Both the ancient oracle called the Chinese I Ching & modern DNA exhibit the same model of nonlinear computation. This model combines sequential binary chunks & exponential analog flow in an amazing hybrid format. These two functions of number together provide 64 nonlinear equations that make up the 64 hexagrams of the I Ching, and also the 64 DNA swatches of the genetic code. Their common root can be found in chaos theory. Number is the key to this meeting of East and West. It provides special fractal "analinear" equations that combine exponential analog and linear binary functions. Learning from this model can even provide insights in how to make new and better kinds of computers.

This paper presents an innovative model for computation. Present-day computers are usually binary and do not take nonlinear aspects into account. They certainly cannot perform both binary and analog computation in the same process. But such things are possible. How?

Two variations on the underlying paradigm may be found in two very different places: in modern DNA and in an ancient Chinese document called the I Ching. Each system combines sequences of linear, binary/digital unitizing along with analog, exponential flow to develop a peculiar hybrid mathematics that exhibits a sturdy survival package. This parallel also allows the I Ching symbols of trigrams and hexagrams to act as a succinct shorthand for DNA and RNA.

Each model manifests the new science of patterned chaos. An early key was the bifurcation tree that tracks the bifurcating dynamics of "periods" of data. James Yorke and Tien-Yien in their famous article “Period Three Implies Chaos,” published in 1975, showed mathematically that when a horizontal period 3 window (hp3) appears in a bifurcation tree (or b-tree for short), the branches above it abruptly no longer track random chaos but rather, the dynamics of patterned chaos.

In this patterned chaos, events develop in an nonlinear dynamic that is able to sustain and replicate its basic pattern with variations. it is predictable in its general form or pattern, but not in its changing specific details. The human body itself exhibits the result. Each of us is human in our overall pattern, but we are also unique in our specific details, and indeed, even in the events of our ongoing lives that are enacted in space, time, matter, and energy. The scope of this underlying paradigm is as big as the universe itself.

Consider the horizontal and vertical period 3 windows (hp3 and vp3) in two very different systems - (1) the genetic code found by Watson and Crick in 1953, and (2) the 5,000-year-old Chinese mathematical system of the I Ching. If we polarize the b-tree into a p-tree (or polarized bifurcation tree), a new kind of period 3 window emerges. In just three levels of polarized bifurcation, the p-tree develops not just one period 3 window, but instead eight…but each is a vertical period 3 window (vp3) signifying resonant waves in exponential relationship. However, the eight trigrams can also do binary counting.

Mathematically speaking, the codon triplet of DNA and the trigram of the I Ching are parallel structures. Each exhibits chaos patterning as both kinds of period 3 windows—hp3 and vp3. Its dynamic is nonlinear, combining both exponential, analog number and linear, binary number to synthesize a transcendent third condition where the system escapes to a higher order of organization. This codon holds nonlinear chaos patterning…and when two codons or two trigrams are bonded together, the resulting six-pack holds nonlinear complementary chaos patterning, or co-chaos for short.

The DNA six-pack and the I Ching hexagram can each be demonstrated to hold two polarized period 3 windows in a bond. The pair of triplets are counterpoised against each other to form a fail-safe six-pack of co-chaos patterning. In each system, its two threesomes may be arranged into six-packs…as 64 polarized pairs of triplets set in every possible pairing configuration...which naturally yields the 64 codons bonded into DNA swatches of the genetic code and the 64 hexagrams of the I Ching.

The DNA swatches or six-packs are strung in succession along the double helix. Probably DNA took this form because it offered the sturdiest possible mathematical structure. [See Javier Amaru Ruiz Garcia's page here to examine the Four Color theory in regard to codon development.] The DNA double helix uses the co-chaos paradigm where a chaos dynamic is bonded to a second chaos dynamic to develop the fail-safe supersystem of co-chaos that provides a remarkable blend of dependable stability in basic pattern along with constant change in specific details.

Paper Highlights:
Most contemporary gene sequencing programs in computers handle only binary aspects, avoiding nonlinear aspects altogether. But to quote "Hacking the Genome" in the April 1992 issue of Scientific American: "The clarity of the answers will depend on asking the right questions." Asking the right question to hack DNA can include looking at the I Ching to ponder, "Why does this I Ching mathematical shorthand show the same nonlinear math that is used in the genetic code?  Answer: both are based on the paradigm of co-chaos patterning!"

Probing the import of that answer can reveal much about basic patterns in life's physical and mental systems. It can also offer us a new way to build computers that imitates the number framework hidden in life itself—its mind and its matter—perhaps even leading to the discovery of a master code at the root of universal nature itself.

Number is at the root. To dig down to this deep root, we need not discard traditional scientific linearity, but instead just add something new that mathematics and physics became empowered to explore from the 20th century onward—nonlinear dynamics. Exponential analog number plus linear binary number give us the uniquely analinear kind of number that develops co-chaos. It combines the chunky lumps of binary units along with the flowing proportions of exponential progression to birth a new third way. Some call this kind of processing nonlinear, but Stanislaw Ulam liked to point out that the word nonlinear is rather uselessly silly, since most of life's problems are nonlinear. He said that calling something nonlinear is akin to saying that most of the animals in the zoo are non-elephants. It clarifies little.

Thus I prefer to use the word analinear for the patterning dynamic of co-chaos, since it implies both kinds of number functioning together to develop a special and synergistic nonlinear third condition.

Linear binary number seeks a goal, a solution that is the end target of quantified units in a string. It is discrete and end-stopped at that goal that determines the answer. But exponential analog number does not emphasize a solution, a goal, a final  summation. Instead, it explores the quality of relationships along the way. It brings up all kinds of resonant associations that open new doors to further processing rather than closing them down into a final answer.

That's the trouble with analogs, in fact, from a traditional point of view. They tend to proliferate rather than end-stop. They engender messy resonances that sound engineers and computer programmers do not want to deal with, not wanting to trigger a network of related resonances that reinforce entrainment, not while they are trying to stay tidy and neat and hurry to an answer.

Entrainment is the main signature of exponential analog number. It does not care about quantified goals, but rather, about the resounding analog qualities of relationship emerging along the way. It compares the ranges and degrees of resonance, of consonance and assonance, not striving for a goal but rather for the consummate trip, so that finally it never ends, because the end is no goal at all. Instead it just wants to keep on traveling.

When both these functions of number combine into analinear dynamics, the result can both determine straight-line solutions and also keep iterating in evolving cycles. The result becomes the dynamic spiral of change that we see, for instance, in the double helix of DNA.

The ancient Chinese I Ching provides an astoundingly clever and complete shorthand for DNA that employs both linear binary/digital sequencing and exponential/analog flow. That shorthand can readily symbolize binary number, a fact that was made evident to the West when the German polymath Gottfried Wilhelm Leibnitz received a copy of the I Ching sent from a Jesuit missionary in China at the turn of that century into the1700s.

Leibnitz realized to his shock that the I Ching symbols called hexagrams could be read as binary numbers counting from 0 through 63. The binary code that Leibnitz thought he'd just invented? It had actually preceded him by at least 3,000 years! Other scholars began to concur in Leibnitz's observation.

More recently, Western scientists recognized that the I Ching's yang and yin can be cross-coded in a binary way with the genetic code. Gunther Stent discussed this procedure in The Coming of the Golden Age, published in 1969. Martin Schoenberger explored it in The I Ching and the Genetic Code, published in 1973. A Scientific American article called "Functions and Models of Modern Biochemistry in the I Ching" appeared in 1978. In 1991 came Johnson Yan's book DNA and the I Ching. I discussed the interface between the genetic code and the I Ching in Tao of Chaos, published in 1994. Each author considers various aspects of the genetic code/I Ching parallels.

How could the ancient East and the modern West, so far apart in space and time, come upon the same mathematical paradigm? The ancient East saw it as an oracle that codes for the flow of universal mind, the way of the Tao. The modern West found this same mathematical structure and called it the genetic code that builds organic matter. Mind and matter—could this shared math code revel a common taproot that plunges even deeper than biochemistry in nature…even down to a master code that builds the universe itself?

To explore this, one must consider the I Ching's nonlinear, fractal aspects. Tao of Chaos discusses in detail the ability of both DNA and the I Ching to combine analog, exponential number and linear, binary number into fractal patterns. Both the genetic code and the I Ching code are based on the same paradigm of co-chaos, which can predict a trend or pattern without being able to specify its exact details. It can determine an overall pattern, but it cannot specify the exact time or manner of its next manifestation. Thus a mathematician can predict the ongoing, determined general form of a fractal but not the iterating details of its specific contents.

Patterned chaos has its own special signature:

  • Order in the midst of apparent disorder.
  • Cycling that repeats with continual slight variation.
  • Scaling that fits one level into another like nesting boxes
  • Universal applicability

Starting in the 20th century, chaos theory enabled us to find pattern within apparently random events. With it, we can rise to a new level of vision to discover simplicity within a seemingly random complex flux of events. Long ago in China, this ability was called following the way of the Tao.

Patterned chaos was explored mathematically in the West to increasing extent during the 1960s, sometimes on makeshift analog computers that charted peculiar cyclic patterns. Chaos theory developed an odd vocabulary where fractals, the Mandelbrot set, Julia sets, the butterfly effect, and strange attractors suddenly opened up a new "nonlinear" reality.

More recently, it became evident to me that co-chaos patterning can be found in the DNA structure that Watson and Crick discovered in the 1950s. Amazingly, the same co-chaos patterning also exists in the I Ching, which originated according to apocryphal Chinese history in 3322 BCE when emperor Fu Hsi wrote down the eight trigrams.

Here is a synopsis of parallel structures in the I Ching hexagrams and DNA double helix:
Each of the eight I Ching trigrams holds the signature of of chaos patterning as a vertical and horizontal period 3 window in a polarized bifurcation tree, or p-tree. It can be read horizontally across the branches of a bifurcation tree in typical Western fashion to provide a linear, binary result, and it can be read vertically going up the forks to provide an exponential, analog result. In other words, this system can read by both methods simultaneously, resulting in an analinear equation that combines both functions. This is quite remarkable!

Furthermore, one period 3 window (a trigram) can bond with a second period 3 window (another trigram) to create the 64 hexagrams. These may be read as 64 co-chaos patterns that combine linear binary/digital units and exponential analog flow. Each of the 64 hexagrams describes its own unique co-chaos dynamic.

Likewise, the DNA double helix is structured in much the same way. It uses the same plan where the 64 possible pairs of polarized codon triplets bond into DNA swatches or six-packs along the double spiral. Thus, any hexagram may be read to encode DNA, and by applying the rules of genetics and the I Ching in parallel, the I Ching figures can also code for the 64 codons of RNA, which can then be further expanded to reveal the 64-hexagram philosophical messages relating to their concomitant amino acids.

In other words, the I Ching and the genetic code not only use the same mathematical structure, but they also cross-correlate according to their basic philosophical/amino acid directives. For example, without finagling or cheating mathematically, the Stop codon of the genetic code actually equates mathematically to Hexagram 12, Standstill. And other hexagrams and amino acids correlate, too, as can be easily seen in those amino acids and traffic directives that are made by only one or two codons.

Each system—genetic code or I Ching—gives a variant on the much larger issue of co-chaos. Fortunately, these two models, ancient and modern, provide a means to explore the foundational mathematical paradigm that is inherent in the fabric of the universe itself…in the makeup of its physical structure of space, time, matter, and energy. Instead of merely bonding two trigrams into an I Ching hexagram, or two codons into a DNA six-pack, the master code joins two bubbles of 3D space and 3D time into a double bubble universe, and the ubiquitous arrows of ½D time and ½D space act as its bonding glue. It lives along the ever-emerging  solution of its reality.

Yes, mind and matter merge in the master code. Its paradigm builds our bodies and our minds. It is bone-deep in our species, archetypal in our minds. It also builds the huge living organism of our universe, its enormous physical body and its ever-exploring protean mind.

© Katya Walter


A New (and very Old) Model
for Nonlinear Computation

by Katya Walter, Ph.D.

Summary for the Presentation in Tokyo - July 1995
at the Sixth International Conference on Human-Computer Interaction