8: The Thologram

How the tholonic structure is defined

Synopsis: This chapter provides a deeper technical understanding of the thologram by examining its geometric foundations and mathematical properties. Through detailed analysis of how tetrahedrons, octahedrons, and cubes embody the N-D-C pattern, we establish a concrete framework for understanding how abstract concepts become physical form. The chapter demonstrates that the tholonic model is not merely philosophical but reveals measurable patterns in nature—from quantum experiments and cosmological ratios to the structure of the periodic table—validating that the same fundamental geometric relationships operate across all scales of reality. By distinguishing between real and virtual tholons and connecting ancient wisdom (Plato, Aristotle) with modern systems theory (Fuller), we show how the thologram serves as a universal template for understanding how order emerges from chaos through recursive self-similar patterns.

Keywords: fractals, geometry, duality, emergence, entropy, dimensionality, cosmology

This chapter describes the thologram, how it came into being, and details about its properties. As this book focuses on the practical and technical aspects of the origins and applications of Awareness and Intention rather than purely philosophical or metaphysical aspects, some technical content is unavoidable. I have kept it as simple and brief as possible. Even if you are not drawn to geometry or mathematics, the visual patterns and examples will illuminate how these structures operate across all scales of reality.

Inner, not Outer

We have seen how the creation of a single trigram will automatically create additional trigrams as part of the same creative process. This would imply an infinitely expanding realm of trigrams, as previously shown, and again shown in more detail in the image to the right. When limited to 37 points, this pattern is often called the flower of life.

The flower of life design dates back to 1300 BC, with the oldest known instance carved on the temple walls of the ancient Egyptian temple “Osireion”. Again, we see the same concepts and patterns appear over and over again in Man’s search for understanding using the power of reason.

The tholonic model sees this expansion a little differently. If we are claiming that the first instance of creation, that of the 1 dot becoming 2, and the 2 becoming 3, etc., even if we are speaking metaphorically, then what we are also claiming is that this first trigram represents the very first polar duality created by the first non-polar duality of somethingness and nothingness. This would imply that every duality and trigram that follows must exist within the first trigram, not outside of it. Every succeeding duality can only be a subset of this first duality and certainly cannot exist outside the bounds of somethingness and nothingness.

We see a very different pattern if we draw our expanding trigrams such that the children are always contained within the parent.

The NDC points represent the first trigram in this example diagram above (Fig. p1). Fig. p2 shows where a child N-State naturally forms along the CD spectrum. Fig. p3 shows the same process as Fig. p1, but as a child of the parent. This inwards expansion (Fig. p3) of the child N-State naturally terminates precisely at the limits of the parent’s boundaries, creating three additional trigrams. Fig. p4 shows the final form as a tetrahedron.

The flower of life represents an external expansion of the singularity. In the thologram, the singularity creates a trigram which then creates infinite trigrams. It is the expansion within that 1^st trigram where movement and growth happen, as there is nothing outside that trigram. From the scientific perspective, a singularity is a 0-dimensional point where some property is infinite. In the case of the Universe and the Big Bang that created it, that property is gravity. From the tholonic perspective, that property is Awareness and Intention.

In Fig. p1, the parent trigram, the order of NDC is clockwise, but the same three points of the resulting inner trigram are counter-clockwise. These end points have not been reordered, as the original clockwise NDC is still there in the parent, but the child trigrams represent completely new creations with their own ordering. The reason the children have a different order is because the new generation was spawned by the new N-source between C and D, and as D is always the first to follow N and always on the left of N (left, or portside, to be more accurate, is arbitrary, but we need to remain consistent). This newly spawned D will appear on the opposite side of the parent’s D, and so for the C as well.

We started with one parent trigram at generation 0. When the N-child forms and creates its own trigram, this produces four new trigrams at generation 1. The number of trigrams generated at each subsequent generation follows the pattern 4⁰=1, 4¹=4, 4³=64, 4⁴=256, 4⁵=1024… (note that 4² is skipped). Here is what the generations look like.

Looking at the trigram formation more closely, we see interesting asymmetries in how colors interact. Consider how Contribution (red) is introduced into Definition (green) on the right side . Can we say the same thing about how green is introduced into red on the left side ? No, because there is an order to creation: blue → green → red. Green existed before red; therefore, it is the red that is introduced into the already existing green point, not the other way around.

In other words, the reality of a thing contributes to the definition of a thing, and the definition of a thing is limited by the reality of a thing. This sounds obvious when stated, but we are dealing with the most fundamental properties of creation, which are necessarily the simplest and most basic laws.

While this is the basic idea of the model, it is only half the story. The more complete version requires us to recognize that between the red, green, and blue corner points, there exist three intermediary points: yellow (between red and green), cyan (between green and blue), and magenta (between blue and red). This creates a dual trigram system where one trigram has RGB corners with CMY intermediaries, and another has CMY corners with RGB intermediaries.

This is a critical concept, so it is worth a demonstration. We have specifically chosen the RGB/CMY color model because the RGB colors are additive and CMY colors are subtractive.

In the first case (Fig. t1), we see the external expansion pattern (like the flower of life). The generative process follows this sequence:

• Blue spawns green
• Green spawns red
• Green and red (additively) create a new yellow
• Yellow spawns a mixture of each of its component colors (red and green) with its parent color, blue, making cyan (created by blue and green) and magenta (created by blue and red)
• Magenta and cyan (subtractively) create a new blue
• Repeat

In the second case (Fig. t2), we see the internal expansion pattern where each new generation creates a new layer that sits within the parent layer. This layer is a mirror or inverse of the parent (Fig. t3), creating an oscillation over generations. This oscillation has its own contextual version of frequency and wavelength. The oscillation is both generational and fractal, much like genetics which shares this same pattern of oscillation (detailed in “The Tholonic I-Ching”). We also see how all of the additive colors move in a counterclockwise direction and all of the subtractive colors move in a clockwise direction showing that each generation moves in the opposite direction of its parents.

This is an important point to keep in mind, but going forward, we will only show the additive, primary, or dominant (to borrow a word from genetics) colors of RGB and ignore the subtractive, secondary, or recessive colors of CMY.

In the images and examples above, we use 6 colors to explain the thologram. These colors are based on the 3 natural primary and additive colors of RGB and their complementary and subtractive colors of CMY. These colors highlight many patterns. One significant pattern shows that everything can ultimately be reduced to 2 colors: blue and yellow. When we additively mix the left and right colors (red and green) that emerge from center blue, we get yellow, the center color of blue’s child. When we subtractively mix the left and right colors (cyan and magenta) that emerge from center yellow, we get blue. This creates a cyclical oscillation between blue and yellow across generations. This blue-yellow oscillation is specific to our model because we chose blue as the originating N-source. The same principle applies to any primary color and its complement: red oscillates with cyan, and green oscillates with magenta. The mathematical pattern is universal; blue-yellow simply emerges from our choice of blue as the origin.

While the choice of blue as the originating N-source is mathematically arbitrary, it carries remarkable symbolic resonance across multiple traditions. Most significantly, in Hindu cosmology, Vishnu (the Preserver) and his avatar Krishna are depicted as blue, representing the cosmic force that maintains balance and order between creation and destruction, precisely mirroring the N-source’s function. Psychologically, blue is universally associated with calmness, stability, trust, and mediation, qualities essential to negotiation and balanced states. In the natural world, blue dominates human experience as the omnipresent sky and the life-sustaining ocean, serving as the background field within which all else exists. Esoterically, the blue throat chakra governs communication and authentic expression, functions necessary for negotiation, while blue represents the transition point between earthly and cosmic consciousness in many traditions. Even scientifically, blue light carries the highest frequency among visible colors and blue stars burn hottest, suggesting a connection to primal, high-energy states. These convergent associations across mythological, psychological, natural, esoteric, and scientific domains suggest that blue’s selection as the origin point, while technically arbitrary, may reflect deeper archetypal patterns embedded in human understanding.

Below, we show the trigrams a bit differently, with the RGB as the points and the CMY as the lines, with no oscillatin g bweet RBG and CMY points

Below, we show the trigrams differently: RGB appears as the points and CMY as the connecting lines. This representation creates clearer three-dimensional models.

Row 1 shows both the 6-color version and the simplified 2-color version. Row 2 displays the patterns formed by the connecting CMY lines, revealing three basic grids: left-handed cubic, right-handed cubic, and hexagonal. The left and right-handed cubic grids are offset by half a “cube” from each other. The added shading helps conceptualize their difference. The rightmost image shows all grids overlaid together.

Row 3 shows the CMY lines along with their related RGB NDC points. Row 4 shows the same patterns in the simplified 2-color mode.

Row 5 presents a 4-generation, 2-color, two-dimensional thologram and compares it with the traditional Flower of Life. While both use the same pattern and reasoning, the Flower of Life expands externally without the thologram’s inward reflection and oscillation.

The four circular patterns above all show the RGB points with different connectors: the first shows CMY connectors (as the circles’ circumference), the second shows Negotiation-Definition connectors, the third shows Definition-Contribution connectors, and the fourth shows Negotiation-Contribution connectors.

In the world of math and geometry, this internally expanding trigram is similar to something called a Sierpinski Triangle, a fractal pattern that appears throughout nature and has practical applications in broadband antenna design, musical composition algorithms, fractal analysis, and molecular chemistry.

The difference between the thologram and the Sierpinski Triangle is that in the Sierpinski Triangle, there are two types of trigrams: those capable of self-replication (the black trigrams) and those that are not (the white trigrams), like this:

In the Sierpinski Triangle construction, only the black trigrams are actively calculated and replicate at each iteration. The white trigrams are merely byproducts, the negative space left behind as the black trigrams subdivide. The white areas are not computed or generated; they simply emerge as gaps in the pattern.

In the thologram, we consider the trigram not as an object but as the boundaries that constrain the movement of energy. The fact that the center trigram is a reversal of the three outer trigrams highlights the significance of this inner space.

While the mathematics of the thologram and Sierpinski Triangle are essentially the same, the conceptual difference is profound. The thologram treats the negative or empty space as equally valid as the filled space, though with different attributes. In other words, both the black and white trigrams are active participants in the pattern, unlike the Sierpinski Triangle where only the black trigrams replicate and the white spaces are merely passive gaps.

With each generation, we have more and more patterns, such as Fibonacci and harmonic series, but perhaps the most dominant pattern is the hexagon. In addition to being the crystalline structure of hydrogen, it is one of nature’s most persistent and useful shapes. It can be seen in beehives, Saturn’s north pole, dragonfly eyes, rocks, bubbles, snowflakes, organic chemistry, memory function in the brain, neuronal firing patterns, etc., as well as being the most structurally sound of all shapes. The success of the hexagon is because it is a synergetic form, meaning it is a form that can self-assemble and hierarchically self-organize. All synergetic systems, such as all organic life (and arguably the Universe and all its creations), are built upon synergetic principles.

The mathematical power of the thologram extends beyond these geometric patterns. The recursive nature of the thologram allows us to calculate fundamental constants including π, φ (phi), √2, and Euler’s number e. This is detailed and mathematically proven in the book by the same author, Introduction to the Tholonic I Ching.

Beyond the mathematics, the thologram’s visual patterns reveal important structural insights. Notice the little blue dots in the middle of the yellow lines: these mark the center of lines that can create new children (stable patterns) or new N-states that form new N-sources. Significantly, these lines only exist on the outer ring of the hexagon, suggesting where new growth and expansion naturally occur.

This theoretical hexagonal pattern finds its most elegant physical manifestation in graphene. A single layer of hexagonally oriented carbon atoms forms a material called graphene. Graphene is superconductive¹, can convert light into energy², can propagate electromagnetic signals³, and given its organic structure, easily integrates into organic systems such as plants and animals.^4,5 The hexagonal structure is not only the basis for life but also the basis for integrating organic life with technology.

From the tholonic view, organic life is a form of technology, and what is commonly referred to as trans-humanism, the merging of the biotechnology of nature and the technology of humans, represents a natural phase of evolution. Nature has precedent for such integrations: mitochondria, now fundamental to cellular energy production in all complex life, likely originated as parasitic bacteria that formed a symbiotic relationship with early cells and eventually became fully integrated organelles. Whether this particular evolutionary path of human-technology merger will prove successful or sustainable remains to be seen. What seems obvious and inevitable is that the merging of biotechnology and digital technology will have dramatic effects on the future of humanity and on our understanding of Awareness and Intention.

The tholonic structure is not simply a geometrical metaphor but a functioning model. One real-world example is plasmonics. Plasmonics, sometimes called nanoplasmonics, exploits the interaction between light and metal nanostructures to manipulate electromagnetic fields at nanoscale dimensions. These geometric patterns, particularly the hexagonal structures found in materials like graphene and gold, enable the redirection of light, concentration of optical energy, and transformation of light into electrical signals at scales as small as 10 nanometers—approximately 10,000 times smaller than a human hair. Below is a diagram of a plasmonic antenna showing how tholonic geometric patterns directly correspond to the atomic structures used for energy conduction and transformation. This antenna can receive energy and information and can be integrated into biological systems. This represents one of the closest physical implementations of a tholonic device currently achievable.

In the tholonic patterns, we can see how the N-sources converge to form the center of every hexagon and how opposite colors always form pairs on the outer edge. This arrangement provides insight into the nature of 0-dimensional N-sources, which are, unsurprisingly, at the center of order. These patterns offer clues for answering the question, “Where did the first N-source come from?”

As we saw earlier, the N-source (blue) is the progenitor of the subsequent D (green) and C (red) dots and therefore must always precede D and C. However, D and C, having all the attributes of the N, can create a new child version of N.

Why can only the D and C pairing create a new N? The N and D pairing cannot create a new N because this combination lacks the attributes of C (form). Similarly, the N and C pairing cannot create a new N because it lacks the attributes of D (definition). Only when both D and C are present can a new N-state emerge, because N-states have no form or dimension: a prerequisite for creating something out of nothing.

An N-state is simply an idea or concept with zero dimensions. In contrast, Cs and Ds are no longer merely concepts; they have dimension and form. D (Definition) introduces one dimension, creating a line or boundary. C (Contribution) introduces a second dimension, and with two dimensions comes form: an area or plane. s Understanding this dimensional progression is significant because it reveals where new instances can emerge. Even though archetypes can appear across the NC and ND spectrums, they cannot create children. Therefore, no N-state will ever appear across these spectrums. However, the CD spectrum, which is the axis of cooperation or conflict and its opposing point of negotiation, can create N-states.

As the trigram self-replicates, the hexagram naturally rotates 60° with each iteration, allowing every side of the thologram (any configuration beyond the first generation) to generate new N-sources. The graph above (Fig. h1) shows the lines connecting the N-sources and their respective N-states (the blue dots). This reveals that anything that exists will have three N-state axes: three expressions, types of instances, or perspectives that together represent the complete concept. For example, the concept of atomic structure manifests through three fundamental particle types: protons (positive charge/contribution), electrons (negative charge/boundary), and neutrons (neutral/negotiation). Similarly, matter itself expresses through three states: solid (defined), liquid (negotiated), and gas (expansive).

Everything Begins with the Trigram

Just as hydrogen is the fundamental building block from which all heavier elements are synthesized in stars, so is the trigram the fundamental building block from which all thologram generations emerge.

A curious observation emerges when considering hydrogen (atomic number 1) as the simplest material expression of the trigram, and carbon (atomic number 6) as the simplest material expression of the hexagon. These are the two most fundamental geometric forms in the thologram. When hydrogen and carbon bond together, they form hydrocarbons, the basic molecular building blocks of all organic life. This is not mere coincidence: the geometric principles that govern the thologram also govern the atomic structures that create life.

The numerical relationship between carbon and hydrogen reveals an even deeper pattern. Adding their atomic numbers gives us 6 + 1 = 7, a number laden with significance across countless cultures and traditions. If we consider 7 as representing life and 1 as representing unity, then dividing unity by life (1/7) produces a remarkable result: 0.142857142857…, an infinitely repeating decimal. This sequence is conspicuously missing the digit 3 and all its multiples (6 and 9). Yet when we sum the digits in this repeating pattern (1+4+2+8+5+7), we arrive at 27, which equals 3³. The number 3, absent from the sequence itself, emerges as the cubic foundation of the sum: a perfect cube with each side having a dimension of three. Remarkably, two tetrahedrons interpenetrating in opposite directions define the vertices of this cube, their edges aligning with the cube’s face diagonals. The volumetric relationship is equally elegant: the cube’s volume relates to the combined volume of these two tetrahedrons in a ratio of 3:2. The two tetrahedrons occupy exactly two-thirds of the cube’s volume, while the remaining one-third is the space occupied by the cube but not the tetrahedrons (and which forms an octahedron). The essence of life (7) thus contains the hidden structure of the trinity (3) at its core, expressed through the fundamental tetrahedral geometry of the thologram.

We can also apply the same associative reasoning to the thologram as a whole to show how its structure appears as a pattern of growth in nature. An example of this can be seen in the fractal progression below, which shows growth from a simple point to increasingly complex self-similar patterns. The images illustrate how the thologram expands from a 0-dimensional dot to a 1-dimensional line to a 2-dimensional area, demonstrating the process of distributing energy, balancing forces, and increasing complexity through recursive iteration.

We can say that the 0D dot could be infinitely self-similar if we consider that there could be an infinite number of 0s within 0, much like how 0 raised to any power remains 0 (0^n = 0). Similarly, expanding this concept into 1 dimension could create an infinitely self-similar 1D line if we consider that there could be an infinite number of 1s within this 1D line, just as 1 raised to any power remains 1 (1^n = 1). However, the expansion into 2D space fundamentally changes this property. When we raise 2 or any higher number to successive powers (2^n = 2, 4, 8, 16…), each iteration produces a different result, breaking perfect self-similarity. This is where we see fractal self-similarity emerge: the pattern repeats at different scales, but each iteration creates geometrically distinct structures rather than maintaining perfect identity. For each dimension that follows, these children unfold into increasingly complex patterns, exhibiting fractal self-similarity rather than the perfect mathematical identity found in 0D and 1D spaces.

This mathematical property provides profound support for the tholonic model. The N-state (Negotiation), existing in 0D where 0^n = 0, cannot grow on its own: it represents pure awareness and balance without expansion. The D-state (Definition), existing in 1D where 1^n = 1, cannot grow on its own: it creates boundaries and limitations but remains constant. Only the C-state (Contribution), existing in 2D and higher dimensions where 2^n produces exponential growth, is capable of actual expansion and multiplication. This explains mathematically why only the combination of D and C can create new N-states: Definition provides the boundary (1D) that constrains where, Contribution provides the area and form (2D) that enables multiplication and growth, and together they create the dimensional framework from which a new 0-dimensional point of Negotiation can emerge. The C-state, as the realm of integration and contribution, is literally where mathematical growth happens, validating its role in the tholonic trinity.

The image on the right shows a reference to the Sierpinski triangle example from earlier, where the outer trigrams were solid black, and the center was white. As a 2D parton, this white space represents the center trigram. As a 3D parton, it represents the base of the tetrahedron. As a 2D holon, the white represents all the centers of all the children of all the generations. As a 3D holon, the white space forms the 3D shape of an octahedron (much more on this later), which remains empty.

In the diagram above, we see the progression of fractal complexity correlating with increasing entropy. Using the Sierpinski triangle again, we show only the borders of the trigrams, which implicitly incorporates the “empty” center trigram. This negative space is just as full of implicit patterns as the positive space is full of explicit patterns. However, these implicit patterns exist as a result of the explicit patterns, which are continually expanding inwards.

In every generation of a 3D tholon, exactly and always 50% of each tholon is the nothingness of empty space, and 50% is the somethingness of the four children it creates. This perfect 50/50 split occurs because when a tetrahedron subdivides, each child has half the parent’s side length, and since volume scales with the cube of side length (s³), each child occupies one-eighth of the parent’s volume. Four children therefore occupy exactly half the parent’s volume (4 × 1/8 = 1/2), with the remaining half forming the central octahedral void.

This dimensional property reveals a profound truth about why three-dimensional reality exists at all. In two-dimensional space, the subdivision creates a 1:3 ratio between void and form (25% empty, 75% filled), an imbalanced state. In three-dimensional space, the ratio becomes 1:1 (50% empty, 50% filled), achieving perfect equilibrium. This perfect balance represents a lower energy state, as balanced systems require less energy to maintain than imbalanced ones. Therefore, three-dimensional reality exists because it achieves the minimum energy configuration through perfect equilibrium between nothingness and somethingness. This is not arbitrary: it is the natural consequence of energy minimization, the fundamental principle driving all physical systems toward their most stable states.

The octahedral space is never divided, but the tetrahedral space is always divided by four. This creates a fundamental duality between two realms: the tetrahedral realm of form, which is dynamic, ever-subdividing, and subject to increasing entropy, and the octahedral realm of void, which is static, unchanging, and possesses zero entropy. The void has zero entropy not because it lacks energy, but because it lacks the duality necessary to create the imbalance that drives movement. Entropy requires energy movement, movement requires imbalance, and imbalance requires duality. The void contains energy as pure undifferentiated potential, similar to how a magnetic field or photon can exist conceptually at zero energy: present as an archetype but without measurable kinetic expression. Entropy exists only in the realm of form, where duality creates the imbalance that sets energy in motion. As the tetrahedrons produce form (the tholons that produce reality), it is form and reality that forever approaches the nothingness of zero, approaching the octahedral void that has already achieved perfect zero-entropy equilibrium. The ever-shrinking tholons correlate with the ever-decreasing energy availability (increased entropy), which both lead to the same destination: total dispersion and balancing of energy, the state that the void eternally occupies.

2D to 3D

Now we are getting into the three-dimensional aspects of the thologram and its self-similar grid of 3D tetrahedrons, so let’s explore that further.

A note on the colors used: When coloring two-dimensional trigrams, the color choices are straightforward because the RGB/CMY model is tertiary. When we move the 2D trigram to a 3D tetrahedron, a new dimension appears, represented by the white dot created by the three pure colors. This naturally adds a dimension of lightness to our colors, making all sides of a tholon different.

As a tetrahedron, rather than two colors mixing to make a new color (which happens on a single edge in 2D), three colors mix with white, creating a lighter shade of each color on each face. The reader may notice this difference between the coloring of 2D and 3D models. To get a broader perspective of the color scheme and the forms, the image below shows all four sides formed by one tetrahedron. The right image shows how two tetrahedrons, one RGB and one CMY, separate and combine.

The image below demonstrates how a 0-dimensional dot expands into a trigram.

Below is a more complete, step-by-step description of how this trigram expands into the thologram.

Note: For clarity, each generation of tholon children is referred to as generations, while the tholons that are created from the peaks of these original tholons, and their children, are referred to as iterations.

Tholons model the movement of energy and consider the existence of form as the consequence of that movement. With that in mind, let us examine the tholonic explanation using the figures above.

Fig. a. We start with a simple trigram. This defines the most fundamental structure of how instantiations of archetypes come to exist and, therefore, the flow of energy and their resulting forms.

Fig. a1. The movement of energy inside this trigram will cause a new trigram to form by the expansion of N-sources due to the flow of energy. The most likely place for this new form is opposite the energy source and in the balanced center between the two limiting poles created by the source. Expanding this new N-source further divides, defines, or limits the parent’s scope, resulting in four new trigrams.

This division process is the basis of the thologram and, we claim, the basis of all creation, so it is worth further explanation.

In the image above, Holon View shows this division from the parent’s perspective, revealing how the three outer positions relate to the central N-source through 180° rotations. Each of the three outer trigrams rotates 180° around a different axis relative to the center, while the top (fourth) trigram maintains the parent’s original orientation. The Transforms view demonstrates these rotations visually.

The Parton View shows the divisions as four independent trigrams, each with its own three points of NDC. The 3rd Generation view shows the resulting fractal pattern after multiple iterations. As a holon, the core trigram serves as the center, and each outer trigram represents a different context of Negotiation, Definition, or Contribution for that tholon.

However, as only blue N-sources can create the red and green children, the trigrams operate differently as independent elements, resulting in different colors and properties. This shows us that the same instance shares different properties as holons and partons.

When all the partons work together as integrated partons (not as a holon), they contribute to the holon in a specific way. The dots and result in two yellow dots.

Why yellow and not brown (++=) and dark green (++=)? Because we are not considering the quantity of the properties but rather the qualities: the relationships between the properties. As was previously shown, the ⟷ relationship is one of definition and contribution, or limitation and form, imposing on one another.

In this example, the color yellow represents challenges, trials, testing, contention, and cooperation. Not coincidentally, this fits well as the opposite of yellow is blue, which represents unity, 1, the singularity.

In 3^rd Generation, this same process takes place in each of the child trigrams, and we get the 1^st instance of a hexagon, but not a single hexagon, but a trigram of 3 hexagons. It is difficult to see in the image above, but if we zoom in (right image) on these 3 primal hexagons, we see that the core of the top hexagon is a mixture of blue and yellow, which is white, and the 2 lower hexagons have a core that is red and green, which is yellow.

Note: Colors are used in this example because colors are another way to see the rules of energy in action, but we are not claiming these colors have these properties, as other expressions of energy could have also been used in this example, but as they are invisible, it would be difficult to show the relationships.

As was shown above, the arrangement of the colored points (N, D, & C) in each outer trigram is the exact reverse image of the center trigram. One way to think about it is the original trigram (top-most) reflects and reverses itself and, by doing so, creates a new trigram (center), which also reflects and reverses itself in the 2 new trigrams (left and right). In this way, the center trigram acts like a reverse reflection trigram, and the 2 new trigrams are a reversed reflection of a reversed reflection. This will be an important point a little bit later.

Note: We will use colors (RGB) rather than types (NDC) because it is easier to understand the geometry of a tholon using well-understood concepts like colors. You can remember that N=blue, D=green, and C=red, but it does not really matter for these descriptions. When it does matter, we will return to NDC labels.

Fig. a2. Because energy always follows the path of least resistance and always seeks balance, this complete tholon will automatically become a self-sustaining structure (tetrahedron) when all the conditions are met, as it would represent the most efficient form. These conditions appear to require 4 types of points; BBB, RGB, RGR, and GRG. Apparently, B points only integrate with themselves and with an R and G together, while the R and G can integrate with each other.

We see the above pattern when any 2 things interact with each other ,and can form more energetically stable, efficient, balanced relationships. For example, hydrogen and oxygen atoms form a stable state that requires less energy or conserves energy than both require to maintain a separate existence.

Fig. a3. Each of these child trigrams goes through the same process as the parent, with slightly different parameters determined by their parent’s limitation. We can now see 4 trigrams of 4 trigrams in their 2D form.

Fig. a4. This is a 4^th generation tholon map. To get an idea of how many sets of tetrahedrons it holds, the trigrams have been color-coded to make it easy to see. The darker colors represent what will be the base of a tetrahedron, with the light shade of the same colors representing their sides. You’ll notice that there are six sets of trigrams that are yellow (or a lighter shade of gray). These are identified a little differently because when they form a tetrahedron, they do so by going in the opposite direction because they are a reversed reflection image.

Fig. a5. If we go ahead and form the tetrahedrons, we end up with a series of tetrahedrons connected at the corners, some pointing forward and some pointing away (the more faded ones). You’ll notice that every odd-numbered row (rows 1,3,5,7) are all forward trigrams (trigrams with blue N-state parents), which we’ll refer to as real, a term taken from the world of holography to indicate that the image is projecting in front of the film. Likewise, all even-numbered rows (rows 2,4,6) point in the opposite direction (trigrams with yellow N-state parents). We’ll call these virtual tholons, a term also taken from the world of holography, referring to an image that virtually exists behind the holographic plate. For every pair of rows (1&2, 3&4, 5&6) there are an equal number of real and virtual tholons.

This structure may look familiar to a mineral chemist, as this is the same structure as the silicon dioxide crystal tridymite (shown below), a form of quartz. Other silica-based oxides also share a similar structure. This detail is mentioned later in this chapter when we look at how water also shares this structure.

Fig. b1. This is a side view and elevated view of 5 generations folded into tetrahedrons. Those pointing up are real, and those pointing down are virtual. This is where we see the first instance of oscillation, with the tholons alternating in their movement away from and towards their originating plane or between the virtual and real states.

But what is happening in Fig. b2? Why are there more tholons stacked on top and on the bottom? If you managed to slug through up to this point, congratulations! Here is where we get to see the engine of creation. Each of these new trigrams formed by the peaks of the previous children acts as the base for new, larger tholons! The process that created the first tetrahedron starts over, not only for each one of these trigrams but for every face of a tholon or trigram, including its children (which can’t be shown here as it is too complex and detailed for a 2D print image).

How can new tetrahedrons appear? You’ll notice that the peaks of the original tetrahedrons are the white dots, which is to say, each peak has the attributes of red, green and blue, allowing them to act as a red dot, green dot, or blue dot, and if they can act as R, G, and B dots, they will. When they do, they naturally create a new generation of white dots. If we start with a finite set of folded tetrahedrons, eventually, this ever-growing iteration of tetrahedrons will end, as each iteration’s number of real tholons is less by one generation, and the number of virtual tholons is less by two generations.

Each iteration builds on the one before it, making larger and larger tholons, as you can see in Figs. b1, b2, b3, which shows the progression of real and virtual tholons.⁶Interestingly, the growth pattern of tholons is the inverse of a recursive Fibonacci sequence. Inverse, because we are multiplying instead of dividing, recursive because we do that for each trigram, children of trigrams, children of children of trigrams, etc.

It is difficult to imagine how each side of a tetrahedron can go through this same process, but we can imagine the final state. Fig. c0 shows an orthogonal view of the tholonic matrix on all four sides of a 2D tholonic map, as in Fig. a1. Fig. c1 shows the perspective view, with real tholons above and virtual tholons below. Fig. c2 shows the same perspective view, but where the sides have begun to fold up to form a tetrahedron. Fig. c3 shows the final view with the real tholons creating a tetrahedron and the virtual tholons projected out from each side of the tetrahedron.

It looks like we have broken the rule that no children can exist outside their parent’s limits because we have a lot of virtual tholons sticking way outside those limits. Virtual tholons are reverse reflections of the parent tholons. In one sense, these virtual tholons do not exist any more than your reflection in the mirror puts you on the other side of the wall, but it does exist in that the reflection is a perfect mirror-image representation of you, and this has significant value as mirrors have a measurable effect on energy (light). It is an illusion, but an illusion that affects reality.

We started with a 2D map that was folded into a 3D form, and in the 3D space, no more folding can occur, but the 3D form can be folded into a 4D form, which can be folded into a 5D form, etc. We end up with a thologram that is a perfect tetrahedron made of an (approaching) infinite number of smaller tholons. On the 4 outer-most surfaces of this tetrahedron is a single layer of the tholonic matrix of real and virtual tholons, with the virtual tholons facing outwards into nothingness.

While the outer form of real tetrahedrons looks like a tetrahedron, it consists of 4 child tetrahedrons, each made of 4 child tetrahedrons, ad infinitum. This tetrahedral stacking leaves gaps. The shape of that gap is that of an octahedron, or a square bipyramid. This is mentioned because the tetrahedron and the octahedron can be converted into each other by means of the Golden mean (phi, φ, 1.618), and together they form a pair of shapes that can fill every part of space (tessellation). The only other shape that can do that is a cube, which is itself defined by 2 tetrahedrons. In other words, the only archetypal forms that can occupy all space are the forms that are created by the tetrahedron.

Besides being an interesting factoid, the tessellating abilities of these forms are surprisingly informative, at least regarding the thologram. For example, in the thologram, there are tholons (tetrahedrons) and octahedral gaps. These gaps represent nothing, or rather, nothingness, as there is nothing there. Perhaps that is why the octahedron is the least “dense” of all polyhedrons (“dense” is the geometry term that refers to the number of times a polygon boundary wraps around the center). The only difference between the void of nothingness and the nothingness of this gap is the difference between limitlessness and the limits imposed by the tholons that define the shape of this nothingness. As this limited nothingness is in the form of an octahedron, it is the octahedron that represents nothingness in the context of somethingness. The tessellating ability of the tetrahedron and the octahedron is a representation of the reality that everything that exists is composed of somethingness and nothingness. The cube, the only single shape that can tessellate, is the basis of both nothingness and somethingness, from the tholonic perspective.

A pair of tetrahedrons make a cube, and if we connect the centers of each face of that cube, we get an octahedron. So, the octahedron is the result of the cube, and the cube is the result of the tetrahedron. This defines the natural order of progression as tetrahedrons → cube → octahedron . In addition, a complete tholon (which is described in more detail later) is both a real tholon and its virtual counterpart, so from a tholonic perspective, the cube, which 2 tetrahedrons can create, is actually created by a single complete tholon.

The Shape of Reality

What does this have to do with reality and the Universe? The cube describes the structure of our three-dimensional material reality, but the tetrahedron describes the structure of concepts, which precede form. Understanding how concepts become form teaches us something about the energy behind reality: tholonic energy.

We have seen that a tessellated tetrahedron can be formed with 4 tetrahedrons and 1 octahedron, which is precisely the first generation of a tholon: the parent creating 4 children with an octahedral void between them. This exact configuration is the unit cell that tessellates all of three-dimensional space, making it the geometric foundation of both somethingness and nothingness. These tessellated structures illustrate that both are self-similar and fractal.

If we assign the length of a side to 1, a tetrahedron has a volume of 0.1178 (V^t1) and an octahedron has a volume of 0.4714 (V^o1). 4 tetrahedrons total 0.4714 (V^t4), which equals exactly the volume of 1 octahedron. This 4:1 ratio of instances creates a 1:1 ratio of volume. Tholonically speaking, the balance between somethingness and nothingness is exactly 1:1, a perfect equilibrium, but it requires 4 instances of somethingness to balance 1 instance of nothingness. Perhaps we can understand this by recognizing that 4 instances of somethingness define 1 instance of nothingness.

This pattern also appears in the two-dimensional thologram where 6 trigrams surround and define 1 central hexagonal space, creating a 6:1 ratio. This hexagonal structure is fundamental to physical reality: it defines the crystal structure of hydrogen, the molecular structure of carbon (including graphene and benzene), and appears throughout nature in beehives, snowflakes, and countless other forms. The 6:1 pattern means that one-sixth, or 16.666%, represents the unified central structure while the surrounding six represent the defining instances. This ratio might sound arbitrary until we compare it to cosmological observations:

“In January 2015, the results of the Planck satellite of the European Space Agency (ESA) were published. The high sensitivity of Planck allowed us to estimate the amount of matter and energy in the Universe with a precision of just a few percent. In particular, we now know that all forms of matter as we know them (atoms, radiation, and all sorts of exotic particles) only account for 16% of the Universe’s matter, the rest being in an unknown form that is usually referred to as dark matter.”⁷ Note: this a does not include dark matter.

If we consider these 6 trigrams as parent tetrahedrons, each creating 4 children, we have 24 child tetrahedrons (somethingness) with a total volume of 2.828. These same 6 parents also create 6 octahedral voids (nothingness) with a total volume of 2.828. When these equal volumes are multiplied (2.828 × 2.828), the result is 8, which is both 2³ (the cube of primal duality expressed in 3D) and the number of vertices in a cube and faces in an octahedron. This is a geometric confirmation that when somethingness and nothingness are in perfect balance, their product defines the fundamental structures of three-dimensional reality: the cube and the octahedron. These shapes are fundamental because they are geometric duals of each other (connecting the face centers of one produces the other), they define how approximately 90% of all elements organize as crystal structures, and the cube is the only single polyhedron that can tessellate all of space by itself.

We could call this a fascinating coincidence, or we might wonder if the shapes that can describe all matter and occupy all space might have a relationship to not only all matter and all space, but also the energy that allows these patterns and concepts to exist: tholonic energy.

These geometric relationships reveal how tholonic energy operates. It functions as an organizing principle that creates perfect balance between opposing or contrasting concepts, manifests identically from the quantum to the cosmological scale, and follows specific mathematical ratios such as the 6:1, 4:1, and 1:1 patterns we have observed. The fact that these same patterns appear in atomic structures (hydrogen, carbon), molecular formations (graphene, benzene), and universal composition (ordinary matter to dark matter) demonstrates that tholonic energy is the mechanism by which ideas, patterns, and geometric principles become physical reality. Geometry is not merely a description of reality but the primary expression of the organizational energy that creates it.

But there might be much more to this incredibly coincidental relationship. For example, we live in a Newtonian world where cause is followed by effect, mass and energy follow the rules, and nothing travels faster than the speed of light, thereby preventing “spooky actions at a distance”, to quote Einstein. Everything we think, believe, and build is based on these laws of reality, and millions upon millions of tests prove that reality is built on these laws.

Then quantum physics crashed the party. The Bell Inequality Theorem makes the quite reasonable claim that if the laws of reality can be broken, then reality, truth, and facts as we know them are not what we think they are. That was in 1964, and since 1972 scientists have been using reality, truth, and facts to prove that reality, truth, and facts may not be real, true, or factual.

Without getting into the details, an experiment was set up with conditions where the results of 2 tests could only be 0 or ±1 according to Newtonian and Relativistic laws of realism and locality. If the test results went past ±2, that would prove that reality as we know it is not reality as it actually is. The test results were in the range of ±2.828. This test is called the CSHS Witness Experiment⁸.

I personally ran this experiment using quantum spin on IBM’s quantum computer (image below), not that I doubted the claims of brilliant scientists. I just wanted to see it for myself. Discovering new realities is fun and interesting, but the real miracle of this test is that humanity has created the tools and infrastructure that allows anyone to access a quantum computer and reality-testing experiments from a flat with limited water, stolen electricity, and a collapsing ceiling, in a poor neighborhood in an economically and politically devastated South American country, which is where I happened to be when I ran this test.

Besides proof that reality is not what we think it is, it also shows that the difference between our Newtonian and quantum models of reality was between 2 and 2.828. What jumps out is how this ratio of 2.828/2 equals √2. This is interesting not just because the two different realities are based on the same pattern (2×√1 for Newtonian results, and 2×√2 for quantum results), but also because we see this same pattern in the volumes of 6 octahedra (= 2.828) and 6×4 tetrahedra (= 2.828).

Why does this matter? The CHSH test was specifically designed to distinguish between two fundamentally different views of reality. If the universe follows classical principles where physical properties exist independently of observation (realism) and nothing can influence distant objects faster than light (locality), then the test result must be 2.0 or less. This is not debatable; it is mathematical certainty. However, quantum mechanics predicts 2.828, and experiments consistently confirm this value. This means at least one of our fundamental assumptions about reality must be wrong. Either physical properties do not exist until they are measured, or entangled particles can instantly influence each other across any distance, or both assumptions are false. The 2.828 result is not an interpretation or opinion but mathematical proof that the universe operates in ways that violate our most basic intuitions about existence, causation, and separation.

How does this connect to tholonic geometry? The volume of 6 parent tetrahedrons (each creating 4 children) is 4√2, or approximately 5.657. The volume of their 24 child tetrahedrons is 2√2, or approximately 2.828. The volume of the 6 octahedral voids (nothingness) is also 2√2, or 2.828. This reveals a remarkable pattern: every measurement is a multiple of √2, the same constant that distinguishes quantum reality (2√2) from classical reality (2). The volumetric relationship creates a 2:1:1 ratio (parents : children : voids), demonstrating that while the 6 parents have twice the volume of their children, the children’s volume perfectly equals the voids’ volume, creating an exact balance between somethingness and nothingness. Furthermore, when comparing individual shapes of the same size, 1 octahedron has exactly the volume of 4 tetrahedrons, which is why 6 octahedrons equal 24 tetrahedrons. This 4:1 ratio explains why it takes 4 instances to create unity: it is the fundamental relationship between the void (octahedron) and form (tetrahedron).

Interestingly, the tetrahedron is the only 3D shape that has the same number of vertices and faces, with 4 of each, making a 1:1 ratio of faces to vertices. This is yet another example of its balance and stability via simplicity. In 2D space, the trigram has a vertex:face ratio of 3:1, suggesting that the progression from 2D to 3D achieves greater symmetry and balance.

These geometric patterns are not merely mathematical curiosities but fundamental expressions of how reality itself is structured. Where do we see all these patterns come together? In the tholon, where a tetrahedron is made of tetrahedrons.

We also see the octahedral/tetrahedral patterns of the tholon in other ways. For example, in the two-dimensional pattern shown below, octahedral sections (green) display 8 triangular elements while tetrahedral sections display 4 triangular elements, maintaining the 2:1 relationship.

Now, let us compare the relationship between tetrahedra and octahedra patterns.

a) is the simplest possible pattern from which tetrahedrons and octahedrons can be made.

In b) and c) we see the pattern that creates 6 octahedrons. These are the only 2 unique patterns that can create an octahedron, as any other pattern is just a reversed and/or flipped version of these patterns. Outside of the context of an octahedron, we can see that b) and c) are the same pattern but shifted by exactly 2/7 of the pattern width. The pattern naturally divides into 7 sections because the 6 trigrams that define the width create 8 vertical divisors, and 8 divisors create 7 divisions. We refer to these patterns as left-hand and right-hand octahedrons. Curiously, we also see this natural division in the periodic table of elements, which classifies all elements into 7 periods.

In d) we see we can create 12 tetrahedrons from this same pattern. This pattern is perfectly symmetrical.

In e) we combine all three patterns and their colors (additively).

We see how 12 tetrahedrons can become 6 right-hand octahedrons or 6 left-hand octahedrons. This creates a triad of a tetrahedron with a pair of opposing octahedrons. To put it tholonically, somethingness creates 2 types of nothingness. We can tie this back to the earlier subject of chaos: nothingness is chaos, and as there are 2 types of chaos (low-entropy and high-entropy), we can say that the existence of something creates either low-entropy chaos or high-entropy chaos. This is compatible with the archetypes just shown as well as reality itself.

In the same way, a cube can be defined by 2 tetrahedrons. A cube has 8 vertices, and the 8 vertices of 2 interpenetrating tetrahedrons perfectly match those of the cube. However, this vertices relationship, which is very efficient and tidy in 3D, becomes complex and less obvious when represented in 2D.

There are multiple ways to observe the duality of the tetrahedron and the cube, and depending on which dimensional space you are working in, these relationships can be either readily apparent or remarkably obscure. When we project a cube onto a 2D plane as unfolded faces, we can identify the vertices that define two interpenetrating tetrahedrons, one represented by red lines, one by blue lines, with green lines marking their overlap. In this 2D representation, identifying the tetrahedrons is difficult precisely because we are attempting to describe a 3D object in 2D space. However, this 2D pattern reveals a remarkably consistent structure found throughout nature, genetics, and mathematics: perfect bilateral symmetry through inversion. This may be obvious to the reader, but it’s worth a visual demonstration. When we split this 2D pattern in half, flip one half both horizontally and vertically, and reverse its colors, we obtain the exact mirror image that combines with the original to create the complete pattern of two interpenetrating tetrahedrons. This is not merely a visual trick but reveals a fundamental property of the system. Each half, when reconstructed in 3D, creates one complete tetrahedron but only half of the cube’s structure, demonstrating that the tetrahedron is the more fundamental component. The persistent duality manifests in the numbering as well: one half contains vertices 0-6 (missing 7), while its inverse contains vertices 1-7 (missing 0). The missing values, 0 and 7, represent the extremes of the system, the boundaries of nothingness and completion, suggesting that each perspective of the whole necessarily excludes one pole of the complete spectrum. This pattern of complementary halves that create a unified whole through inversion and color reversal mirrors the fundamental tholonic principle: two opposing states that through their interaction define a third, more complete reality.

There is another less obvious but more profound relationship revealed by this pattern. As we have just seen, we can create one half of a cube (or one tetrahedron) by using only half of our 2D map. When we flip that half and join the two halves, we have two tetrahedrons which form one complete cube. We now have a pattern which can be represented mathematically as shown in the grids in the image below. The pattern shows us that there are two additional patterns or values that can be surmised or implied. In this case, our lower left-hand corner of the 2D pattern equates to the 10 part of the binary grid (2 in decimal), and the upper right-hand corner equates to 01 (1 in decimal). We can see that 10 and 01 are flipped in the same manner that our two half segments are flipped, but now with this grid we can see that we can also create 00 and 11, which are not explicitly shown in the pattern for a complete cube but are implied. In order to get the full picture, we need to recognize these other hidden yet implicit quadrants. In the two grids, the upper grid shows the binary values and the lower grid shows the decimal values. The lower grid gives us the sequence 0, 1, 2, and 3, but only the numbers 1 and 2 were explicitly present in the original pattern. The values 0 and 3 are the implicit boundaries, the missing states that complete the system. This reveals that the visible duality (1 and 2) necessarily implies an invisible duality (0 and 3) that frames and defines the entire structure. The pattern demonstrates that any manifest duality contains within it the seeds of its own completion, requiring only the recognition of what is implicitly absent to reveal the full quaternary structure. As previously mentioned, a complete tholon can be represented by two tetrahedrons, one real and one virtual, and this equates to a single cube. But now this pattern reveals an additional layer: there is a real cube created by the two explicit patterns that are visible (1 and 2), and a virtual cube created by the implicit patterns (0 and 3), a hidden, invisible cube that must exist in order to complete the pattern of 0, 1, 2, 3. This recursive duality, where two tetrahedrons create one cube and two cubes (one real, one virtual) complete the quaternary system, demonstrates the fractal nature of tholonic structure across multiple scales of organization.

Returning to the three-dimensional relationships, we can see that a tetrahedron has a fundamental, more primitive relationship with an octahedron than with a cube. However, the octahedron has a closer relationship with the cube in 3D space. This is because the cube and octahedron are geometric duals: if you connect the center points of a cube’s 6 faces, you create an octahedron, and vice versa. Their face-to-vertex ratios are inverted (octahedron has 8 faces and 6 vertices, cube has 6 faces and 8 vertices), making them two expressions of the same underlying structure.

This duality extends to their fundamental connectivity patterns. In the cube, each vertex connects to its three nearest neighbors through the shortest possible edges, creating a local, nearest-neighbor network that distributes forces through adjacent points in a compressive structure. In the octahedron, each vertex connects across the maximum distance to its opposing vertex at 90° angles, creating a long-range tensile network that establishes spatial boundaries through maximal separation. This distinction reinforces their tholonic roles: the cube manifests Contribution through short-range material bonding and local interactions, building up form through nearest-neighbor relationships. The octahedron embodies Definition through long-range constraints and opposing forces, establishing the spatial framework within which manifestation can occur. The cube creates through proximity, the octahedron defines through distance. These connectivity patterns are not merely geometric curiosities but the fundamental mechanism by which entropy manifests: the octahedron’s long-range constraints create rigidity and order (low entropy), while the cube’s local nearest-neighbor bonds allow flexibility and distributed energy states (high entropy). In this way, the cube and octahedron serve as contextual representations of the fundamental concepts of somethingness and nothingness, not as absolute states but as relative expressions that shift meaning depending on the scope and scale of observation.

We end up with a 6-step progression from dot → line → trigram → tetrahedron → octahedron → cube. This sequence follows the N-D-C pattern perfectly. When the tetrahedron (Negotiation) replicates to create children, the octahedron (Definition) emerges as the void space between those children, representing nothingness. Only after the octahedron has defined the spatial boundaries can two parent tetrahedrons combine to create a cube (Contribution), representing somethingness.

This sequential order is not arbitrary but reflects a fundamental principle: the octahedron must precede the cube because Definition must precede Contribution. The octahedron requires only 3 perpendicular axes to establish the complete dimensional framework, while the cube requires 12 edges to materially manifest within that framework. This efficiency difference reveals a profound distinction between conceptual and material reality. In the conceptual realm where space-time has no material instance, the octahedron is maximally efficient, crystallizing from infinite possibility into defined constraints through a process where entropy moves from high to low (chaos → order). In the material realm where space-time requires physical instantiation, the cube is maximally efficient through nearest-neighbor bonding, but entropy moves in the opposite direction from low to high (order → dispersion). The octahedron serves as the bridge where conceptual entropy (decreasing, ordering) transitions into material entropy (increasing, dispersing). This bidirectional entropy flow explains why Definition must establish boundaries before Contribution can manifest form.

It is important to remember that the tholonic model is primarily focused on this conceptual plane of existence, the realm where patterns, archetypes, and relationships define reality before they manifest materially.

The octahedron and cube are therefore complementary but sequential expressions that emerge from the tetrahedron through different processes: first through subdivision (creating voids and constraints), then through combination (creating form and manifestation). These two processes are mathematically equivalent to division and addition, which in logarithmic space corresponds to division and multiplication, the two fundamental operations that define all mathematical relationships. For this reason, the cube, which incorporates all previous components, defines the crystal structure of 90% of the elements, which are cubic-based. There are only 3 hexagonal elements. If our reasoning above was correct, we would expect these 3 elements to be more fundamental than the cubic elements, and they are. These elements are hydrogen, carbon, and nitrogen, 3 of the 4 most common elements on this planet (oxygen, the simplest of all cubic structures, being the fourth), which are the required elements to create life as proteins. Of course, hydrogen is the most common element in the Universe and the source of all other elements.

The tetrahedron, octahedron, and cube form a triad of complementary structures that represent the next level of the N-D-C pattern, demonstrating how the tholonic structure continually builds upon itself, creating new scopes and contexts that emerge from the N-D-C process in the parent context.

How do these geometric forms map to tholonic properties? The tetrahedron represents Negotiation because it is the most fundamental balanced structure in 3D space, defining the potential and vertices from which all other forms emerge. It is the point of balance and stability that makes further development possible. The octahedron represents Definition because it is the void, the nothingness, the low-entropy space that defines the boundaries and limits of the space wherein things can come into existence. It establishes the parameters and constraints that determine what form can manifest by defining what is not form. The cube represents Contribution because it is somethingness, the high-entropy manifestation of maximum material presence. The cube is the only Platonic solid that can tessellate all of 3D space by itself, making it the ultimate expression of form contributing its structure to fill every possible location in reality.

These geometric relationships extend beyond pure form into fundamental principles of energy and organization. As we saw earlier with N-D-C patterns in electrical systems, the tetrahedron’s balanced structure represents the stable potential that enables energy flow, analogous to voltage (electromotive potential). The octahedron’s void space represents the resistive boundaries that define and constrain that flow. The cube’s space-filling nature represents the actual current or contribution of energy manifesting in material form.

Previously, we referred to the tetrahedron/octahedron/cube as symbolic of somethingness/nothingness/form, and while that is true in a symbolic sense, it is obviously not true in the real sense, as the octahedral is an actual shape that is ubiquitous in nature, such as in fluorite, diamonds, and other minerals. Likewise, there is no such thing as a total nothingness or total somethingness. However, we do have the concept of that which is receptive, dark, cold, dead, infinite (nothingness) and that which is creative, light, hot, energetic, and finite (somethingness). These concepts have been around since we grokked the difference between dead and alive, day and night, over 2.5 million years ago. The modern world has its own versions in concepts such as physics’ protons/electrons, technology’s 0/1, finance’s debt/credit, Descartes’ mind/body, science’s reductionism/holism, hierarchy/unity, concrete/abstract, evolution/devolution, discrete/continuum, time/space, etc.

Such modern dualities are as neatly encapsulated in the ancient Taoist concept of yin and yang as were the dualities of our distant ancestors. When speaking about the material world where somethingness and nothingness have no real instances, it is more practical to speak in terms of yin and yang. Tholonically, we distinguish between thermodynamic states and functional processes: the octahedron, though a low-entropy state (organized void space), serves a yin function as the receptive container that allows and defines where energy can distribute toward manifestation and increasing entropy. The cube, though a high-entropy state (complex material form), is the result of the yang function, the creative organizing force that structures low-entropy chaos into ordered form. In this sense, yin represents the movement from order to high-entropy through receptive distribution, and yang represents the movement from low-entropy chaos to order through creative structuring. We can also say that any pattern or structure that allows for the increased distribution of energy is yin, and the actual distribution of energy is yang.

We can see this duality in chemistry when we look at the periodic table, as all the alkali metals (H, Li, Na, K, Rb, Cs, Fr) in the left-most column are body-centered cubic structures and are very interactive with other elements. These would be considered yang in nature, representing the contributive and integrative function as they actively combine with other elements to form compounds. All the right-most column elements are the “noble” gasses (He, Ne, Ar, Kr, Xe, Rn), so named because they stand alone and do not interact with anything unless they have to. These are all face-centered cubic structures and are yin in nature, representing the defining and limiting function as they maintain boundaries and resist integration. Now we can see the entire periodic chart as moving from yang and Contribution (left) to yin and Definition (right).

Further confirming the order of these 3 archetypal shapes is how the first and second shapes, the tetrahedron and octahedron, tessellate as a pair (Negotiation and Definition working together), while the third shape, the cube, represents their Contribution and tessellates all by itself. These 2 tessellating forms are the only tessellating shapes of all 75 uniform polyhedra⁹.

We might conclude from all this that it is the cube that best describes the form of reality, which would be a pretty safe claim considering as we live in a 3D reality. However, we might also then conclude that it is the tetrahedron that describes ideas, archetypes and concepts, and the octahedron that describes the limitations of context. This view aligns with Buckminster Fuller’s work in Synergetics, where he identified the tetrahedron as the minimum structural system in the universe, the most fundamental building block from which all other structures derive¹⁰. Fuller saw the tetrahedron not merely as a physical shape but as a conceptual prototype representing maximum structural efficiency with minimum components, making it the geometric embodiment of pure structural concept before physical manifestation. This perspective echoes ancient philosophical traditions: Plato’s Theory of Forms in Timaeus associated the tetrahedron with the element of fire and explored how perfect geometric forms exist as eternal archetypes before their physical manifestations¹¹. Similarly, Aristotle’s doctrine of hylomorphism, which states that Form + Matter = Substance, finds its most fundamental expression in the tetrahedron as the simplest form that can actualize matter into three-dimensional substance¹².

For practical purposes, one perspective we can better understand is to look at the real tholons and ignore the virtual tholons. While virtual tholons are just as significant, they are also far more complex, and, for the most part, we will avoid delving into them. As we demonstrated earlier with the cube, where explicit patterns (1 and 2) necessarily imply implicit patterns (0 and 3), when we look at a real tholon, there is always a virtual tholon implicitly existing alongside it. This being the case, it is more effective to deal with the structures of real tholons, which are identical to a 3D Sierpinski fractal. Below are 5 different views of a 6-level (iterations) thologram. The most basic pattern is that of a tetrahedron composed of infinite iterations across infinite generations, within which infinite patterns emerge. The Sierpinski pattern is the most fundamental of all patterns, as it is composed of only the most basic of all forms. This makes it a very stable pattern that will define how energy moves, and therefore a pattern we expect to see many instances of in reality.

Existential Foam

There exists another model whose structure provides a useful conceptual analogy to the thologram. It is called quantum foam, a concept in theoretical physics that describes how space and time break down into chaos at the quantum level, but from this breakdown, higher orders emerge. While quantum foam operates at the Planck scale, the thologram applies this same principle of order emerging from chaos to all of existence at all scales. Below is an image from the essay “Order emerges out of Chaos: the fundamental insight of science”¹³ that demonstrates the fractal nature of quantum foam and its conceptual parallels with the tholographic model of chaos giving rise to structured form.

*Note: More in “What π, 2, 5, and e have in common” in Appendix C, “Tholonic Math”.0D