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d.  DNA: I-Ching

DNA Ė IíCHING

Dow

[Editor's Note:] This paper was presented to the 9th International Conference on the Studies of I-Ching, University of Hawaii, Hilo, Hawaii, August 17-21, 1992. See the original at www.thomehfang.com/suncrates4/2Dow.html

I. Introduction

An article entitled "The Unusual Origin of the Polymerase Chain Reaction" by Kary B. Mullis disclosed a surprisingly simple method for making unlimited copies of the DNA fragment.[1] On February 16, 1992, the New York Times attributed to Mullis's 1983 invention the revolutionizing of molecular biology and genetics.

What is surprising is Mullis's graphic presentation of this simple binary mathematical formulation obviously corresponds to, if it is not identical with, the hexagram progression sequence in China arranged by Shao, Yung. (See appendix). Is this striking similarity between these two merely an accident? In view of the fact that the I-Ching sequence is based on binary mathematics and its identity with the binary table of Leibniz has been well established, the identicalness between the DNA replication sequence and the hexagram progression sequence would seem difficult to dismiss as pure coincidence.

1. The DNA replication sequence is restricted to genes, a causal event except for human inducement or manipulation, and is relatively precise, automatic and derived from scientific experimentation.

2. The hexagram sequence, derived from the observation of nature for the purpose of showing the way for human action, is all inclusive and comprehensive in its prediction of events in uncertainty. An attempt can be made to see the DNA sequence as a microscopic empirical discovery to complement the macroscopic I-Ching sequence in searching for a unified description and understanding of world phenomena?

Advancement in quantum mechanics, fractal geometry and the chaos theory, etc., tend more to reaffirm the values of the basic philosophical assumptions in I-Ching's formulation of the hexagram progression sequence. The DNA replication sequence serves as one example. However, science describes what it is but does not answer the question of why. Therefore, a philosophical approach is necessary to search for an answer to why these two phenomena are similar.

In view of the fact that both sequences are derived from the observation of nature and both deal with the problem of human beings, the significance of this search for an explanation of their similarity may shed some light on the efforts to reach a unified theory which is able to explain the world.

We shall proceed in three steps to examine the issues.
They are: (1) An evolutionary view of the world with a unique position of man or the material unity of man and the world;
(2) A twofold complementary-contradictory process of being and becoming of the world phenomena;
and (3) A binary representation of the world process in our understanding of and predictions about the world.

II. The Material Unity of the World and the Position of Man

1. The Concept that Quarks and Leptons are the Basic Constituents of Man and the World.

Aside from the admission of an omnipotent, omnipresent and omniscient external creator of this world and man, the prevailing scientific view of the origin of the universe is the theory of the "Big Bang" initiated by George Gamow in 1948 and subsequently confirmed in 1964. It described that not only were human beings part of nature, but also originated from the same source which is beyond the scope of science. This is known as the singularity. Nevertheless, it is believed to be composed of undifferentiated material, a "primordial soup," or "quantum chaos," etc. The advancement of quantum mechanics asserted that the basic constituents of this material unity of which man is a part, consist of the fundamental particles of quarks and leptons. The originator of this concept of quarks and leptons, Murray Gell-Mann, conceded that they are mathematical conclusions rather than those made from direct observations or laboratory experimentation.

Obviously the composer of the hexagrams in I-Ching relied on direct visual observations of nature in order to arrive at their statement that the world including mankind originated from the same source indescribable but implied to be a "primordial chaos or fusion" (hung tung) the "great void (tai xu)" "oneness" (tai yi, tai su), or "yuan", etc. Despite its antiquity, this speculation of singularity by I-Ching surprisingly anticipated modern science. In view of this convergence regarding the origin of the universe and its human beings in the concepts enunciated in the I-Ching and the prevailing concepts of modern science, we may venture to say that the similarity which exists between the hexagram sequence and the DNA sequence may be based upon the same assumption of a material world unity.

Assuming that man and nature have evolved from the same stem and have the same physical constituents, then, the patterns of their evolution and the processes of their manifestation in diverse and complex entities on various levels would have to bear some similar marks though it may be to a greater or lesser degree. Recent discoveries of the phenomena of both the simple and yet the complex in fractals reveal such a process. In the words of Benoit B. Mandelbrot, the originator of fractal geometry, "a major and surprising impact of fractal geometry and the chaos theory is that in the presence of a complex pattern there is a good chance that a very simple process is responsible for it."[2] This discovery is known as the principle of self-similarity, which seems to underlie the structures and evolvements of all natural entities.

The best example of self-similarity as revealed by fractal geometry is in cauliflowers. The head of it contains branches or parts. Compared with the whole, the branches are very much the same, only smaller. These clusters again can be decomposed into smaller clusters which again look very similar to the whole as well as to the first generation of branches. This process can continue but with limitations. That is what is known as iteration which is synonymous with the concept of feedback and dynamic laws. In other words it is the same operation carried out repeatedly, the output of one iteration being the input for the next one. Macroscopically we see a shape or form which is best described as a cluster. It is composed of smaller ones which look almost identical to the entire cluster however scaled down by some factor. On a microscopic level in dealing with subatomic particles one can determine the location and velocity of a particle at one instant from the preceding instant whether or not the process is discrete or continuous.[3] The technique of video feedback and multiple reduction copy operates on this principle with great success.

In view of the discovery that the infinite and complex phenomena of nature can be seen as a result of the interactive process of the most simple elements or events, the arrangement of the hexagrams known as "double-up (chong gua) of the lines" in I-Ching would seem to be part of the same process as the DNA sequence. The similarity between the two arises naturally since both systems base their statements upon objective observations of nature with the exception that the latter rests upon a more scientific ground than the former. Nevertheless, the relationship between the structural arrangements are modeled after the relationships that are recognizable as those existing in nature. In this sense the statement in I-Ching that every thing or event in this world contains a Yin and Yang would be consistent with the structures of being and becoming of the world phenomena. This may be due to the iteration process displayed in the I-Ching sequence that inspired Zhu Shi-jieh (1303) during the Yuan time to portray it in an arithmetic triangle (Ssu Yuan Yuan), which computes the binomial coefficients up to the eighth power. Since that time it became the Pascal Triangle (1623-1662). Both of these triangles could be related to the mathematical structure of the hexagrams sequence.

As a consequence of this mathematical enlightenment of the iteration process, the introduction of the Cantor set (George Cantor, 1845-1918) in 1883 in graphics not only magnified the nature of this similarity to that of the hexagram sequence (Appendix) but also reinforced the concept of the ultimate unity of world phenomena as Buddha had long ago contended. In 1913, Ramanujan equaled the square with the circle, one with two. To Ramanujan, the mathematics and metaphysics lay side by side, inextricably intertwined.[4] To equal a square to a circle is what topology has revealed to us since the Cantor set is considered to have contributed to the development of topology, the concept that the complex and diverse world phenomena are temporary aggregations of simple elements would seem to underlie all of these, whether a DNA hexagram sequence or a Pascal triangle, Cantor set or topology for they all are equivalent. In the demonstration of Peitgen (p.157) fractal is in strict correspondence to the decimal system. Therefore, the graphic presentation of the hexagrams sequence in the "Tai Ji Tu" ("Diagram of the Great Ultimacy") of Zhou Dunyi and the mathematical presentation of it in Shao, Yung's sequence or any other arrangements are all ultimately the same. Furthermore, Peitgen showed that the Cantor set is a solution to the problem of equations (p.192). Thus the Confucian concept of zhong (harmonious balance?) which explains the rule of change in hexagram lines in Xici (Appended Commentaries) may have conveyed the same meaning and function as the mathematical meta sign of "=" in mathematics.

(2) Relational Holism: the Seven Stages of Progression and Eight Catgories

Admitting that the Cantor set is truly complex yet also self-similar and simple, then, can the iteration process as shown in fractals be infinite? From a purely linear view of progression on a one-dimensional plane, the iteration process could be infinite. However it would be contrary to the phenomena of recycling observed in nature such as the alternation of day and night, the four seasons and the birth and death of human events and especially, contrary to the manner in which topology and fractal geometry deal with nature. Further discoveries in the phenomena of self limit in fractal geometry and the notion of relations in topology have become reconciled with the problem of recycling.

In the first place, the iteration process has its limit. This is know as fractal dimension. The concept of self-similarity is intimately related to the concept of limit. Self-similarity may hold true for a few orders of magnitude. Below a certain scale, matter decomposes into a collection of molecules, atoms, and elementary particles. It would be too infinitesimal to consider further dissection in scaling down copies of the complete object. It has been proved notably by Gestalt's contention that parts can never be exactly equal to the whole. Some variations must be accounted for. There are several variations of mathematical definitions of self-similarity. This is what is known as the phenomena of statistical self-similarity. The Shannon limit in computer operation may serve as another example. In addition, the resulting miniature copies may be distorted in some ways. In fractal geometry this is called self-affinity, a sort of approximation of a self-similarity which is not strictly or precisely identical. This principle of approximation in epistemology has been well explored in quantum mechanics. Therefore, the concept of similarity is intimately related to the concept of limit in terms of fractal dimension. Every limit is a negation which leads to new qualities. In this sense, negation can be seen as creation or positiveness as in the Chinese saying "bu po bu li" (no destruction, no construction). For example, the weight of a tree is proportional to its volume. If the stem is increased beyond a certain limit the strength of the wood will not be sufficient to bear the corresponding pressure and the existence of such an entity as a tree will be at stake. The tension between volume and area also explains why mountains do not exceed a height of seven miles, or why different creatures respond differently to a fall. A mouse by his own weight may be unharmed by a ten story fall but a man may be injured in just falling. As volume increases the hazards of falling from the same height increases. The phenomenon of nonproportional growth is at the heart of fractals. In the I-Ching it is said that the way of heaven is to diminish the strong and augment the weak so that all can survive.

Secondly, in topology a straight line can be bent into a circle, pitched into a triangle or pulled out into a square. However, not everything is topologically changeable. The basic relationship between things does not change. For example, an intersection of lines remains an intersection which is invariant mathematically. A sphere may be transformed into the surface of a horseshoe but never into a doughnut. For topology deals with forms and shapes from a qualitative point of view. Qualitative features will not change when the objects are transformed. If the quality of an object is changed the object will be no longer the same and it will become something else. While the basic objective of I-Ching is not a mathematical excursion such as topology, the main efforts of the arrangers of the hexagram sequence centers on the relationship of one's action within the circumstances including time and place or position and one's ability and capabilities to predict the anticipated results. Six lines (yao) were used to depict the relationship of the place value of change to the constancy of the situation in a hexagram. With a consultation to the models provided in the hexagram, one can determine an intelligent choice on which to act. Regarding the hexagram, the second and the fifth lines are the most significant because they occupy the central positions. The second is in the center of the lower trigram, and the fifth is in the center of the upper, while the third and the fourth lines are variable with the first and the last as constant. Thus the structure and functions of the hexagram sequence seem to reflect the same meaning as the limit in fractals and the unchangingness in topology. While molecular biology is primarily experimental and advances are based on observations of a specific biological system, a natural way to organize the sequences is by taxonomy or evolutionary relationships. DNA exists as a complex, three dimensional structure. Thus, the use of topology and geometry in the study of the DNA sequences is well accepted. The famous formula that is related to topological property, the unchanged property under continuous deformation of the DNA with geometric quantities by James H. White, serves as an example.[5] If the existence of humankind depends on the human gene pool, the similarity of the DNA sequence with the hexagrams sequence may signify a profound meaning for all of us to explore.

Since the DNA polyinerase chain reaction reveals a cyclic process as pointed out by Mullis,[6] it can be seen to display seven cycles or seven stages. For example, the first copy may be identified as the first cycle or the first stage; with the two copies, the second cycle or stage; the four copies, the third cycle or stage; the eight copies, the fourth cycle or the fourth stage; the sixteen copies, the fifth cycle or stage; the thirty-two copies, the sixth; and the sixty-four copies, the seventh. Assuming there must be an initial stage and assign it as zero (0) and count the initial or the original one from which the first copy derives, the sequence can be categorized as an eight cycles or an eight stage process. In a seven stage process the sequence will naturally consist of 64 copies or hexagrams in simple mathematics because the initial one is zero (0). However, Mullis did not engage in this sort of analysis.

What is puzzling is that these seven or eight stages of this evolutionary process in the DNA sequence and the hexagram sequence, correspond to the seven stages of evolvement in the origin of the universe as conveyed by the theory of the Big Bang. Why is this true? The Big Bang theory does not attempt to explain nor to infer what came before or after the sudden explosion. Genesis attributed it to the creation of the world in six days plus one for resting while the 24th hexagram, Fu (Returning) in I-Ching depicts the recycling process in nature as running seven days (qi re yi lai fu). What justified the seven days cycle? The I-Ching left that for speculation. However, the hexagram Fu explicitly conveys the meaning of it as a turning point. It contends that recycling is the process of a continuum. It occurs as the result of self-regeneration within the mutual or reciprocal relationship of the twofold structure and functioning of world phenomena. In the arrangers' mind all that is good in nature and society will eventually decline once its limit is reached; when decay has reached its limit recovery will begin. This is what the hexagram, Fu, meant.

Recent studies on subatomic particles seem to reveal a similar seven-stage structure and change though it can not be precisely matched with the DNA or the hexagram sequences. Electrons which orbit the atomic nucleus have a seven-layer shell. If we count the initial one, it can be counted as eight-layer. Each layer can hold limited numbers of electrons. The inner one holds two electrons. Both the second layer and the thirdlayer hold 8; both the fourth and the fifth, 18; the sixth and the seventh, 32. Atoms may join one or more other atoms to fprm the smallest stable particle of an element or compound, the molecule. This process takes place between electrons in each atom's outer shell. The number in the outer shell dictates whether an atom will donate electrons to another, share or seize them, and how tightly the two will bond; if the outer shell is filled, the atom is inert. This simple process of exchange of the positions of electrons in a definite number within seven layers constitutes the base of the periodical table that accounts for the structures and functioning of all elements in the known world. Is it coincidental that the framers of the I-Ching sequence based on their macroscopic observation of nature, derived eight trigrams as the basis for projecting the whole world of change? In view of the fact that Murray Gell-Mann proposed to classify the particles according to Buddha's eightfold path,[7] can it be outlandish if we see the structure of the eight trigrams as an ambitious primitive attempt to interpret the world of change within the general frame of reference in eight categories? It wouldn't be a surprise if the ancient Chinese, particularly during the Han time, had observed through the hatching of eggs the embryo process to acquire a vague idea of a seven stage development, in addition to the original one and thus inferred the eight trigrams. For embryology has revealed such a process.

(3) Material Unity as the Basis of Cognition and the Unique Position of Man

Albert Szent-Gyorgyi who became a Nobel laureate due to his isolation of vitamin C, confessed that in his search for the secret of life, he ended up with atoms and electrons which have no life at all. In his opinion, life is the expression of the harmonious collaboration of all.[8] Therefore, the search for an answer to the question of the origin of life continues. At the present stage, the standard proposal that life originated from a "primordial soup" is equally disturbing. No one has even been able to precisely describe it. The is no clear or viable theory of the chemical origin of life. Modern science may delve into the recesses of the brain to explain the phenomenon of consciousness, yet there is an undeniable distinction between the brain and the mind. While the brain may be measured on physical terms, the mind is difficult to quantify. Assuming that the brain is constituted of atoms and atoms have the so-called "parallel psychic components of some sort,"[9] it is still difficult to explain, as the Gestalt theory has contended, why the brain's consciousness is unified and not merely the sum total of all atomic consciousness. Francis Crick, the co-discover of DNA, even dreamed of a genetic code-carried to earth from another planet.[10]

So long as we cannot describe the phenomena of the mind in terms of nerve fibers, as some of the artificial intelligence researchers would like to do whether or not we can study the mind remains a profound challenge to the world. However, we can learn about the phenomena of the mind by what the mind does. This is involved with the most fundamental question: How do we know and how do we perceive the world and how do we anticipate events? The fact that we have survived and made progress in civilizations proves that we know it, but not completely. The most popular prevailing explanation of this universal success of mankind at the present level of knowledge is, perhaps, the corresponding theory. This is that the mind is structured in a way that corresponds directly with the structure of the universe. In a sense it meant that there is a unified ground state of consciousness in which the observer, the process of observation, and the observed are unified in a structure of self-interacting consciousness. This subjective experience of the unified origin of what we call subjective and objective existence constitutes the essence of the corresponding theory.

Aside from philosophical support of the corresponding theory, the DNA encoding could have been the link between the world and ourselves in knowing. For example, we do not learn grammar. We could not make this up. The ability to organize and arrange words into intelligent sentences and the capacity to identify the patterns of the world and link them to instructions from parents, teachers and others are said to have been genetically set through DNA messages which stay with us throughout our life span and are passed on to future generations. According to Heisenberg, apart from the process of observation, we are finally led to believe that the laws of nature which formulate mathematically in quantum mechanics no longer deal with particles themselves but with our knowledge of elementary particles.[11] Modern physics has postulated the existence of various dimensions or levels of reality which correspond to different levels of conscious perception. In this sense human beings occupy a unique position in their relationship to the world. It is upon this premise that the framers of I-Ching said that the coordinated unity of the way of heaven, the way of earth and the way of man is the essential structure for perception and the basis for cognition.

From a broader perspective the theory of correspondence may imply the correlated positions pronounced in the I-Ching, and particularly expounded by Tung, Chung-shu's (Dong Zhong-shu) of the Han time. Dong's exposition of the correlation theory has failed in its test by experimental philosophy but the rise of quantum mechanics has resuscitated it. In an attempt to integrate the contents of perceptual experience we have learned that the function of particularity becomes effective only through its being correlated with the function of universality (identify in whiteness as the definition of white)! The function of univereality becomes meaningful only in correlation with the function of particularity (perceiving the particular object intuitively). It is the correlation within one and the same structural complex of the two basic functions, that of particularity and that of universality, that make cognition possible. Echoing Leibniz that there is a preestablished harmony between the world and man which effects knowing. Andre-Marie Ampere, the pioneer of the science of electrodynamics has contended that our mind is structured in a way that directly corresponds to the structure of the universe. Otherwise, we wouldn't be able to ascertain any degree of certainty.[12] Science has now conceded to idealism that its objective reality is not given but is constructed by the creativity of the mind. Without assuming a coordinated system existing between the world and man, objective reality cannot be constructed. What we have claimed as the laws of nature are not the elements of nature but the products of the human mind derived from their observation and understanding of nature. It is on this ground that we can explain the pure process of logic which consists of seven stages, (e.g., 1. perceiving the world; 2. selecting the features perceived; 3. discriminating between them; 4. judging them; 5. analyzing them; 6. constructing a theory or hypothesis; and 7. testing them) which is universal. Some researchers have extended a similar seven-stage view to analyze the development of our moral reasoning process from childhood to old age, e.g., 1. simple obedience; 2. conforming to group behavior; 3. peer pressure; 4. duty orientation; 5. legal considerations; 6. conscience orientation and 7. liberty). Karl Jung thought that behind every picture or image of the nature and social behavior as well, there stands an archetype deeply ingrained upon the collective unconscious of humanity and available for the individual to decipher. That is what the hexagrams meant. Mandel-Brot demonstrated that mathematical fractals in fact have many features in common with the shapes found in nature. Indeed some characters such as the Cantor set were already there.

(4) The Twofold Complementary and Contradictory Process in Being and Becoming

Specifically both the DNA sequence and the hexagram sequence are oriented towards practical application. If we examine the predictability of these sequences, they are almost incompatible. The DNA replication must proceed with an accuracy of one error in every iota base pair.[13] The fidelity of the DNA replication is relatively assured. The predictability of the hexagram sequence depends on the capability of the inquirer to decipher their meanings. Therefore, the margin of error in the hexagram sequence is enormously greater than that of the DNA sequence. However, a careful study would show that the difference between the two in predictability is a matter of degree rather than of kind. The tact that the terms of error, wrong or mismatched can be introduced in the DNA replication tells us that all is not absolute. If our heredity is a chance occasion as biologists have stated, then, chance also plays a critical role in hexagram predictability. In three aspects we can illustrate that both DNA and hexagram sequences reveal the same fundamental principles. Their differences in predictability arise not from their basic operation, but from the scope and degree of their applications. These principles are: (1) the twofold structure in continuous creating or change; (2) the twofold polar relationship of complementarity in contradiction; and (3) the reciprocal generation of each other in a spiral progression.

1. The basic twofold structure in continuous renewal.

In their efforts to search for a unified theory of elementary particles and forces with which to explain the material world, physicists have found that all matter is built from just two classes of elementary particles: leptons and quarks. Frank Wilczek notices that "all the fundamental constituents of matter came in matched pairs: For every kind of particle there is an antiparticle. Indeed, a particle and an anti-particle have often been discovered simultaneously when the two were created as a pair by a high energy collision in a particle accelerator."[14] It has become apparent that all particles are coupled more or less strongly to each other. Scientists do know that the twofold nature of a positive and negative charge constitutes the basic nature of the structure and functioning of matter but they do not know why. Except in different linguistic expressions, it is indisputable that both the DNA sequence and the hexagram sequence consist of basic pairs. For example, Yin must pair with Yang in order to exist and function.

Either Yin or Yang by itself cannot exist (gu yin bu sheng; yu yang bu zhang). The I-Ching says that this is the way of the universe. Anyone can infer from the obvious event of the pair-up between male and female for procreating and contemplate that this twofold process of being and becoming is not just for the material world of heaven and earth but also for human events as well, as the arrangers of the hexagram sequence had contended. The advancement of our knowledge at the present stage seems not to disapprove this theory. Microscopically the photons display a twofold phenomena of particle and wave; the DNA structure is a double helix; the 46 chromosomes of a person are in 23 pairs except x and y which specify sex, etc. On a macroscopic level, it has become common knowledge that our brain hap two hemispheres and our thought has both digital and analog processes; linguistic expression consists of semantics and syntax, i.e., yes and no; in the structure of any society throughout the history of the world there has always been the twofold stratification between the ruled and ruler or between command and obedience, enemy and friends, internal and external, etc; in economic activities there is the twofold structure between haves and have not, gains and losses, management and labor, monopoly and small enterprises, diminishing returns and increasing returns, etc.; in politics, the most cherished democracy is predicated on a two-party system; there are conservatives and liberals, subjects or citizens and rulers or government for and against, in and out, etc.; in ethics and aesthetics there is the twofold distinction between the virtuous and the disreputable, the beautiful and the ugly, etc. In civilizational development as a whole, a twofold process has evolved in all things. For example, there are the small vehicle and great vehicle schools in Buddhism, idealists as opposed to realists in early Confucianism and schools of reasoning and the school of intuition in Neo-Confucianism, there are Catholics and Protestants in Christianity, and Sunnis and Shiites in Islam, etc. Sufficient is this example to warrant a statement that the twofold structure of being and becoming is universal and permeates all phenomena of the world. We take this for granted because it is so obvious.

Ultimately it is the primordial relationship of pair-up which constitutes the source of individuality, change and transformation, and finally, our cognition of world phenomena. For instance, logically in order for A to change into B both A and B must be identifiable and individualized entities; however, if either A or B is an absolute entity or all by itself, then, A can not change into B. Therefore, A and B are separate and yet in unity. A and B may vary in their structures and identities but the twofold primordial relationship does not vary. In our cognition of the world the first fundamental step is to classify observable objects into similar categories so that if we know A we will be able to know B through A and so on. If A is similar to B, then, B is not different from A. Therefore, it is the twofold primordial relationship of similarity and difference which provides the bases for classification, or taxonomy, and thus for cognition. Without the twofold relationships of unity in separation and similarity in difference classification and cognition would be impossible. Quantum mechanics reveals that the union of a positive and a negative force or energy generates a new object or a new phenomenon. The difference between the positive and negative aspects is only a sign. Even scientists do not know why this is true. Particles have a property called spin. We can measure for spin in any direction but cannot meaaure in two or more direction. on any one given particle. Any measurement will yield exactly one "up" (+) or "down" (-). The spins are correlated.[15] While the identity of an individual structure or object may vary, the relationship between one individual and another in terms of their union (of positive and negative aspects) to generate a new individual will not vary. The relationship of wife and husband will remain regardless of who the husband and wife may be. This is how Kung Yinda defined the concept of "yi" as having two meanings: changing and unchanging (yi, yi ming er han erh yi). Change and non-change are contradictory [contradistinctive] yet complementary.

Recent studies in life science have further enlightened us through its discovery of simplicity in diversity for the continuous renewal of the evolutionary process. Research in the morphogenetic field reveals how an initial assembly of identical cells can give rise to differentiated cells that are spacially organized and stabilized. In embryology there exist physiological gradients within eggs during their development. The difference in gradients provides the basis for difference in gene positional information which enables the DNA double helix to assume the way in which those forms can recognize and interact with their cognate binding proteins and enzymes. This essential information for differentiation implies the existence of a coordinated system within which cell position is specified and induces their differentiation in neighboring cells.[16] This discovery that the gradients in position can effect change and transformation, would seem to shed light on the contention in the hexagrams of the place value to interpret through the position of lines the anticipated course of development of events. The line 3 and the line 5 in a hexagram have similar functions but are in different positions. Therefore, their results would be different. The line 5 occupies the most influential and advantageous place according to the values of being in the center and, therefore, strong. It is the place of honor. This place of value in the functional sense reminds us of the Chinese saying: There is not so much apprehension about who has the title to govern but who is in the position to take charge (bu pa guan, zhi pa guan). For the sake of safety the driver of a car in which the president of a country is riding is more critically important than the president.

Since science does not fully explain why gradients exist in the relationships of the DNA helix, the primordial relationship between and within the primordial pair which prompts change and transformation of world phenomena seems to have meaning. Scientists, have learned that a profound difference can result from the half integer spin of particles between Fermion and Boson. Several protons are supposedly arranged in close proximity to one another, the strong binding force among them (Boson.) is much greater than the repulsing weak force (Fermion). If one replaces each proton with a neutron, the force remains unchanged. [17]

2. The Twofold Polar Relationship of Complementarity in Contradiction

In his recounting of the race to decipher the structure and the functions of DNA, 25 years old James D. Watson pointed out the reason why the venerable Linus Pauling fell behind him in this discovery because of Pauling's inability to perceive the contradiction that "A" and "Not A" can accommodate each other in the manner of complementarity between two opposite fundamental nucleotides but Watson, in light of quantum mechanics, reasoned that the basic DNA is helical and consists of two intertwining helices; these are not identical but complementary so that if one helix contains a quanine base the other helix contains a thymine base. This pairing-up of DNA in a double helical structure meant that if the base sequence of the two intertwined chains were complementary to each other but yet must run in opposite directions.[18] We see emerging now a general agreement that the functional distinction between the continuous and discontinuous DNA replication is accomplished by a dimeric DNA polymerase. The relationship of donor and recipient, lagging and leading, affects locality and mismatches.[19]

What is amazing is that the fundamental principle upon which the hexagram sequence is formulated is the same as the DNA sequence, though it is not in one-to-one correspondence. By virtue of the fact that the signs which symbolize the primordial pair, Yin and Yang, with the broken line and the unbroken line reflect their contradictory nature. Yet it is the complementarity of Yin and Yang in their union that generates a myriad of things and events ad infinitum. The apparent order displayed in the hexagram sequence is that any odd numbered hexagram and its even numbered successor form a pair of opposites. The inverse of a hexagram is found by turning the hexagrams upside down, resulting in a pair in inverse. For instance, hexagram Heng (continuity or perseverance) is followed by Tung's retreat. Whether or not the principle of complementarity in contradiction constitutes the ultimate truth of reality, no one can be certain. Niels Bohr, one of the foremost founders of quantum mechanics, proclaimed this as the fundamental principal of quantum phenomena. Bohr considered this power of explanation to encompass not only physics but also chemistry, biology, psychology, sociology, and even economics and history, or for that matter, the whole spectrum of human existence and its manifestation in the development of civilization. In this sense, the attempt of the arrangers of the hexagram sequence to realize an all inclusive interpretation and prediction of events and things may have, philosophically if not scientifically, anticipated the ideas of quantum mechanics.

With the advancement of science we began to realize that the mechanism which deals with the cognition of the world, our thought processes consists of a twofold mode of operation: synthetic and analytic. The former deals with the dynamic and the latter with the static; the analytic thought refers to formal logic while the synthetic deals with dialectical logic. Both serve as complementary methodological tools and are not mutually exclusive. Bohr contended that our consciousness is a stream of mental phenomena (analog), the subject being our experiences of life, which can not be concretized and atomized yet there exists a discontinuous aspect (digital) from one moment to another which is tightly coupled with every given object. These two aspects complement each other though they are contradictory. He noticed that this principle of complementarity in contradiction is strikingly similar to that which William James has contended all along.[20]

Due to the fact that economic activities comprise the most critical and also a very large portion of our daily activities, we may rely on some economic aspects to illustrate how the principle of complementarity in contradiction operates in human events. The function of the exchange of goods and labor through a market mechanism in sustaining our life, involves not only the twofold structure of buyer and seller, supply and demand, as well as both the use value and exchange value, but also the twofold relationship of productivity and production in addition to diminishing and increasing returns. For example, in an agrarian and natural resource oriented economy the negative feedback of diminishing returns dominates. The high price of a product may lead to a drop in the price in the long run because it stimulates production and discourages consumption. In a highly technological and knowledge oriented economy, the positive feedback of an increasing return may occur to corner the market. The general trend is that a free market often leads to monopoly and monopoly requires a free market for its remedy. Buyer and seller, monopoly and free market are contradictory yet they complement each other to bring about overall economic development in terms of the exchange of goods and services to sustain life. In the production process the twofold relationship between those who are in command and those who are in obedience or between management and labor is indispensable for attaining the objectives of production regardless of whether the relationship is within a family or in a social unit, etc. The only difference which may occur, is that the potential to achieve the production goal is tar greater in relationships of the heart than otherwise. The relationship between command, (generally a minor elite) and obedience (usually the ordinary majority), is contradictory yet they are complementary from the functional or relational point of view. This principle can be extended to explain all phenomena in politics, history, etc., as Bohr has contended. For that matter the similar principle of being contradictory yet complementary (xiang fan er xiang cheng) underlies the formulation of the hexagram sequence, particularly in its interpretation in the Appendices during the Han time and would seem to anticipate the ideas of Bohr if not quantum mechanics. Thus, indirectly there may' be some grounds for the thread of identicalness between the DNA sequence and the hexagram sequence via quantum phenomena from the relational holistic point of view.

3. The Reciprocal Self Generation (of "Each Other") in a Spiral Progression

While Pauling had difficulty in visualizing the complementarity in contradiction, Watson went beyond the boundary of classical physics to prove this in DNA replication and pointed out the polar relationship as the process for self-generation in an evolutionary world. For DNA replication in Watson's words requires the formation of a comple-mentary (negative) image whose shape was related to the original (positive) surface like that of a lock to a key. The complementary negative image would then function as the mold (template) for the synthesis of a new positive image.[21] Subsequent studies have shown that in a typical gene, a segment of DNA encodes a protein that has a specific sequence of the base pair that makes up that segment of DNA and corresponds to the specific sequence of the amino acids that make up the protein. Decoding a gene requires first that it be transcribed into a molecule of the messenger RNA (,RNA) and then that RNA molecule be translated into protein. These proteins are activators (helpers) or repressors, (hinderers) depending on whether they increase or decrease transcription. Genes are turned "on" or "off" by enzymes called RNA polymerases which transcribe genes into DNA.[22] The presence of two DNA bound repressors affects transcription in two ways: one negative, and one positive. The advancement of this technology has reinforced the claims of psychophysicists and has even prompted the contention of socio-biologists who see human behavior in part as an adaption encoded in the genes. The opening of this new vista in our attempt to study human events has shed new light on the approach to the I-Ching with its holistic endeavor.

Whether or not the symbolic description of the primordial pair, Yin and Yang can be seen as a "template" of each other, as with DNA, is a perplexing question in view of its antiquity and ontological nature. There is no doubt that the predictability of the hexagram sequence is incompatible with the high fidelity of DNA replication. However, the phenomenon of the reciprocal generation of each other between the pair as shown in the DNA replication would seem also to support the principles which underlie the formulation of the hexagram sequence, particularly if we take its graphic presentation in the Tai Ji Tu by Zhou Dunyi as the starting point.

First, both sequences reveal an orderly non-linear spiral progression, knowable except in their predictability. However, the difference in predictability, between the two is a matter of degree rather than of kind. For the value of predictability lies in the degree of approximation. The recent discovery of the principle of uncertainty ascertained through quantum mechanics has postulated that in the process of change the phenomena of certainty, order, symmetry, determinant, and predictability go hand in hand with uncertainty, disorder or chaos, indeterminant, and chance or unpredictability. The enduring influence of the hexagram sequence is derived not so much from its function of fortune telling, but rather from its philosophical implications notably among thinking people that have shaped the Confucian-Taoist world-outlook towards life. In its non-linear mode of progression, regularity in human events is seen as the way to hold a harmonious balance in steering the course within and among emergent situations. The fifth line in a hexagram indicates regularity as being advantageous due to its central position within the twofold structure. Yet its regularity is subject to the intelligent choice of an actor (li jian da ren). For example, in the hexagram 3, Tong, it says that movement in the midst of peril gives rise to great progress and success through firm correctness; the feeling that rest and peace has been secured, should not be indulged. This certainly indicated neither a linear progression nor a mechanistic determinant. But history bears witness to the fact that it was over-indulgence in luxury and comfort together with the abuse and corruption of power which decimated the conquering Mongols and not the battlefields. Alexander the Great died in the arms of Persian princess.

Secondly, both sequences display a process of generating each other in their interactions, though they are not precisely in correspondence. The DNA replication starts with the separation of its two identical chains. One chain serves as the template for the synthesis of the other chain. The two new strands of the offsprings are made on the two parental templates. The essential task of gene replication comes from the requirement that each base in the newly synthesized chain is always composed of hydrogen bonds: an adenine-thyinine pair held together by two hydrogen bonds was identical in shape to a guanine--cytosine pair also held together by at least two hydrogen bonds.[23] The DNA replication stands out as the consequence of a double-helical structure of the base sequences complementary to each other. Given the base sequence of one chain, that of its partner was automatically determined.

Comparing this process of replication in DNA with that which "Tai Ji Tu" presented in hexagram progression such as that: (1) when Yang reaches its zenith Yin emerges and vice versa; (2) within the sphere of Yang there is an element of Yin and vice versa; (3) when Yang waxes Yin wanes and vice versa, the resemblance between the DNA sequence and the hexagram sequence in the principle of reciprocal generation and identity of the base pairs throughout the process of change would seem apparent. The discovery of self-similarity in fractals tells us that the output of one iteration of Yin and Yang is the input of the next. The interation process carried out in the DNA sequence follows the same pattern as the hexagram sequence. However, the chart of the DNA structure and the chart of Tai Ji Tu appear radically different. The DNA hydrogen bond served as both an intermediary to hold the nucleotides chains together and the catalyst protein to set it in motion. The graphic presentation of the Tai Ji Tu stressed the polarity of Yin and Yang in the most simple form and nothing more.

Again the difference between the two seems to reflect a philosophical approach in Tai Ji Tu against the scientific approach to DNA in dealing with the same reality of the twofold structure in the self-progression process. Since topology and fractal geometry indicate that a square can equal a circle and the decimal system can be converted to graphics, the difference between the charts of DNA and Tai Ji Tu would seem superficial. In this sense the square arrangement of hexagrams ascribed to King Wen and the circle arrangement ascribed to Fu Xi formed the same sequence. Therefore, the identity of mathematical sequences between DNA and hexagrams remains standing. But the philosophical value of the Tai Ji Tu even provided some explanations of ultimate reality which science is, so far, unable to address. The most simple question of how the first catalyst originated continues to challenge science. In 1989 Thomas R. Cech and his associate wan the Nobel prize for their discovery that certain types of RNA could act as their own enzymes. But their success was secured with a great deal of human manipulation in the laboratory. Could RNA have initiated itself at the beginning of the universe? No one is sure.[24] This reminds us that Thomas Aquinas asked the world: "Who set the world in motion first?" Since 1982 the "immortality" of genes began to hold true and gene profiles now have become a booming business. Not only is the reforming of the Darwinian theory of evolution necessary, but also can the justification of I-Ching granting a unique position to human beings in the unity of heaven, earth and man in a trigram be dismissed as nonsense?

Thirdly, both sequences display the principle that one helps is to generate the other and in turn, regenerates itself. In this process of reciprocal generation one or the other participant is involved in its own negation. Microscopically in particle physics every particle has an anti-particle with which it can annihilate itself in order to be synthesized into a new self. The study on genome imprinting shows that when individuals produce sperm and eggs, the old imprints are erased and new ones specific to the sex of the individual are imposed on all the gamete chromosomes. Philosophically, the occurrence of erasure in gene imprinting can be seen as the negation of self or the extinction of ego. In genome imprinting, the parents may contribute identical genes to their offspring, but if the genes have been differently imprinted they will not have identical effects. Sex specific gene imprinting may influence normal development and trigger diseases. The mismatching of genes may be abnormal occurrence. Otherwise, human beings would have no health problems.[25]

Not only do Mendelian genetics and the Darwinian theory of Evolution need revision due to these recent discoveries, but also these events have reinforced the concept of the non-permanence of ego which was advocated by Buddhists, echoed by Hermann Weyl[26] and employed by Confucians for their justification of martyrdom. This erasure of parental imprinting ensures the gene profile which otherwise would have no meaning. Obviously the offsprings have inherited their parental traits in order to continue the genetic line and heritage but in the meantime, have their own uniqueness because genes define the unique characteristics of every living being. This process in a sense, reflects the twofold structure of development in continuity. Following the same line of reasoning, the hexagram sequence would parallel this phenomenon. For example, hexagrams Qian and Kun are derived from their parents Yin and Yang but the meanings of the two hexagrams are no longer the same even though they retain their primordial lines, etc.

Macroscopically every thing or event identifiable in this world must have its boundary in order to be individualized. In this sense boundary means limits. Every thing or event in this world has its own limit. Once the development of a thing or event reaches its limit it is turned around to become its opposite. Thus birth is the limit of death and vice versa. The events of death and birth make room for each other to go on indefinitely. The food chain may serve as another example of self-negation which generates the other. A fattened pig signals a slaughtering. Recent discoveries in science are increasingly noticed in which order and disorder (chaos) go hand in hand without one excluding the other in the process of change. The Xu Gua of Appendix X in the Zhouyi Zhengyi (Correct Interpretation of the I-Ching of the Zhou Dynasty) clearly stated this principle of negation in the formulation of the hexagram sequence. The advancements in computer science would seem to reinforce this principle. The computer seems especially vulnerable to chaos. But it is through computer usage that the theory of chaos gained strength due to the tact that chaos tends to follow certain very stable patterns.[27] Fractal geometry has shown that similarity is not restricted. The infinite and finite in a polygonal spiral are not precise.

(5) A Mathematical Description of the Twofold Polar Relationship of the World: O:1— Unity of DNA and Hexagram Sequences

Assuming that the world phenomena of being and becoming in things and events can be conceived of as unity in diversity in a twofold relationship of complementarity in contradiction then, mathematics, the conceptual crystallization of our cognition of the world, would be able to describe it in the most simple and easily understood mathematical language. If we are part of this universe, then, from a relational holistic point of view, the ultimate material base of our life, the genes and their linkage with the world would have a twofold mathematical formula. The binary mathematical formula of 0:1 would be appropriate to describe the relationships of the world and could compute the possible course of world development. Martin Gardner observed this. Therefore, the power of 2 turns up everywhere in mathematical and physical structures and the arrangers of hexagram sequences apply it to almost everything from crystal structures to the solar system and the cosmos. In a momentary flash of intuition Z. D. Sung perceived the way to generate the eight trigrams at the corner of a cube. With 0:1 indicating the distance from the corner of origin for each coordinate direction, he showed that the eight members correspond to the eight trigrams with complementary trigrams at diametrically opposite corner of the cube.[28] As a consequence Sung divided the cube into 64 smaller cubes, linked the 64 hexagrams with six-digit binary numbers at the six-dimensional hypercubes and identified them to the 64 "modes" of the classical syllogism. The major premise, the minor premise and the conclusion of a syllogism can each be of four different forms giving 64 possible "modes." This is what, according to Gardner (p.110), C. Howard Hinton's work on the Fourth Dimension, reveals as well as his discovery that cells correspond to traditionally valid syllogisms that the anomalous syllogism is valid after all if one applies a more liberal interpretation to the syllogism.

Not only has the binary basis of computer operation inspired new thinking in mathematics leading to a rekindled interest in I-Ching but also it has resuscitated the mathematical value of the simplicity in the Yin-Yang iteration process. Since quantum mechanics has provided the means to see the microscopic world beyond our ordinary sense of perception and fractal geometry has discovered the process of iteration in which the phenomenon of extreme simplicity can evolve into the phe-nomenon of extreme complexity, no one now can ignore the hexagram sequence as pure fortune telling or magic nor can one dismiss it as too obvious. A major and surprising impact of fractal geometry is that within the presence of complex patterns there Is a good chance that a very simple process may be responsible for it.[29] The compatibility between the Cantor set and the evolving hexagram sequence can be easily discerned. It is at the microscopic level that the DNA sequence parallels the hexagram sequence. Prigogine argues that the complexity which involves the interacting cells in a myriad of organisms can be appreciated if one realizes that even the most advanced organism issues from a single cell, the zygote.[30] Gregory J. Chaitin of IBM concedes that the regulatory genes of a developing embryo are in effect a computer program for constructing an organism.[31] Therefore because both the DNA and the hexagram sequences have the same twofold base of binary structure, their similarity is natural.

After all, Claude E. Shannon's contribution to the world was his successful application of the principle of simplicity in complexity by switching on and of f the electric current in computer operations. Basically the unit of information in bits contains an extremely simple choice between two opposite yet complementary numbers: yes or no, absence or presence. Billions of calculations can be made if the object to be counted is measurable or quantifiable. On a single microchip millions of components can be stored and retrieved in a billionth of a second. Computers analyze the most essential molecular phenomena such as DNA which is mathematically elementary and involves straightforward algorithms because of the microscopic simplicity of nature.

The problem which the hexagram sequence has is its texts that, as noted by Gardner, are so ambiguous that no matter what hexagram is selected and matched, it is always possible to interpret it in such a way that it seems to apply to the question. For basically in I-Ching any relevant event which may happen is not causally linked, but it is causally related in the sense of being a part of the vast cosmic pattern. The essence of I-Ching's approach is to compute the future of an event in question and not to rely on prayer (ji shu er zhi lai). Chinese fortune-telling is described as calculating one's decree from heaven (suan ming or suan gua). Though the fidelity of DNA replication is assured to a large extent, the reliable prediction of its protein structure from the amino acid sequence is still beyond both current mathematical methods and computer science.[32] Both the DNA and hexagram sequences represent the attempt of human beings to understand nature as well as human events so that we may act according to the possibility that we may manipulate events or things to suit our purposes whatever they may be.

Ultimately, mathematics is the creation of the human mind. Yet ultimate singularity is beyond the ability of any mathematical language to describe. Even the proposal for the term "quantum fuzz" to explain the Big Bang as well as many others, is merely a proposal. The intro-duction of the concept of zero, with its relationship to concrete reality designed as "1" (one) is perhaps a necessary conceptual medium to convey the being and becoming of world phenomena in the most simple terms. In our cognition of the world it has turned out that the solution to certain mathematical problems requires the use of assumptions essentially transcending arithmetics.[33] With the assumption of zero to reflect the wholeness of the universe not only can the oneness of the universe be ascribed to each and every identifiable object in the world of reality without losing itself, but also the interation process can be described for the continuous generation of things or events in this changing world. For zero itself does not change. It can not be created or destroyed. It does not have contradiction in a twofold positiveness or negativeness. Yet zero is not nothing. It can operate as the "hidden correlation" in what Bernhard Riemann referred to as the Zeta function which fits any smooth function locally yet is globally chaotic.[34] With this the statement of "wu ji er tai ji" in Tai Ji Tu can be explained. Numbers begin with zero and are open ended to add on more zeros. This may be why the basic Cantor set corresponds to the hexagram and DNA sequences, because it is an infinite set of points in the unit interval O:1, which can be interpreted as a set of certain numbers. A binary tree can be constructed for the Cantor set which explains the meaning of the equation in mathematical functions.[35] Therefore, with zero everything can be done; without it nothing can be done. Thus have declared the inventors of the theory of zero both in India and in China.

(5) Conclusion

Since the scientific DNA replication sequence reveals a binary mathematical structure and the non-scientific hexagram progression sequence also reveals the same, the value of the latter can not be easily ignored as mystical or magical as some have recently attempted. Despite the differences in the contents between the two sequences the similarity between them would seem to strengthen the shared ontological ground as follows: (1) the diverse and complex world phenomena derive from a singular unity, be it named as quantum fuzz, quantum chaos, primordial soup, hung tong, or wuji etc.; (2) from this unity the world evolves in a twofold complementary and contradictory process; (3) in this twofold structure every identifiable entity contains a twofold polar relationship of being and becoming, e.g. stable-dynamic, symmetry-asymmetry, etc.; (4) this process exhibits an hierarchical order in simple iteration; (5) in the polar relationship one generates the other which in turn, generates itself in a recycling spiral; (6) the meta-sign zero best conveys this order; and (7) for practical applications the reformulation of the hexagram sequence may be necessary in order to gain viability in the contemporary world.

In view of the recent claim of the genetic algorithms, modeled after the DNA sequence in the binary mathematical structure of 64 bits (strings) or possibilities for computer programming to solve problems both physical and social, a question naturally arises: Would this new development in computer science lend credibility to the ancient attempt of the hexagram sequence?[36]

With the contention that the genetic algorithms for computer application rely on the principle of implicit parallelism for simulating the events in question to project their evolving course of development one may wonder if the concept of "implicit parallelism" could coincide with the concept of correlation upon which the hexagram sequence rests.

Notes

[1] Scientific American, April 1990. pp. 56-65.

[2] Heinz-Otto Peitgen,Fractals (New York: 1992), Springer-Verlag, p.20.

[3] Peitgen, p.20; Michael F. Barnsley, Fractals Everywhere, Academic Press, N.Y. 1988.

[4] Robert Kaniel, The Man Who Knew Infinity (New York: MacMillan, 1991), p. 65-66; G. H. Hardy, ed., Collected Papers of Srinivasa Ramanujan (1887-l’920) (New York: Chelsea, 1962), p.22.

[5] J. H. White, "An Introduction to the Geometry and Topology of DNA Structure," Michael S. Waterman, ed., Mathematical Methods for DNA Sequences (Boca Raton, FL , CRC Press, 1989), p. 255.

[6] Mullis, p.64.

[7] Michael J. Long, Fundamentals of Elementary Particle Physics (New York: McGrew Mill, 1973), p.170.

[8] Biology Today, San Rrancisco, California: CRM Books,1972), "Preface."

[9] Bernhard Rensch, Evolution Above The Specie Level (New York: Columbia University, 1960), p. 355.

[10] Life Itself (New York: Simon-Schuster, 1981), p.88.

[11] Werner Heisenberg, "The Representation of Nature in Contemporary Physics," Daedalus, Vol. 87, No. 3, 1958, pp. 95-108.

[12] L. Pearce Williams, "Andre-Marie Ampere" Scientific American, Jan. 1989, p.27.

[13] Charles C. Richardson, et al, Molecular Mechanism in DNA Replication and Recombinatio (New York: Wiley, 1990), "Preface."

[14] F. Wilczek, "The Cosmic Asymmetry between Matter and Anti-matter," Scientific American, Dec. 1980, pp. 82-102.

[15] Charles H. Bennett, "Quantum Cryptography," Scientific American, May, 1989, p. 28.

[16] G. Nicoles & I. Prigogine, Self-Organization in Non-Equilibrium Systems (New York: Wiley, 1977), p. 411.

[17] Gerald Holton, "Gauge Theories of the Force between Elementary Particles," Scientific American, June 1980.

[18] James D. Watson, The Double Helix (New York:New American Library, 1969), pp. 125.

[19] Charles C. Richardson, et al, Molecular Mechanism in DNA Replication and Recombination (New York: Wiley, 1990), "Preface."

[20] Gerald Holton, Thematic Origins of Scientific Thought (Cambridge: Harvard University Press, 1973), p. 139.

[21] The Double Helix, p. 84-141.

[22] Mark Plashrine, "How Gene Activators Work? Scientific American, Jan. 1989, p.41.

[23] Watson, p. 125.

[24] John Morgan, "In the Beginning", Scientific America, Feb.1991, p.119.

[25] Carmen Sapienza, "Parental Imprinting of Genes," Scientific American, Oct. 1990, pp. 52-53.

[26] Philosophy of Mathematics and Natural Science (Princeton, 1949), p.75.

[27] Fractals, p.62.

[28] Martin Gardner, "Mathematics of the I-Ching," Scientific American, Jan. 1974, pp. 108-113; A. K. Dewdney, "Computer Recreations" Scientific American, Nov. 1984, pp. 19-28.

[29] Fractals, p. 20.

[30] Prigogine, p. 409.

[31] "Randomness in Arithmetic" Scientific American, July, 1988, p.85.

[32] Michael S. Waterman, et al, Mathematical Methods for DNA Sequence, Boca Raton, FL.: CRC Press, 1989), "Preface."

[33] Kurt Godels, in Gregory J. Chaitin, "Randomness in Arithmetic", Scientific American, July, 1988, p. 83.

[34] Martin C. Gutzwiller, "Quantum Chaos," Scientific American, Jan. 1992, p.84.

[35] Fractals, p.79.

[36] John H. Holland, "Genetic Algorithms," Scientific American, July 1992, pp. 66-72; Rick L. Riolo, "Survival of the Fittest Bits," Scientific American, July 1992, pp. 114-116.

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