3 “perspectives”, or “faces”, or partons that are required for a tholon to exist? Here we are shown that those 3 partons are the values [1,2,3], which means that the (tholonic) definition of the number 1 (as well as 2, and 3) is an infinite, self-referencing, circular recursion.

What π, 2, 5, and e in common

This topic probably qualifies as an appendix; however, it’s too significant not to place front-and-center, as it is fundamental later in the book when we consider where all universal knowledge is stored and how to access it.

Note: For a detailed mathematical exploration of how π emerges through tholonic recursion and its connection to perfect squares, see the earlier section “Mathematical Foundations: Tholonic Recursion and the Emergence of π”. This section approaches π from a geometric and visual perspective.

A fascinating pattern in the thologram shows not only that of π, 3.1415924, but also the transcendental part of π, or 0.1415924. This value is the part left over when you take 3 perfect trigrams and curve their outer edges to form a half circle (shown below). Because a curved line is an infinite number of points, as opposed to a straight line, which is 2 points, the transcendental part of π represents the difference between the archetype of a singular instance of the simplest and most balanced form that can exist and the archetype of the form that is created by an infinite number of trigrams. The same applies to that of a tetrahedron and a sphere, the most common form in the Universe, but as spheres are 3D, the difference is π².

Interestingly, when we square the transcendental part, (π-3)², we get the value 0.02004847955, which is 5/10,000^ths off from 2/100. As $\sqrt{\frac{2}{100}}$ = 0.14142135623, we can approximate π with $3+\sqrt{\frac{2}{100}}$ or, if we just use prime numbers, $3+\sqrt{\frac{2}{2^2\times 5^2}}$. This approximation is off by 0.00017 (0.00546%), which is far better than the traditional approximation of $\frac{22}{7}$, which is off by 0.003 (0.04%), but not as good as $\frac{333}{106}$, which is only off by 0.0026%.

(Apologies to math purists, but as there is no symbol for the transcendental part of π, we’ll be using the capital omega character, Ω, a catch-all symbol for the “end”, to represent that value, and also the small omega character, ω, to represent the component values derived from the fractions that lead to Ω .)

There have been, and are, many creative interpretations of π; books have been written on its mystical properties, calendars have been created based on π, and π even has its day, March 14 (3/14, obviously), that is celebrated globally (usually involving some actual pie) and is even sanctioned by the U.S. Congress and UNESCO. If you appreciate cosmic coincidences, Albert Einstein was born on Pi Day, 1879 (as well as famous porn star Sasha Gray, 1988). In this case, Ω represents the infinite part of π, and 3 represents the finite part… nothing too mystical or philosophical there, but it is curious that π starts with a finite value (3) and ends with an infinite value(Ω).

In example a), if we alternately subtract and add the inverse of the number of tholons for each row (1, 3, 5, 7, 9,…) and multiply the result by the number of sides (4), we end up with 3.141592, which is the value of π. Leibniz originally discovered this pattern which is appropriately called the Leibniz formula for pi.

In example b), we use the same formula as a), but with swapped numerator values of 1 and 4. Why? Because 1 as a numerator (in a) describes the unity of a tholon with 4 contexts (sides). This is the perspective of a holon with its 4 contributing partons. In b), the 4 describes each contributing element to the tholon, i.e., the perspective of the partons relationship to the holon. In the context of a), π represents the holon, which, being tetrahedral, is a tholon. As a complete tholon requires 2 tholons, it would be represented as 2π, and, in fact, 2π radians is a complete circle and represents the defining or limiting context for its partons.

From a geometry perspective, if we look at the basic 2D map of a tholonic matrix, we can see that it is a 2D trigram as well as the makings of a 3D tetrahedron as 4 trigrams so that we can consider this in both perspectives; each individual trigram as one side of an existing tetrahedron, or as four trigrams that join together to form a potential tetrahedron. Example a) considers one side to an existing tetrahedron. The formula counts the tholons by the rows of the entire trigrams (as a side), then has to multiply by 4, the total number of sides needed for a tetrahedron, to get π. With this formula, each of the 4 sides contributes 1/4 of the solution, which we would expect as we are looking at only 1/4 of the tetrahedron. Example b) considers the tholon as one whole made of 4 sides and therefore need only multiply by 1, and in this way, it arrives at Ω.

So what is this suggesting? That the instantiation of π, which never ends, begins with the first instance of the primal archetype of the original 0-dimensional dot of A&I when that 0-dimensional dot became two to form both the concept of a line and the 1^st dimension. We will see this again in the next chapter when we explore the tholon’s relationship with Euler’s God Formula, +1=0, the “most elegant formula in math”.

Note: As more detailed (and somewhat circuitous) description for those who enjoy math and numbers but is not necessary for the overall understanding of the thologram can be found in the following section titled “The Entropy of Numbers”

The Entropy of Numbers

If you were asked what pair of numbers, when added together, make 4, you’d probably say 2+2 or 1+3. You would probably not say √10.6933136688+0.90038532821³ because, even though it is correct, it makes no sense, adds no useful information, and requires a lot of work to provide. You would naturally provide an answer that was easiest to arrive at, easy to comprehend, and has a chance of being useful information. This might sound like common sense, but only because it is common to do things that are more efficient, practical, and useful than inefficient, impractical, and useless. But why do we even care about that in the first place? The answer is that any movement of energy always travels the path of least resistance. Thinking requires work, and ideas and concepts require work to be maintained. This is common knowledge, but we tend to only apply the concepts of energy in a thermodynamic or electrical context. The instance of the laws of energy outside of those contexts can be found in anything and everything that is the result of energy, which is every context possible.

The movement of energy is the balancing of an imbalance. The first law of energy is that it will always move toward balance. Thought is energy. Ideas are energy. Concepts are energy. How, then, does this law of balance manifest in ideas, thoughts, and concepts? First, we need to ask how ideas, thoughts, and concepts came to exist in the first place. The traditional view is that our meta-awareness results from our evolved neurology. While that is true, the tholonic view is that our neurology is an effect, not the cause. What is the cause? The movement of energy, which forms patterns, which become partons of yet more complex systems, living systems. If the Universe, and everything in it, is the result of a 0-dimensional concept whose only attributes are A&I, then not only are all things that exist instances of, and have the qualities of, A&I, but the expression of A&I is the very reason for their existence and the driving force behind evolution, growth, and direction.

For us humans who have managed to evolve a ventrolateral frontal cortex, we have been able to discover and invent the miraculous tools of thought, ideas, and concepts, which has brought humanity to a new level of discovery and creation. Of all these advanced tools we have managed to evolve, the great under-sung hero, and by far the most powerful is that of curiosity. All creatures may be curious to a degree, but humans stand alone in their ability and capacity to ask “why?” and “how? and then find an answer. Curiosity is the most powerful driving force for the self-aware being, more potent than even the need to survive, as curiosity has been the end of many an explorer.

Why is there a $1,000,000 reward for finding the solution to a^x + b^x = c^x? Why do we ask “What if?” or the proverbial question, “What happens when I press this button?” Why have we spent years, effort, and money to discover ever-larger prime numbers (currently at 2^82,589,933 -1, a number that has 24,862,048 digits)? What drives us to calculate π to the 2,000,000,000,000,000^th place? Curiosity. This may sound overly simple, but there is nothing simple about curiosity. Curiosity is hardwired into our being. It makes the brain more efficient, expands our options, and leads towards the unachievable balance between the 3 states of knowledge; what we know, what we know we don’t know, and what we don’t know we don’t know. Our brain rewards us in the form of dopamine when we have learned something, found an answer to a question, or solved a problem, especially when the motivation is internal rather than external, learning for the sake of learning vs. learning to pass an exam. The degree to which our aggressively unnatural modern society has altered and/or manipulated these evolutionary circuits is a conversation worth having but is not relevant to the point that all life, being an expression of energy, will replicate the patterns of energy across all contexts. Curiosity is not born of a human or animal trait; it is the movement of energy in the realm of archetypes, and as we explore the world of abstract thought, we become the conduits for that energy. Curiosity is an instance of the energy moving through the circuits of life in the context of a self-aware mind. It is that same energy moving through every context, making all that exists active elements in a living circuit.

“Knowledge is light, ignorance is darkness” is a very common phrase and metaphor, but why do we naturally equate light with knowledge and that which is “good”? Perhaps it is because we know at the cellular level, and after 4 billion years of evolution, that light is life. Although less poetic, it would be more accurate to say, “Energy is life”, given what we have learned about light and energy. Still, as we only discovered this fact a little while ago, starting with the works of Isaac Newton, and taking another leap with the works of Albert Einstein, I assume that the phrase predates those giants by millennia in one form or another. We could go further and say, “Energy is existence”, and if energy is existence, then existence, i.e., all that is; life, planets, plants, aliens, etc., is not only subject to the rules of energy but is formed by the rules of energy.

If the above is true, we should be able to identify the fundamental laws of energy, such as entropy, resistance, potential, charge, etc., in any context. One obvious example is heat. It is one instance of energy that is the same for all contexts and things, as there is nothing in this Universe that has no heat, even the empty void of deep space.

So how does this apply in the context of concepts or numbers? Let’s take the concept of entropy. Entropy is defined as a value of the amount of energy that escapes a system and is proportional to the number of microstates available. The number of microstates is defined as “the number of ways system components can be arranged without changing its energy”. In other words, the higher number of microstates, the more ways energy can dissipate and the higher the entropy. Can this concept apply to other types of energy, such as the energy of the concept of numbers? Using the same definitions, we can then say the entropy of a number is “the number of ways a value’s components can be arranged without changing its value”. For example, when dealing with only natural numbers (excluding 0), the value of 3 can be “arranged” as 1+1+1 and 2+1, which would be the equivalent of 2 microstates. The chart below shows the “microstates” for the numbers 1-10 using only prime numbers (the inset is only meant to make the patterns more pronounced).

There are 23 ways (or “microstates”) to make the value of 10 using unique combinations of prime numbers, starting with 1+1+1+1+1+1+1+1+1+1, which requires the least amount of “work”, ending with 3+7, which requires the most amount of “work”.

Besides this form of numeric constructions being the same process used by the Pythagoreans when analyzing the properties of a number, the reasoning behind how these number towers are formed is straightforward; create each “microstate” starting with the “simplest”, “least resistance” or “lowest entropy” numbers, and keep adding until you get to the final value. In the example above, we only use prime numbers, but the same process works when using all numbers or when using only single instances of numbers. It’s debatable which is more ‘accurate’. As this is mainly a conceptual exercise to demonstrate that we can apply energy concepts to numbers, it may not matter. However, the sort order shows an interesting fractal pattern of trigrams, each representing a new level or value. The order is defined by 3 simple rules meant to emulate the natural world: 1) The smallest numbers must first be used (from left to right). 2) When all possible combinations are exhausted, a new number will be created using the least amount of “work” possible. 3) Only numbers that have already been created can be used.

Using the unique prime number arrangements (like above, but only using any number once in a set), the chart below shows the 53,041 possible configurations that can produce the number 300. We won’t get into the details of the graphs, but there are two points worth a closer look: the persistent presence of both trigram relationships and patterns (left) and binary expansion (right). The most interesting regarding the binary expansions is that the first 7 or 8 layers show a clear and consistent binary pattern. The next 7 or 8 rows begin to show changes emerging. Beyond that, we begin to see the beginning of the fractal nature of these “microstates” with patterns and colors replicating and flipping every 2ⁿ columns becoming more and more chaotic, like leaves of grass flowing in the breeze.

Note: In both charts, the colors follow the spectrum from dark-red to fuchsia (bright violet), with dark-red presenting the lowest number in a set that sums to 300 and fuchsia representing the largest number in that same set. Each dot also has a depth, creating a Z-axis which is not shown that looks similar to the vertical/Y-axis

Applying the concept of microstates to numbers suggests that large numbers “dissipate” faster than small numbers. In one way, this is true, as the dissipation of energy is actually the energy balancing itself out by distributing itself evenly across its environment via the microstates. The number 2 can be distributed only as 1+1, but 300 can be distributed in 53,041 ways. How many ways can the number 12,345,678 be distributed? We can’t know that answer without counting every possibility as there is no way (I believe) to predict this value as it depends on when prime numbers appear, which are themselves unpredictable. Technically speaking, 1 can’t have a microstate because, by definition, 1 is unity; it has no sup-parts because it was created from nothing, 0. Therefore, 1 can only have a macrostate, and that must be the only macrostate that can exist.

A more concrete example of applying entropy to numbers is that of water pipes. The larger the pipe, the lower the entropy. A water pipe with 0 entropy, or 0 resistance, would have to be as large as theoretically possible. This would equate to the number 1, or the singular 0-dimensional dot of A&I, or the entropy of the Universe at the moment of the Big Bang (which was actually 10⁸⁸ k_B, so, while not 0, it was the lowest amount of entropy that will ever exist, as that value today is 10¹⁰³ k_B and will continue to grow over the next 10²⁰ years until it maxes out at 10¹²³ k_B). The first finite water pipe, or the first instance of resistance or limitation, would be equivalent to the value of 2. We can see this concept in how all numbers are divisible by 1, but only half are divisible by 2. More resistance gives us 3, which $\frac{1}{3}$ of all numbers are divisible by, etc. In this water example, the “size” of the water line is the inverse of the value. So, a water pipe of 1, where 1 represents the unity of all that exists, means that the water pipe is as big as unity, as unity÷1=unity. A pipe as large as unity÷2 is half the size, etc.

Just like water and electricity, if we double the resistance (or make the pipe half the size), we get half the current (or half the water), so as far as efficiency is concerned, two 2s are identical to one 4… at least this was the reasoning behind using only prime numbers.

The value of 1 alone has 0 microstates, and as such, its energy can’t move anywhere within itself and therefore has 0 entropy. We know from classic entropy that this suggests maximum pressure, maximum order, and maximum impetus. As there is nowhere for 1 to expand to within itself, the “energy”, which must move, is forced to create a new state or value outside of itself that requires the least amount of “work”. This would be the value of 2. The value of 2 has one microstate and therefore has higher entropy, slightly less pressure, and order and represents a more balanced, distributed condition of the “energy” of 1.

We see this same type of expansion in simple math when we have to create a new “place” when we add 1 to 9 to get 10, and we also see it in electrons when they get energized and jump to their next higher orbit of the 7 orbits that exist in every atom (image right). This is also the source of all light because when the electrons drop from a higher orbit to a lower orbit, they emit the excess energy that is no longer needed to maintain that higher energy state as a photon of electromagnetic energy (of course, the reality of these orbits and states is quite a bit more complex than the image suggests).

We took the long way to get to the point where we can demonstrate that binary counting is the most efficient way to count and that counting naturally occurs in the movement of energy. OK, maybe not the most efficient way for humans, but the context of humans is quite different and far more restricted than that of the Universe. Surprisingly, no culture used binary counting until 1963, when IBM came up with the idea to use it for computers. The one exception to this is the binary-based, metaphysical Taoist writing titled the “I Ching” (“Book of Changes”), which dates back to ancient times. Still, this material was not considered practical or valuable (by Western culture) and was relegated to mysticism and lore.

Note: To avoid confusion between the bases, binary numbers are preceded with a “b”.

The binary system of numbers is not only the most efficient, but it is the only counting system that contains the most efficient recipes for creating any value. For example, let’s say we have a result of some addition that equals 10. We could have arrived at that value by adding 4+6, or 9+1, etc. However, in the binary system, the value of 10 is represented as b1010, so we automatically know that the most efficient way to create 10 is by adding b1000 (8) and b10 (2). Any other combination of values would require more “work”. Here are all the possible combinations of creating 10 with pairs of numbers. They are listed in ascending order of “work” required. As we would expect, the most efficient formula is that which is based on 2, as in 2¹+2⁴.

Consider the binary number b11010011 (211). How long would it take to discover the most efficient numbers that sum to this? In binary, the number itself shows us the answer. Using any other base is considerably more “work”, as shown in the image below, and we know this because 2 is the first (pure) primary number and, therefore, the first root of all following numbers (in the binary system). Were there a more efficient system other than something and nothing, that would be the system reality would be based on because that would be the way energy moves.

We can easily divide all numbers into 2 groups of odd and even. How do we create 3 equal groups for all numbers? This is done in the thologram (and explained in the Appendix C, “Tholonic Math: Prime Numbers”), and those groups are:

• All numbers divisible by 3 (33.3333…%)
• All numbers divisible by 2 exclusively, i.e., excluding the previous numbers divisible by 3 (33.335%)
• All prime numbers and products of primes numbers, excluding the previous prime numbers of 2 and 3 (33.3317%)

We now have 3 groups of equal size that can describe every number’s type, and by assigning each group to a side of the tholon, we can see some fascinating patterns (see appendix).

In the same way, for the values 0-7, we can assign a 0 or a 1 to each of the 3 sides to describe each value with the 3 sides of the tetrahedron (shown below for numbers 0-7). We can only use the values 0-7 as they are the only values that require only 3 places

So, using the sides of a tholon only, we can hold 8 values (the center trigram of the tholon holds the total value). This organically makes the tholon a base-8, or octal, number system, which is the same base used in all digital processing.

Note: To avoid confusion between the bases, octal numbers greater than 7 are preceded with an “o”

That’s fine for 0-7, but what about larger numbers? What about the number 8? As this is octal, there is no 8, per se, as 7 is the largest single value that exists, so 8 in octal is written as o10, which is the same as o010 which is the same as b1010. We can’t fit 4 values to 3 sides, but we can refer to the parent tholon representing b111, which is 7. Below is the same chart as above but using octal numbers instead of binary.

How, then, do we notate o10? As we already have a 7, which is a parent tholon, we only need to add a child tholon that has a value of 1, as shown in the left-most image below:

The center image above shows the same technique for 14 (7+7) and the right image for 17.

As you can see, this is not a very human-friendly way to count, especially when we get to numbers like 300,000,000. We’d have to start with a trigram that stretched from Istanbul to New York City to be able to read the last trigram. Fortunately, the point of this is not to introduce a terrible way to add (although if the Department of Education wants to give me billions of dollars to develop their NEW New Math program, please get in touch with me), but to show the fractal nature of numbers. It also shows how numbers act like energy in the same way we demonstrated above with the number towers because a trigram must be filled before a new trigram is added. Finally, it shows that, at least tholonically, there is an optimum set of parton numbers that creates a holon value; 7+7+3 = 17 is more optimal than 2+3+5+7, and in the tholonic system, it would not even be possible to notate that because the tholonic system organically only permits the most optimum set.

Further on, we will see how this same tholonic process is why cells and organs come together optimally.

You probably already know, or figured out, that using base-2 numbers limited to 3 places results in a base-8 system because 2³= 8, but the less obvious implication is that the sides of the tholon can act exponentially to whatever is applied to them.

Just for fun, here is what the same process above looks like if we start at the center, move outward, and then change the trigrams to circles.

An interesting example of how the same patterns appear across different scopes is how the xyz coordinates of the tetrahedron, cube, and octahedron relate to these binary values. For example, in the image below, we can see that the simplest xyz coordinates for each shape have a pattern, which is then expressed in 1s and 0s. If we interpret those 1s and 0s as binary values, we can convert them to a decimal value and then map that value to the 3D location of a point.

Note: Negative values are in red, odd values in blue, and even values in green. 0 is yellow, as it has both odd and even qualities.

There are several interesting relationships, but especially interesting is the octahedron because, to ensure all the coordinates are integers, we have to use 2 as the dimensions of the sides, unlike how we have to use 1 for the other shapes.

This is an interesting details to add to the pattern, which now looks something like:

What this coordinate values also show us is that each point has a qualitative value and order, which tells us there is a direction, which looks like this, respectfully:

The Paths represent the closed form we get when we connect the vertices in their binary order. 1 can only connect to 2, 2 to 3, etc. It is a closed path, or closed form, because when we get to the last point, we connect it to the 1^st point (indicated in red). An open path occupies 3D space but is itself only 1 dimensional and cannot define a 2D or 3D form. Columns 2 and 4 show closed forms. Columns 3 and 5 show these closed forms after that have been transformed via bi-cubic uniform B-splines (Catmull-Clark algorithm, specifically), which is intended to show the most probable flow of energy, using the vertices of the original forms as control points. Whether or not this is the most accurate way may be debatable, but regardless, we see some fascinating patterns, such as in Column 5. Also interesting is how in Col 2/3, we see the tetrahedron produces 1 “saw-tooth” form of a cycle, the octagon produces 2 cycles, and the cube produces 3 cycles. All these images are orthogonal projections of 3D shapes onto 2D space, so these cycles are not actually cycles but just appear as cycles. However, the same could also be said about our 3D cycles of light, energy, sound, etc., if we consider them as 3D projections of a multidimensional source.

Column 6 shows a colored 3D version of the original form when the vertices are connected in their binary order and can be considered archetypes of the 3D forms in Column 1 because the binary paths are simpler than the 3D shapes they create. It’s worth pointing out that Col 6/tetrahedral is a hyperbolic paraboloid, which most of us know as the shape of a Pringles potato chip. The Pringles shape (and the canister to hold them) was designed by Fredric Baur, an organic chemist and food storage scientist who was keenly aware of the hyperbolic paraboloid’s remarkable durability. This shape is familiar in architecture for the same reason, especially in the military and for roofing. So, the next time you get to the bottom of a Pringles can and the last chip is still intact, you can thank the tetrahedron.

If we apply these forms to the trigram, we get the following:

There is another relationship between these forms that is worth looking as well, but we will want to look at all 5 Platonic solids for this.

We can add the degrees of the angles of each of these forms to get their total number of degrees. For example, a tetrahedron is made of 4 triangles and each triangle as 3 angles of 60°, so 60×3×4 = 720. Applying this to all the platonic solids we get: tetrahedron = 720°, octahedron = 1440°, cube = 2160°, icosahedron = 3600°, dodecahedron = 6480°, and totaled all together = 7920°.

As a reminder…

• Tetrahedron: 4 faces of triangles (@ 60°×3=180°)
• Octahedron: 8 faces of triangles (@ 60°×3=180°).
• Cube: 6 faces of squares (@ 90°×4=360°)
• Dodecahedron: 12 faces of pentagons (@ 108°×5=580°).
• Icosahedron: 20 faces of triangles (@ 60°×3=180°).

You can see they that are all a perfect multiples of 10 of commonly known values, but more interesting is that they have the same relationship to each other as the numbers 1, 2, 3, 5, 9, 11.

As well, every total number when divided by 60 = 12. This might lend a clue as to why we have 60 minutes in two 12 hour periods to define a day, as per the ancient Babylonian system that our time system is based on.

Now, here is some crazy coincidence… the mean diameter of the Earth is 7920, the total of all the angles of the platonic solids, and the mean diameter of the Moon is 2160 miles, the angles of the cube! What’s really crazy about this coincidence is that the mile was a totally arbitrary length that started out as 1000 paces of a Roman soldier, so we would never suspect that a human’s pace would relate to planets sizes… but they do because the same patterns that create the platonic solids, are the patterns that create the planets, and life. Contrary to accepted history, the base 10 counting system (which the 1000 paces is based on) is not because humans have 5 fingers on 2 hands. Humans have 5 fingers and 2 hands because 5 x 2 is one of those fundamental patterns that we see on all of life, and the base 10 is the most efficient system of counting simply because 0-9 is the most contiguous set of values that can be defined with 2 trigrams.

“Arbitrary” is like “coincidence”… neither actually exists.

We can also see that the dodecahedron is the only form that is the perfect multiple of 3 of the cube, which is itself a perfect multiple of 3 of the tetrahedron, and if we add the degrees of these 3 forms, we get 9360, which is the degrees of the tetrahedron, or 720, × 13.

And this finally brings us back to …

With all that said, let’s return now to the tholonic formulas for π, which are based on the Leibniz model for π. This model provides a way to measure the difference between each generation of tholons. While Leibniz used odd numbers, the tholonic perspective is to consider the generations of tholons because the thologram is a model of creation, growth, and the movement of energy from an originating point (the parent tholon) to a destination point (the child tholon). As such, we are using the size of each sequential pair of parent and children, or family of tholons, i.e., 1 tholon generates 2 (real) child tholons, hence, a family of 3. These 2 child tholons generate 3 children for a family of 5 (real) tholons, etc. The quantitative values are the same in both equations, but the qualitative values are radically different. For the record, while we are calling them a family, technically, they are more like siblings as they were all created in the same generation and by the same parent. Still, there is a natural hierarchy within this generation simply by their positioning relative to one another.

In applying formula b) we begin with 4÷family-size for the first tholon, so we have 4÷1, or simply 4. The next value, with a family-size of 3, would then be 4÷3, or 1.333, the next family-size of 5 is then (4÷5), or 0.8, etc. Then we combine millions of these values, 4 - 1.333 + 0.8 -…, to arrive at Ω. Not surprisingly, this series of values creates a pattern of a damped harmonic oscillator (top, image right), an oscillation that gets reduced due to energy being drained from a system. Of course, in this case (and all cases), the energy is not being “drained” but rather distributed. This progression goes on forever, with ever-decreasing amplitude.

What jumps out are the perfect, or exact, values for ω that only appear for family-sizes that are 5-based (image right, in yellow), and the fact that these values are progressing in binary order; 8, 16, 32, 64, … or 2³, 2⁴, 2⁵… but at the same time is divided by 10 (or in terms of prime numbers, 2×5) at each level, as they appear as 0.8, 0.16, 0.0032. While reality is often fuzzy, measured in degrees of truth, these binary values are absolutely exact, and they are the only exact values that appear. It’s also interesting that in this example, the growth of the thologram results in smaller and smaller values, just like the size of the tholon gets smaller and smaller in the binary counting example… which I guess is what we would expect if the thologram was internally expanding, but I wonder if there is a correlation.

In this example, the mathematical reason for this is that 1 can only be divided by numbers created by 2 and 5 (i.e., 2, 4, 5, 8) to produce an exact value. Still, as our family-size will always be an odd number, only numbers created by 5 will produce exact values. The more curious result is their continuous shrinking by a factor of 10 at each occurrence. To be clear, the very same thing happens with the other values, the difference being that only 5-based values produce an exact rational number. These 5-based values also integrate the base-2 binary and base-10 number systems perfectly.

On closer examination, we discover the relationship between 2 and 5 is very close and can be seen in their simple relationships $\frac{1}{5}=0.2$, $\frac{1}{2}=0.5$, $\frac{2}{5}=0.4$ or $\frac{1}{5}\times(2\times5)$, $\frac{5}{2}=2.5$, plus the more elaborate $\frac{2\times 5+\frac{2}{5}}{2}=5.2$. 5 and 2 just can’t get enough of each other, and given that 5 just happens to be $\frac{1}{2}$ of 10, which is the base of our number system, this 2/5 relationship is even more apparent.

There are solid arguments why base-12 or base-8 is superior to base-10, so why do we use base-10? The common argument is because that was the base system that was in common use when the Indians/Arabs introduced the popular place-holding system, and base-10 is presumed to have originated from our having 5 fingers, just as the Mayan base-20 comes from fingers and toes (making counting a form of exercise, I would imagine), and the base-8 of the Yuki tribe from California from counting the spaces between their fingers. Although, for many, 10 was a sacred number, which we can see in the divine languages Arabic, Hebrew, and Sanskrit, all of which were heavily biased toward base-10. “Divine” refers to the belief that these languages were believed to be delivered by God. Sanskrit, called the “Language of the Gods”, is said to have been created by God before he created the Universe. Hebrew comes from the Torah, which is said to be “given by God” and “was made of an integument of white fire, the engraved letters were in black fire, and it was itself of fire and mixed with fire, hewn out of fire, and given from the midst of fire”⁶. Arabic is said to be the language God revealed in the final revelation of the Quran. What is important here is not whether they were of divine origin but that the people believed they were.

In addition to the base-10 bias, the number 10 had specific significance, for example, the 10 commandments, 10 plagues, Jesus’ use of 10 in many parables, 10% tithing, etc. In the Hindu Vedas, which date back 7,000 years, the numbers 1, 3, 7, and 10 are most used and are considered the most sacred of numbers, with 10 also being symbolic of “many” and “fullness” and is (perhaps) why the Rig Veda (ancient Indian collection of Vedic Sanskrit hymns) is 10 volumes⁷. In the Islamic Hadith, Mohammad states that good deeds are returned 10-fold. These are just some of the many examples of the significance of 10 that came from the dominant and unquestionable cultural influence at the time when modern math was being invented.

For this reason alone, we would expect the Hindu-Arabic and Judeo-Christian number systems to be base-10 because that is what their respective gods gave them. Even if a better number system existed, no one was keen on committing heresy by pointing that out. We can easily imagine how base-10 counting made its way into these sacred texts if we presume that their origins, although perhaps divinely inspired, were written by men with 5 fingers on 2 hands.

However, the tholonic view is that we have 5 fingers and 2 hands because the 2/5 relationship defines a fundamental pattern in nature. This is common among most species of mammals and exists in many birds, fish, and insects, going back to the ancestor of all mammals, reptiles, amphibians, and birds, the pentadactyl tetrapods of the Devonian Age 400,000,000 years ago (pentadactyl mean “having 5 fingers”, and tetrapod mean “four-limbed vertebrate animals”). This is solid evidence that this 2/5 relationship is a very efficient pattern; otherwise, evolution would not have stuck with it for 400,000,000 years, at least where tetrapods are concerned. It would be natural, then, that we would use a base-10 system. But why 2 and 5? Because they are the values, the forms of energy, that are both in the domain of Contribution/form, i.e., interaction and integration with the environment, and both values represent the opposing sides of the same axis, as 5 is the “complement”, and in a way, the “opposite” of 2. This is because the 1^st generation of 2, i.e., the 1^st “child” of 2, is 5, just as the child of 1 is 4, and most telling, the child of 0 is 3. In the thologram, 0, 1, and 2 are points, and their children of 3, 4, and 5 (each child is always +3 of the parent.), are lines, and their children are points, and so on. The natural pairing becomes that of parent and child or a point and a line.

This may sound like “numerology” or some other form of irrational symbolism and superstition, but it is based on the movement of energy, which we claim numbers represent as they must follow the same laws of energy. As for the 5-finger phenomena, the traditional explanation from the Leading experts in the field of human anatomy, developmental biology, and evolution goes something like this:

“… we simply don’t 100% know the ‘why’ or ‘how’ of the narrowing of hand morphology to five digit rays in the Devonian. [Any hypothesis] would be highly speculative without a lot of data on the types of selective pressures organisms at the time were facing… ⁸

The tholonic explanation provides a very “real” answer, and by applying the rule of science, which says “something is real when it is a necessary ingredient of a theory that correctly describes what we observe”, these qualitative powers of numbers are as “real” as the never-actually-seen black holes and quarks.

Interestingly, we see a similar pattern of the Contributing 2/5 pair in the other Defining pair of 1/4:

$\frac{2}{5}$= 0.4 and $\frac{5}{2}$ = 02.5 Contributing pole

$\frac{4}{1}$ = 4.0 and $\frac{1}{4}$ = 0.25 Defining pole

This suggests that the spectrum where new N-states appear can be seen as an octave or a magnitude of difference between the poles, as the Contributing 0.4 is one octave higher than the Defining 4.0 whose 0.25 is an octave lower than the Contributing 2.5. This creates an octave of the size of 3.75, or (3×4)+$\frac{3}{4}$, or perhaps more relevant, 15 parts of $\frac{1}{4}$ or $\frac{15}{4}$. Equally interesting is how the 1^st N-state = 0 (i.e., $\frac{0}{3}$), but the 2^nd N-state = 0.5 (i.e., $\frac{3}{6}$), right smack in the middle of the nothingness and somethingness of 0 and 1, which is what we would expect as the 1^st trigram only defines the boundaries of the children and is not itself a child. The children, which are the tholons that instantiate and exist within the 1^st tholon, only begin after the first trigram.

The natural number system in the thologram appears to be binary, so binary math and binary logic are probably the most applicable for tholonic calculations. If all creation exists within the thologram, and the thologram is binary in nature, then this might lead to some insights into the “reality is a simulation” hypothesis.

Note: More on the significance of the number 5 in the section of this appendix titled “Prime Numbers”.

1,2,3, Points and Lines

In a 6-generation bifurcating tree, we have a total of 63 points, as we start with 1 point that splits into 2 points 6 times, so 20+21+22+23+24+25 = 1+2+4+8+16+32 = 63. The connecting lines of each parent-child relationship begin at 2, so 21+22+23+24+25 = 2+4+8+16+32 = 62.

If we look at these two numbers, 63 and 62, we find a fascinating pattern that is perfectly mathematical that is typically tossed aside as a mere curiosity.

Both numbers are created by the numbers 1, 2, and 3. If we look at the products created by these 3 numbers, among other interesting details, we see that the ‘point-based’ numbers are multiples of 3, and the line-based numbers are multiples of 2. There is also a 3rd set of numbers (23 & 32) with no multiples; of these, 23 is the only number that is both prime and has no multiples. 23 is the 9th prime number (or 10th if, like Euler, you consider 1 a prime number). Most interesting is that all the sums of each subset of numbers is a multiple of 11 (the 5th or 6th Euler prime number), and all the number totaled is 53.

What does this tell us when we consider that points are 0-dimensional non-existent concepts only, and the relationships between these non-existent concepts are 1-dimensional lines? Among other things, it tells us that anything with a dimension (i.e., everything that exists) results from a relationship, suggesting that reality is the relationship between 0-dimensional non-existent concepts.

Even though there are some patterns when we do that same calculation in base-8/octal, it is base-10 only that shows the most interesting patterns.