Part I

1: CHAOS

Everything begins and ends in chaos.

Synopsis: Chaos is the first principle of existence, the fertile void from which all things arise and to which all things eventually return. This chapter explores chaos as a spectrum of predictability extending from absolute nothingness (zero) to infinite somethingness (infinity). Both extremes represent perfect unpredictability. Between these poles exist various degrees of what we perceive as order—states where patterns become discernible and outcomes grow more predictable. The chapter examines how zero and infinity operate as two faces of the same unknowable state, how nothingness is defined across physics, philosophy, and cosmology, and how chaos generates both the patterned complexity of nature and the unpredictable behavior of dynamic systems. Through mathematical paradox, cosmological theory, ancient myth, and modern chaos science, it establishes that order is not chaos’s opposite but rather a manifestation of chaos at scales where predictability increases. Chaos is revealed as both the emptiness that precedes form and the overflowing fullness that dissolves it, the beginning and the end of every cycle within the Tholonic worldview.

Keywords: chaos, randomness, pattern, duality, creation, zero, infinity

What do we mean when we say everything begins and ends in chaos?

The word chaos carries a peculiar burden. It must describe both the absolute void that preceded creation and the turbulent complexity of systems spiraling toward unpredictability. This is not linguistic confusion but ontological necessity. Chaos occupies both ends of existence, the zero and the infinity, the empty and the overflowing.

Consider the Oxford English Dictionary’s three definitions:

• The formless matter supposed to have existed before the creation of the Universe.
• The formless and disordered state of matter before the creation of the cosmos.
• A state of extreme confusion and disorder.

The first two definitions circle the same paradox. Formless matter existed before creation. How can matter exist before creation? The contradiction reveals our linguistic limitations when confronting the pre-existent state. Ancient definitions described chaos as matter existing before energy, but modern physics has dissolved that distinction. Matter is energy in another form. The old boundary vanished under scrutiny.

The Ancient Greeks understood chaos more fundamentally. In their cosmogony, χάος (Khaos) denoted the void state preceding the Universe, personified as the primordial deity from whom emerged Gaia, Tartarus, Erebus, and Nyx. Chaos was not disorder but the pregnant absence from which the first forms crystallized.

Physics offers a different perspective. In complexity science, chaos describes “the property of a complex system whose behavior is so unpredictable as to appear random, owing to great sensitivity to small changes in conditions.” This is the famous Butterfly Effect, where minuscule perturbations cascade into vast consequences. Yet even here, agreement fractures. Science writer James Gleick, in his foundational work on chaos theory, notes that the scientists pioneering the field could not settle on a unified definition.¹ Philip Holmes characterized chaos as “the complicated aperiodic attracting orbits of certain, usually low-dimensional dynamical systems,” while Chinese theoretical physicist Bai-Lin Hao described it more poetically as “a kind of order without periodicity.”

Order without periodicity. This phrase captures something essential. Chaos contains structure, but structure that refuses to repeat, that evades prediction while remaining deterministic.

Cosmology contributes its own vocabulary, namely the primal void. Modern Big Bang theory, for all its mathematical precision, begins with this concept. Before the Universe erupted into being, there existed only void, an absence so complete it precludes even the existence of time itself. Stephen Hawking argued that time could not exist in absolute nothingness, for time requires change, and change requires something to change.²

This void has accumulated many names across cultures and disciplines. These include the gap between heaven and Earth, the abyss, ex nihilo, and primordial waters. Each name attempts to gesture toward the ungraspable, an immeasurable nothingness that may exceed the mind’s capacity to comprehend. How does consciousness, which arose from somethingness, grasp the absolute absence of all things?

Despite vast differences in worldview and methodology, scientists and mystics converge on a singular narrative. Before there was something, there was nothing. Then, through mechanisms still hotly debated, somethingness appeared. Perhaps it was a singularity spontaneously erupting into expansion. Perhaps energy achieved sufficient intensity to warp spacetime into mass. Perhaps reality is a multidimensional projection from a source beyond our perceptual access. The specifics remain contested, but the pattern holds. The transition from nothing to something marks the fundamental mystery at existence’s root.

Nothingness

If nothingness preceded everything, we must ask uncomfortable questions. How large was this nothingness? Does nothingness have limits, or does it extend infinitely? If it has limits, what defines those boundaries? If it has no limits, does that make nothingness itself infinite? These questions may sound like philosophical games, but they represent active research debates in science, philosophy, and mathematics.³

The concept of nothing proves surprisingly complex. There exist multiple types of nothingness, each with distinct characteristics. Similarly, there exist many types of infinity, possibly infinitely many.

Physicist Sabine Hossenfelder identifies up to nine distinct levels of nothing.⁴ These levels organize into three general categories, which further collapse into two fundamental concepts: relative nothingness and absolute nothingness. These two categories provide the framework we use throughout this book.

Consider the statement “I have nothing in my bank account.” What level of nothingness does this represent? A balance of zero places your empty account at Level 6 in Hossenfelder’s taxonomy. Because money measures the non-physical entity of credit, it also occupies Level 7. Although financial destitution may feel like absolute nothingness, money functions as a value derived from material things. It represents energy transfer between entities such as people and businesses. We might even conceptualize it as a “field.” This makes one’s poverty a form of relative nothingness rather than absolute nothingness.

Relative nothingness depends on context. Absolute nothingness transcends all context because nothing exists to provide contextual reference. This creates a profound limitation in language and thought. We cannot meaningfully discuss absolute nothingness because any act of description imports concepts foreign to it.

Even the phrase absolute nothingness imposes artificial constraints. The word absolute means “to the largest degree possible,” which presupposes concepts of size, measure, and time. None of these can exist in true nothingness. Adding intensifiers like super-duper-ultra-unlimited merely compounds the error. Perhaps the only honest statement about true nothingness is that nothing can be said about it.

This linguistic impossibility resonates with Lao Tzu’s description of the Tao:

The Tao that can be described is not the eternal Tao.

The name that can be spoken is not the eternal Name. The nameless is the boundary of Heaven and Earth.

The named is the mother of creation.

We might translate this into the language of nothingness:

The nothingness that can be described is not the eternal Nothingness.

The nothing that can be spoken of is not the eternal Nothing. The eternal Nothingness is the boundary of all that can ever be and all that is.

Nothing is the mother of creation.

If two fundamental types of nothingness exist, those being absolute and relative, does this imply two fundamental types of duality? Yes. Each of Hossenfelder’s nine levels exists within some form of duality (with the possible exception of Level 9). However, the two types of nothingness generate two corresponding types of duality.

The first is primal duality, which describes the duality of the first thing that existed within primal nothingness. The second is relative duality, which encompasses every duality that exists because of, or within, the primal duality.

We symbolize primal duality as 1 and 0. From these two symbols, all other numbers emerge. One might object that the primal duality should be represented as infinity and zero rather than one and zero. This objection has merit. Infinity represents the totality or unity of all numbers yet is not itself a number. The number 1, by contrast, does represent unity and is even defined by that term. Within the context of unity, we can write 1 = ∞. The unity of 1 represents the unity of all. This equivalence explains why mathematics and computer programming commonly map the range 0→1 onto 0→∞.

All dualities following the primal duality exist within the realm of relative nothingness. We name these relative dualities according to their manifestations: positive and negative, yin and yang, or any difference that distinguishes one somethingness from another. Measurement itself emerges as a byproduct of relative nothingness. This relative duality structures our reality. It is the mundane duality that permits things to exist, allows energy to move, and creates states of balance and imbalance.

Here we must clarify what balance means. In this context, balance refers to equilibrium within movement, not the balance of stillness. Dancing, electricity, natural systems, cosmic dynamics, etc. exemplify balance within movement.

Just as nothing must precede something, zero precedes one. Everything begins with zero. Because we will use the concept of zero and nothingness extensively throughout this book, we should examine this supposedly “simple” concept more carefully.

Zero

We represent the general concept of nothingness with the number 0, yet this seemingly simple symbol emerged surprisingly late in human intellectual history. Even after its appearance, acceptance took thousands of years and provoked intense debate.

The historical record shows 0 first appearing in Mesopotamia around the 3rd century B.C. However, the ancient Indian Bakshali manuscript, whose dating remains contested but may extend back several centuries BC based on some scholarly estimates, appears to use a dot symbol (•) to represent zero. From these origins, zero spread gradually across civilizations. It appeared in Mayan Meso-America in the 4th century A.D., then again in the same region during the 5th century. Cambodia adopted it in the 7th century. China and Islamic cultures integrated it into their mathematical systems by the 8th century. Western Europe, remarkably, did not encounter the concept until the 12th century. Even then, European scholars resisted accepting zero as a legitimate number for hundreds of years.

This resistance becomes particularly striking when we consider that honeybees,⁵ various monkey species,⁶⁷⁸ and crows⁹ all demonstrate clear understanding of zero in controlled experiments. Human exceptionalism takes a blow when corvids grasp what took our species millennia to formalize.

Animal physiologist Andreas Nieder identified four developmental stages in humanity’s adoption of zero,¹⁰ stages that mirror the psychological progression through grief described by Elisabeth Kübler-Ross. This parallel seems apt. On an existential level, confronting nothingness resembles confronting death.

Stage 1: Recognition of absence. The awareness that something expected is missing.

Stage 2: Conceptual distinction between nothing and something (5th century BC, Greece).

Stage 3: Understanding that zero precedes one in the numerical order (7th century AD, India).

Stage 4: Development of mathematical rules and properties for zero as a symbolic entity (13th century AD, North Africa).

Note: These dates represent best-guess estimates based on documented references.

The puzzle deepens when we consider the timeline. Humans had achieved remarkable cognitive sophistication long before formalizing zero. Anatomically modern humans emerged roughly 300,000 years ago. Complex language likely developed 50,000 to 100,000 years ago. Yet the mathematical concept of nothing arrived only in the last few thousand years. The discontinuity suggests something unusual about zero itself or about the conceptual leap it requires. Some researchers have proposed external catalysts for human cognitive leaps, whether genetic mutations, alien intervention,¹¹ or psychoactive compounds.¹² While these remain fringe theories, they highlight the genuine puzzle presented by the timeline.

Zero’s arrival coincided with the emergence of the modern world. This timing is not coincidental. Zero enabled modern mathematics and calculus. Without zero, the scientific revolution could not have occurred. As zero’s practical utility became undeniable, it began to erode philosophical objections grounded in the ancient principle ex nihilo nihil fit (from nothing, nothing comes).

This created considerable discomfort for medieval Christian authorities. The concept arrived in Europe through Islamic mathematicians, which alone created ideological resistance. More troubling still, if something could come from nothing mathematically, what did this imply about divine creation? The theological implications of zero threatened established doctrine.¹³

The Complexity of Nothing

Zero presents mathematical puzzles that challenge intuition. Consider the expression 0⁰. Many calculators, including Google’s, return the answer 1. This seems paradoxical. How can nothing raised to no power equal something? The more cautious mathematical position holds that 0⁰ is undefined. Yet the general rule X⁰ = 1 holds true for all X where X ≠ 0.

The logic unfolds straightforwardly. X⁰ = X^1-1 = X¹/X¹ = X/X = 1. This proof reveals something profound about zero. Beyond representing nothing, zero also represents a state of nothing that results from something. The equation 1 - 1 = 0 demonstrates this. Conceptually, we might express it as something minus something equals nothing. Zero functions as both the womb from which all numbers emerge and the destination where numbers vanish when they cancel from context.

Lao Tzu’s Tao Te Ching may address this dual nature when it states the following.

Yet mystery and reality emerge from the same source. This source is called darkness. Darkness born from darkness. The beginning of all understanding.

We might reinterpret this passage through the lens of nothingness.

Yet the unknown and the known emerge from the same source. This source is called nothingness. Somethingness born from nothingness. The beginning of all understanding.

The number 0 stands as the antithesis of all other numbers. Everything that ever existed, exists, or will exist does so because of a paradox. The “existence of nothingness” contains within it the seeds of all being. Mathematicians and philosophers have written countless conjectures and proofs about zero’s relationship to other values. Some argue 0 equals 1, or the sum of all numbers, or infinity (∞), or 1/∞, or many other values, or even all values simultaneously. These are not mere academic games. These concepts generate genuine debate in mathematics and philosophy.

The statement 0 = ∞ initially appears meaningless. Zero is a number while infinity is a concept, so how can they be equal? The resolution lies in recognizing that 0 functions as both concept and number. It possesses a quantitative value of 0 and a qualitative value of nothingness. Similarly, the number 1 has a quantitative value of 1 and a qualitative value of unity. All numbers function as concepts to some degree, and most common numbers carry well-known qualitative associations. But the primal pair of nothingness and unity initiated everything else.

Historical mathematical giants recognized this relationship. Parmenides, Leibniz, and others equated unity with ∞. Their reasoning followed this pattern. All that is collectively defines the unity of existence. Therefore, 1 = ∞. From this equivalence flow other concepts. If 1/∞ = 1, then 1 - ∞ = 0, therefore ∞ - 1 = 0, therefore 0 + 1 = ∞, and so forth. These concepts may fail when applied quantitatively to balance a budget, but they represent fundamental qualitative ideas examined by great minds from Heraclitus to Hegel.

A truth emerges from this analysis, though perhaps not the truth. The numbers 0 and 1 possess remarkable flexibility. They encapsulate the idea of absolute nothingness and the totality of somethingness. These two poles define the arena or spectrum wherein all things exist. What happens inside this arena? In a word, order.

Order cannot exist in absolute nothingness because nothing exists to be ordered. Order cannot exist in the totality of somethingness because that totality lacks form, structure, or sequence. If 0 represents the mathematical concept of nothingness, then 1 functions as the ultimate mathematical attractor, the point where all distinctions collapse. In this sense, 1 does represent ∞. The pair 0 and 1 (or equivalently, 0 and ∞) constitute two states of chaos. Order exists exclusively between these two states. Our understanding, discovery, and invention of order allows us to recognize this fundamental structure.

Key 1: Chaos is a state lacking any order, time, or energy. Total nothingness. Zero.

Key 2: Chaos is a state of total energy and matter. Total somethingness. Infinity.

Order and the Chaos of 0 and ∞

Newton’s work with infinitesimals and limiting processes, which involve ratios approaching forms like 0/0, contributed significantly to his development of calculus. We learn in school that equations involving 0 or ∞ require careful treatment. Calculus recognizes several distinct indeterminate forms, including 0/0, ∞/∞, 0×∞, ∞-∞, 0⁰, 1^∞, and ∞^∞. Many of these can be transformed through algebraic manipulation into forms amenable to techniques like L’Hôpital’s rule. The expression 0/0 holds particular significance because it represents pure indeterminacy, capable of yielding different values depending on how the zeros are approached.

Current research continues to grapple with this fundamental question. Ilija Barukčić, Chief Editor of the publication Causation, authored a paper titled “Zero Divided by Zero Equals One.”¹⁴ The paper opens with careful language.

Objective: Accumulating evidence indicates that zero divided by zero equals 1

It concludes with equal caution.

Conclusion: The findings of this study suggest that zero divided by zero equals one.

Another paper co-authored by Barukčić, published in the Journal of Applied Mathematics and Physics, states the matter more plainly.¹⁵

A solution of the philosophically, logically, mathematically and physically far reaching problem of the division of zero by zero (0/0) is still not in sight.

The ambiguity extends further. Not only can 0 equal ∞, it can equal any number. This is mathematically valid. When an equation produces multiple possible values, mathematicians call it indeterminate. The expression 0/0 has no single or fixed determinable value, despite our intuition or claims suggesting it should equal 1. The same indeterminacy applies to ∞/∞, ∞/0, 0×∞, 0⁰, ∞⁰, 1^∞, and ∞-∞. All are indeterminate.

Indeterminacy functions as the mathematical equivalent of chaos. This supports our claim that 0 and ∞ serve as qualitative representations of chaos. They stand apart as the only mathematical symbols representing concepts without determinate value. Zero explicitly denotes the absence of value. Infinity denotes the opposite of zero yet manifests in many different forms. Throughout this book, we use ∞ as a frame for all positive numbers, expressed as 0 < positive_numbers < ∞.

The mathematical argument for 0 and ∞ representing chaos states finds support in empirical definitions of chaos itself. We can characterize nothingness as chaotic because it possesses no order or periodicity. Yet chaos carries another definition that means precisely the opposite, echoing the ∞ = 0 equivalence.

Consider Northwestern University physicist Adilson Motter’s study of chaos in the Universe’s origins. Motter concluded that 10^-36 seconds after the Big Bang, the Universe existed in a state of total chaos.¹⁶ What conditions prevailed at 10^-36 seconds after the Big Bang? Everything. All matter and energy compressed into an infinitesimal volume at temperatures exceeding one trillion degrees. Motter’s conclusion captures the paradox.

“Now we establish once and for all that [the universe] is chaotic.”

Setting aside questions about the Big Bang theory’s correctness, we face a conceptual puzzle. How can chaos describe both the void of total nothingness and the state of all matter and energy in the Universe existing as total somethingness?

The definition that resolves this paradox frames chaos as “the degree to which order is present in any state.” Both extreme states, total nothingness and total somethingness, lack order, pattern, or periodicity. This aligns with another common definition of chaos as synonymous with unpredictability. The authors of “Introduction to Complex Systems, Sustainability, and Innovations” offer a more dynamic definition.¹⁷

chaos explores the transitions between order and disorder. An order arises from the ever growing disorder of the Universe, chaos and order together.

This definition remains somewhat imprecise for our purposes. More useful is conceptualizing chaos as a spectrum extending from 0 to ∞. More precisely, the spectrum of chaos operates as the inverse of the spectrum of order (which we define more thoroughly later). Just as darkness is not a thing but rather the absence of light, chaos represents the absence of order. We begin with this working definition, understanding that we will refine it as the conceptual framework develops.

Key 3: Chaos as a measure of pattern, order, and predictability.

Chaos Versus Randomness

Chaos differs fundamentally from randomness. A random event is non-deterministic. It cannot be predicted because it has no pattern, rule, or reason we can discover, and no known immediate cause. This defines how we use the term throughout this book. The concept of randomness may simply serve as a label for unexplainable events that are actually chaotic but exceed our current capacity to understand them.

We should note that our use of the term “chaos” differs from classical chaos theory, which applies strict mathematical criteria to dynamical systems. In the tholonic framework, chaos describes any phenomenon that is deterministic yet unpredictable, creating a spectrum from absolute order to absolute disorder.

Consider RANDOM.ORG, a website that generates random numbers. To increase unpredictability, the service uses atmospheric noise as input. At macroscopic scales, atmospheric noise appears chaotic rather than random. It follows deterministic physical laws yet remains unpredictable due to sensitivity to initial conditions. Similarly, computer random number generators use electrical noise and heat as inputs. These sources appear chaotic within our framework because they are governed by deterministic processes yet produce unpredictable outputs.

The digits of π (pi) illustrate this distinction between determinism and unpredictability. The value of π extends infinitely, and the sequence of digits is entirely deterministic, calculable through simple mathematical algorithms. Yet no pattern emerges in the digits themselves. We cannot determine the next digit based on previous digits. This resembles prime numbers, which can only be generated through calculation, never predicted through pattern. Within the tholonic framework, π’s digit sequence appears chaotic because it is deterministic yet produces results that are unpredictable and appear statistically random (what mathematicians call normal distribution). We say “appears” because proving that π is truly normal remains impossible, just as proving that its infinite sequence contains every possible combination of numbers remains impossible. However, empirical evidence strongly suggests this is the case.¹⁸

Chaos contains pattern but remains unpredictable. Randomness contains no pattern and therefore contains every pattern. Here we encounter another instance of the 0 = ∞ concept, where infinity exists within zero.

There’s a beauty to Pi that keeps us looking at it… the digits of Pi are extremely random. They have no pattern, and in mathematics that’s really the same as saying they have every pattern. ~Jonathan Borwein, mathematician

Multiple definitions of “random” exist depending on whom you ask. For this book, we adopt the following definition.

Key 4: A random event is a spontaneous event with no apparent cause.

This definition leaves considerable room for interpretation. Many cases exist where we cannot perceive the cause of an event. What appears random may relate to the butterfly effect, which describes how small changes can generate growing and cascading consequences. The famous question “Does the flap of a butterfly’s wings in Brazil set off a tornado in Texas?” exemplified this idea. This question served as the title for a 1972 talk¹⁹ by Edward Norton Lorenz, mathematician, meteorologist, and founder of chaos theory.

A historical example demonstrates the butterfly effect’s power in human affairs. A chance meeting at a ball in Prague in 1896 set in motion a chain of events that contributed to both World War I and World War II.

Archduke Franz Ferdinand and Sophie Chotek, a duchess and daughter of a Bohemian Count, met at this ball, fell in love, and married. The problem arose from Sophie’s status. As a duchess rather than royalty, Habsburg protocol forbade her from appearing beside the Archduke in official royal ceremonies. Franz Ferdinand loved Sophie but obeyed the rules of his position. However, protocol permitted her presence at non-royal military ceremonies when he acted as Inspector-General of the Austro-Hungarian Army. Seizing this loophole, Franz Ferdinand organized a public inspection of the Bosnian army with Sophie by his side. To ensure visibility, they traveled in an open car. During this public demonstration, Serbian nationalist Gavrilo Princip approached the vehicle and shot both of them at point-blank range, killing them instantly.

Austria demanded an apology from Serbia. Serbia denounced the assassinations but refused to apologize, maintaining they had no involvement in the plot. Austria declared war on Serbia, which triggered treaty obligations. Russia allied with Serbia. Germany, bound by treaty to Austria, declared war on Russia. France and Great Britain came to Russia’s aid. World War I devastated Germany, creating conditions that enabled the rise of nationalism and Hitler, culminating in World War II.

The death and destruction of these wars stemmed from a chance meeting at a ball, combined with countless other butterfly effects. Gavrilo Princip stopped to buy a sandwich, placing him at precisely the right location at the right moment. Every interaction since the beginning of existence contributed to that convergence. Such stories can deceive us by obscuring a profound realization. Every trivial act results from everything that has ever happened. Yes, World War I might have occurred regardless, but in this timeline, these specific events unfolded. You might not have spilled your drink had you not turned your head at a particular sound, but you did, because the entire history of the Universe led to that moment. This sounds like determinism overriding free will, a topic we address toward the book’s end.

Radioactive decay represents a classic candidate for “truly random” events, but only at the level of individual atoms. As a group, decay becomes entirely predictable, which is how we calculate half-lives. This parallels knowing that a 1 in 366 chance of a car accident exists for every 1,000 miles driven while remaining unable to predict any specific accident’s who, when, or where. Insurance companies compile extensive data on accident probabilities involving teen drivers, weather, speeding, impaired driving, distracted driving, and other factors. This narrows the probability range but cannot predict individual accidents.

Individual accidents are not random. Suppose Sally gets drunk after being fired for aggressive behavior. She drives home from a bar in a snowstorm while arguing with her husband on her cell phone and driving 30 miles per hour over the speed limit. She crashes into Bob. Nothing random characterizes this scenario. Sally’s path and Bob’s path both resulted from chaotic event sequences. Each event adjusted accident probability. Sally’s events created a high-probability accident scenario. Bob, a careful driver paying attention and driving slowly, had a very low accident probability, yet he got hit anyway. We might call this a random event (“bad luck”) for Bob but a predictable event for Sally. In this scenario, “random” describes an unpredictable yet deterministic chaotic system (Bob’s life) intersecting another unpredictable yet deterministic chaotic system (Sally’s life). Because neither system is individually predictable, predicting their intersection becomes impossible. Thus “random” often means “theoretically unpredictable.” This accident resulted from the intersection of two chaotic systems.

We should clarify that this discussion of randomness and determinism applies to the scope of macro-scale events, not to quantum-level phenomena. At quantum scales, genuine randomness may exist as a fundamental property of nature. Different scopes of reality operate according to different patterns and laws. This concept of scopes, and how patterns manifest differently at various scales of existence, will be addressed more thoroughly later in this work. For our current purposes, we focus on the macro-scale realm of everyday experience where chaotic determinism provides a useful framework.

Stock markets provide another familiar example. Stock prices are unpredictable but not truly random. Prices result from thousands of buyers and sellers making non-random decisions based on their financial interests. Each cause-effect chain constitutes a system. All these systems combine to form the larger system of the stock market.

Chaos is deterministic because it adheres to rules and follows patterns, yet its effects over time generate unpredictable results. Chaos unfolds through time rather than existing in a single moment. What happens next depends on what happened before, making chaos a self-similar, self-referencing process. Because change occurs over time, two components require consideration: a growth factor and a limiting factor. The image to the right demonstrates this principle. Starting with a simple pattern that never changes but replicates itself in successive generations, after ten iterations the pattern resembles a tree. The growth factor is self-generating. The limiting factor consists of permanently fixed variables (length, angle, color). Weather, economics, and biological growth patterns represent familiar chaotic systems. In reality, everything that grows, moves, or channels energy contains chaotic elements influenced by millions of perpetually changing variables.

The following diagrams help clarify these concepts through visual examples.

Row A demonstrates how identical patterns manifest across vastly different contexts. The top-left image shows a simple recursive form where each line generates a child line at 20° that is half its length. This pattern repeats through successive generations. The top-right image uses the same algorithm but with 90° angles instead of 20°. Placing these side by side reveals a profound insight. The 20° pattern resembles natural plant growth, while the 90° pattern appears in crystalline lattices (how atoms arrange in many solids).

This grid pattern dominates modern urban planning. Thomas Jefferson pioneered its application in the 1700s to systematically organize territorial expansion across North America. City planners adopted this pattern because it proved far more efficient than the organic layouts of European cities. Financial incentives drove this adoption. Not coincidentally, this efficient development pattern emerged and proliferated during the industrial revolution, when city populations exploded. London, for example, grew 600% between 1775 and 1885.

The same patterns appear in biological systems. Grid cells in the brain enable spatial navigation through triangular grid formations of neural impulses. The claim that industrial revolution, capitalism, suburban roads, tissue growth, and zebrafish stripes all follow the same underlying patterns may sound far-fetched. Yet the evidence supports precisely this conclusion.²⁰ Context and available resources differ, but the patterns remain consistent across scales.

Row B illustrates the transition from order to chaos using the simplest possible form, the triangle. A line extends from the previous line, rotated 120° left. At that line’s terminus, another line extends, rotated in the opposite direction. This alternating pattern continues through iterations. The left image maintains perfect consistency across all variables. The middle image introduces 1° (0.27%) random deviation. The right image allows complete directional freedom. These three images represent perfect order, transitional order, and total chaos.

This row also demonstrates relative duality. While primal duality describes the relationship between primal nothingness and primal somethingness, relative duality operates within manifest reality. Here, the lines represent relative somethingness, and the empty space represents relative nothingness. Together they create a perfect form according to the governing rules, an archetype. Perfection derives from perfect rules. Introducing minimal imperfection (1° deviation) produces a form that remains recognizable despite losing perfection. Introducing maximal randomness (100%) generates total chaos, a perfect disorder. This exemplifies the path from order toward chaos. The reverse path, from chaos toward order, receives examination in the chapter on energy.

Row C addresses determinism and variation. Because chaos is deterministic despite appearing random, a chaotic system repeats identically given identical variables. In our Universe, variables remain dynamic. Some degree of variation always occurs. This explains why pine trees, though following the same growth algorithms, display individual variation. This row shows five instances of the same fifth-generation pattern with slight variations in length and divergence permitted.

Growth operates as a chaotic system. Evolution operates as a chaotic system incorporating randomness. Without variation, nothing changes. Reality expands (generates complexity) through chaos but evolves (produces novel forms) through randomness. This premise follows from a fundamental asymmetry. Random events can occur within chaotic change, but chaotic change cannot occur within a singular random event. The Universe demonstrably constitutes a system of change.

These diagrams show only two-dimensional patterns extending fewer than ten generations from their origin, with only length and divergence varying. The three-dimensional Universe we inhabit traces back to existence’s beginning. Every moment since then has accumulated countless variables across innumerable generations. Variable changes range from infinitesimal to dramatic. The impossibility of tracking all details necessary to predict a chaotic system’s behavior defines its unpredictability.