"Computer of Ancient China"
Part I. Mathematical mysteries of "Book of Changes ".
From most ancient times and up to now the "Book of Changes " has the strongest influence on all life of Chinese society. Influence of ideas of "Book of Changes" can be found out at all philosophical schools both ancient and modern China. In its practical application it is regularly used in each Chinese family, and now it has wide application at the whole of world.
Specificity of fortune-telling by "Book of Changes " consists in that with its help one can not to predict the future but can to determine detailed characteristic of current situation and recommendations, following which it is possible to come to the optimum decision of problems and to favorable development of events. Speaking in other words, the "Book of Changes" applies that by its method it is possible to determine properties of any vital situation and tendencies of its development.
According to the theory of "Books of Changes", all world process is the situations alternation which is the result of interaction and struggle of forces of light and of darkness, of pressure and of pliability. Each of such situations is symbolically represented by one of 64 symbols (hexagrams) consisting of two types of lines. One type is the intact horizontal line: it is named "jan" ("light") or "gun" ("tense"). Other type is the horizontal line interrupted in the middle; it is named "in'" ("shadow") or "zhou" ("pliable"). In each symbol (hexagram) there are six such lines placed in various combinations, for example: , ,.
Everyone hexagram consists of two so-called threegram (symbol from three lines). It is considered that the bottom threegram concerns to internal life, to coming and creating, but top - to an external world, to receding, to collapsing.
All known sources attribute the invention of fortune-telling threegrams to the legendary governor of ancient China Fu Si who was in power as it it is accepted to count, since 2852 till 2737 BC. (almost 5 thousand years back!!!). Fu Si has represented these symbols in such sequence:
Various combinations of these threegrams also form all hexagrams in amount 64:
Everyone hexagram has own semantic treatment and own number according to the hexagrams table:
The principle of fortune-telling is simple: thinking about any concrete question (i.e. meditating about it), you toss up a coin or a playing bone six times and draw hexagram from below to upwards (!!!) depending on falling out result, then according to the hexagrams table you find its number and according to the "Book of Changes " you find its semantic value which is the required description of a situation with recommendations of action...
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Here we shall finish the description of fortune-telling procedure and of "Books of Changes" construction principles (for those who suddenly was not still familiar with them) and shall pass on to an explanation of certain "strangenesses" which can be found out by attentive analysis. The first strangeness consists in any "illogical" order of threegrams. Let's remind it:
Really, it would be more clear if threegrams would be settled down, say, in such sequence: etc., i.e. interrupted lines (feature) would consistently replace by continuous lines.
Thus if to take into account that threegrams (as well as hexagrams) are written and read from below to upwards then the following sequence would be much more logical: etc. or something similar...
However we have that we have...
Captious objections concerning such "strangeness" can seem to someone as completely idle: well, it has historically formed so - well and what?.. But not all is so simple...
Let's do small "focus": let's put in conformity cipher "0" for continuous line and cipher "1" for faltering line and write down threegrams in habitual to us horizontal "development":
000 001 010 011 100 101 110 111
And now the reader familiar at the most elementary level with various notations can notice here that the given row of symbols is not other that a numerical row from 0 up to 7 in binary system of numbers record:
The "strange" order of threegrams proves be in "stranger" manner connected with a row of natural numbers from 0 up to 7 located strictly (!!!) according to increase.
Is it accident?.. Theoretically: it can be. But it is not necessary to hasten with conclusions...
Let's look now at hexagrams and apply the same "focus" to them. Then from the hexagrams table we shall receive the "binary" table:
Converting contents of table from binary notation to habitual decimal notation, we shall receive:
So, thanks to "whim" of ancient Chinese we receive numbers from 0 up to 63 located in the table absolutely strictly under the order and without any mistake!!!
It is possible that somebody still considers it as an accident. Then let he will recollect combination theory and will calculate probability of such casual "hit"...
But if we not consider received result as the inconceivable whim of chance it is necessary to draw a conclusion that as long ago as 5 thousand years back ancient Chinese were familiar with position's principle of numbers record and with binary notation!!!
The result seems to even more improbable than casual concurrence of hexagrams with numerical row. But besides it is not necessary to hasten...
Let's pass to other "strangeness". Let's recollect that everyone hexagram has serial number which is determined by hexagrams table:
In this case "strangeness" is that for already described orderliness of hexagrams their numbers are scattered under the table by absolutely chaotic manner as it seems from first sight: any order or symmetry (except for several hexagrams) in arrangement of hexagrams numbers does not visible by "unarmed" eye.
Let's apply the same "focus" again, having compared everyone hexagram with binary "code":
Let's take now hexagram with odd number, for example No 41:
Its binary code is 001110. Writing down this code upside-down (i.e. not from left to right but from right to left) we shall receive 011100 that corresponds to hexagram No 42: .
Having carried out the analysis of all table of hexagrams numbers (that careful reader is capable to make himself), we shall receive a conclusion that in system of conferment to hexagrams by serial numbers there is the principle of inversion (the principle of return perusal). The given principle is shown in the following: everyone odd hexagram "is bound" to following for it (according to number!) even hexagram which binary code is formed from binary code of initial odd hexagram by means of return perusal.
(Let's note that perusal of binary code in the opposite direction, i.e. from right to left, corresponds to perusal of "natural" hexagram not from below to upwards, but from top to downwards.)
All hexagrams submit to inversion principle except for only eight ones:
These hexagrams are characterized by that in their case return perusal (i.e. inversion) results in the same hexagrams. But also for them conferred numbers are not casual: as it is easily visible, these eight hexagrams also are divided into four even-odd pairs: No1 - No2, No27 - No28, No29 - No30, No61 - No62.
In this case mentioned pairs are formed on the basis of principle of addition (or of replacements): in binary code of hexagram "0" is replaced by "1" and on the contrary, that corresponds to replacement of continuous line by interrupted line and on the contrary in "natural" hexagram.
Owing to addition principle given "exceptions" (from inversion principle) form central-symmetric pairs in hexagrams table (relatively the centre of table). Thus only in ones (!) "exception" is in each line and in each column of table.
So, absolutely all hexagrams numbers submit to quite certain laws finding reflection in binary codes of hexagrams and reflecting essence of position's record of numbers.
Unfortunately, till now the author did not manage to find any other laws in system of hexagrams numbering in "Book of Changes" except for division on even-odd pairs. In particular, "chaos" in distribution in the table of even-odd pairs themselves does not be ordered in any way (say, it is not clear - why hexagram No3 is not near to hexagram No1 or No2 but is located almost in middle of the table). Whether somebody can order this "chaos" and whether it possible at all - it is not clear yet...
Without dependence from this it is represented already doubtless that ancient Chinese were acquainted with binary notation and with position's principle of record of numbers when even Egyptians did not know them.
For that who doubts in this till now it is possible to adduce the additional indirect evidence, for that we shall pay attention to one more ancient China curiosity closely connected with "Book of Changes ". The question is the fortunetelling board using threegrams familiar to us.
On given fortunetelling board "bottom" of threegram corresponds to the centre of circle, i.e. it is necessary to read threegrams from the centre of circle to its external area.
It is easy to notice that threegrams are located in such manner that they form central-symmetric pairs by addition (replacement) principle familiar to us: and , and , and , and .
"Focus" with transition to binary code illustrates an addition principle more evidently (for convenience the binary code of threegrams is located in habitual manner, i.e. it is necessary to read it from left to right not inclining head).
Conversion of binary code to habitual decimal notation reveals one more law: threegrams are located on fortunetelling board in such manner that appropriate to them decimal system numbers form two numerical rows. Numbers from 0 up to 3 are located counter-clockwise in ascending order, and numbers from 4 up to 7 - clockwise also in ascending order.
Position of numbers from 4 up to 7 is caused by position of numbers from 0 up to 3 and by principle of addition since in binary system number 4 supplements 3, number 5 supplements 2, 6 supplements 1, and 7 supplements 0.
However orderliness of row from 0 up to 3 on fortunetelling board is not explained so simply any more and induces on ideas about premeditation of actions of ancient curiosity authors familiar with binary notation which has received wide application only in the century of computer machinery.
For received conclusions it arises much more questions than answers, but it is the cause of our interesting to ancient heritage which is full of riddles and of improbable level of knowledge of those whom we have got used to count as primitive peoples...
Part II. "Book of Changes" - code of the Universe?!.
The previous part of article about Chinese "Book of Changes" in electronic variant was already published when the author has acquainted with F.Kapra's book "Dao of physics" where obvious similarity of ancient east philosophies to views of advanced modern physics is analyzed. In this connection the opportunity to add "bare" mathematical analysis of the first part by some reasons has appeared...
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As it was already mentioned in the first part of the given work, Chinese "Book of Changes" was not simply a "fortunetelling board" but represented original symbolical quintessence of all philosophy of Ancient China. The basic idea of this philosophy consists in that all phenomena in the universe have dynamic character and the world is filled with ceaseless metamorphoses and transformations of all real.
"Changes is a book which cannot be avoided. It Dao changes always - Transformations, movement without one minute of rest, Proceeding through six emptiness, Rising and rolling without steady law. Firm and pliable replace each other in one. They cannot be contained in any rule, only changes operate here" (from Dao's books).
Thus any changes, any alterations are considered here not as result of action of any forces but as immanent to all things bent for change.
Peculiarity and originality of such viewpoint of east philosophy were long interpreted only as certain "exotic", as smart logic construction far from the real world. But in second half of XX century such science apparently far from " simple philosophizing " as the physics suddenly began to come to similar conclusions.
"Dynamic character of east philosophy is represented to us by one of its major features. East mystics interpret the Universe as an indissoluble net which webs have not static but dynamic character. This space net is endowed with life, it incessantly moves, grows and changes. Finally, the modern physics also has come to recognition of the world as an original net of mutual relation and, similarly to east mysticism, postulates internal dynamism of this net. With dynamic aspect of a matter we face in the quantum theory describing a dual nature of subnuclear particles having simultaneously properties of particles and of waves, and in even greater degree - in the relativity theory in which... it is supposed that the matter can not exist outside of movement. Hence it is possible to explain properties of subnuclear particles only in context of dynamic picture of the world that is in terms of transferences, interactions and transformations" (hereinafter: citations from F.Kapra, "Dao of physics").
"The quantum theory has found out that particles are not isolated grains of matter but probabilistic models-webs in indissoluble space net. The relativity theory has inhaled life in these abstract patterns, having thrown light on their dynamic essence. It has shown that the matter can not exist outside of movement and of formation. Particles of subnuclear world are active not only because they very quickly move; they are processes in themselves!"
But the affinity of positions of ancient Chinese views and of modern physics is considerably deeper rather than "simple coincidence" of main principle.
Say, if the physics for evident representation and for reflection of gist of own principles has got into the way using mathematical expressions then according to viewpoint of ancient Chinese all real around of us arises due to known models of realization of changes and is reflected in them by means of various combinations of lines inside threegrams and hexagrams. But in the first part of the given work it was shown, that in Chinese tradition the disposition of threegrams and hexagrams is regulated by quite precise and unequivocal mathematical laws!!!
F.Kapra marks that "the maximal manifestation of similarity of east philosophy views and of modern science is observed in the theory of S-matrix. S-matrix is a set of probabilities for all possible reactions with participation of adrons. S-matrix has received such name due to that circumstance that all set of possible adron's reactions can be represented as an infinite cells sequence which in mathematics refers to as a matrix... One of the major innovations of the theory of S-matrix consists in that it transfers accents from objects to events; thus subject of its interest is not particles but reactions between them. Such displacement of accents follows from positions of quantum theory and of relativity theory".
But "Book of Changes " also examines not objects but events (i.e. changes). And that is reaction in S-matrix, in "Book of changes" it is possible event, and appropriate hexagram is some "code" of the given event. Thus, "Book of Changes " is original analogue of S-matrix!..
And this analogy is not limited only to accent on events but traced on a lot of parameters.
"...theory which units quantum theory with relativity theory should refuse an exact site of separate particles... Theory of S-matrix solves this problem by means of indicating of exact values only for pulses of particles and of holding back about that site of space in which there is an appropriate reaction".
But "Book of Changes" is accented not on concrete situation (initial or final) but only on change tendency of this situation too that is similar just to pulse of particle ...
In theory S-matrix and in its development - theory of bootstrap (there is also such one) - each particle accepts the most active participation in existence of other particles. Each particle helps to derivate other particles which, in turn, derivate it. That is all set of particles is interconnected and mutually dependent.
But according to "Book of Changes" in each situation there is each another, each event ("change") contains elements of all other events too...
Thus it appears that "Book of Changes" is the same S-matrix, only ultimate and two-measured. But also it is six-level (you see it is constructed on hexagrams - figures with six elements).
"In "I tszin" [i.e. in "Book of Changes"] threegrams and hexagrams represent those Dao patterns which are derivated by dynamic alternation In' and Jan in various situations, both in space and in people's life. These situations are ones of infinite course and modification. All things in this world are subject to variability and fluidity. The same is typical and for their symbolical images - threegrams and hexagrams. The latter constantly stay in a condition of transformation and formation: one figure flows in another, continuous lines cave in and broken off half-and-half, and two fragments of broken line aspire to be pulled together and to accrete together with each other. "I tszin" with its doctrine about dynamic patterns which is produced by changes and transformations is the closest analogy of east thinking and the theory of S-matrix. Both these systems of views pay prime attention to processes but not to objects. In S-matrix theory processes are particles reactions underlying all phenomena of adrons world. In "I tszin" processes have the name of "changes" and are considered as the notion which is necessary for description and explanation of all natural phenomena".
F.Kapra pays attention some more to one amazing analogy between S-matrixes and "Book of Changes":
"In S-matrix theory the notion of reaction channel has more fundamental value than notion of particle. It is determined as the set of quantum numbers which is inherent to various adrons combinations, and frequently - and to separate adrons. What combination will pass through this or that channel - it is determined by probability and depends first of all on available amount of energy ".
"Probability of reaction in case of two adrons collision depends on amount of energy participating in collision... However for some values of energy stock the probability of reaction grows rather sharply; for such values reaction will occur much more often than for all others".
"Changes in adrons world derivate structures and symmetric paterns which can be symbolically represented as channels of reactions. The physics is not inclined to attach fundamental importance neither to these structures nor their symmetry, perceiving them as logic consequence of dynamic nature of particles from their tendency to transformations and changes".
"In "I tszin" we also deal with generations of changes - special structures, threegrams and hexagrams which, as well as channels of particles, represent symbolical images of possible directions of changes. If channels of reaction are filled with current of energy, then a stream of "changes" run between lines which form hexagrams..."
It is possible to illustrate this process as follows: throwing bones, the subscriber (i.e. guessing by "Book of Changes") for full maintenance of conditions should think about a problem (or about situation) i.e. should meditate, speaking in other words. Thus, meditating the subscriber directs his energy to matrix of "Books of Changes" and causes the certain resonance with some variant of events. In case of resonance growth of energy provides "choice" of the most probable variant of events development. Analogy with S-matrix is practically full...
And moreover. That till now was considered as "lack" of the theory of S-matrix turns around to its advantage, namely: dependence of S-matrix structure on consciousness! In philosophy of "Book of Changes" importance of a role of subscriber consciousness not only is taken into account, but also is one of basic theses. In the given case the consciousness is that additional energy source which provides a resonance of appropriate "change" (i.e. of reaction channel, accoding to the theory of S-matrix).
Further. In S-matrix theory there are some main principles, limiting opportunities of construction of S-matrix elements. In particular, according to one of these principles: "outcome of this or that reaction can be predicted only in terms of probabilities, that is the sum of probabilities of all possible outcomes - including that case when interactions between particles do not occur at all - should be equaled to unity".
But in Chinese "Book of Changes" possible variants of events development are considered just in terms of probability. It is obvious that the sum of probabilities of various variants of real life also is equaled to unity. It is including case of loss of dies during hexagram construction.
If we shall examine "pure" variant of dies throwing then naturally the probability of construction of concrete hexagram will be equaled 1/64. But if we shall assume that process of meditation (reflection about situation) transfers to matrix of "Book of Changes" a certain energy, changing distribution of this energy between various variants, then (according to methodology of "Books of Changes") probabilities of loss of various hexagrams will have to change, as though "redistributing" probability of construction for the benefit of that hexagram which corresponds to "resonant channel of reaction" (i.e. to the most probable variant of events development). In this case the sum of probabilities of all hexagrams, according to specified principle of S-matrix (and according to simple logic), also should remain to equal unity.
Other principle which limits opportunities of construction of S-matrix elements "says that probabilities of reactions (and consequently elements of S-matrix) do not depend on arrangement of experimental equipment in space and in time, on its spatial orientation and on condition of observer movement".
Some analogy with this principle also can be found out in "Book of Changes". As it was specified in the first part of the given article, numbers of hexagrams form "even-odd" pairs on a basis of "principle of return perusal". Since hexagram is "one-dimensional" object (there is only one axis "top-bottom") the change of "observer location" can be expressed only in change of direction of hexagram perusal relatively to this "dimension": either from the top downward or from below - upwards. The fact of formation of hexagrams pairs during this change of "orientation" just will reflect the fact of "changes" probability independence from "observer position". (Though this principle in "Book of Changes" is not strictly kept: in spite of all "paired relationship" however hexagrams turn out different...)
So, the hexagrams table of "Book of Changes" and principles of its construction find out rather strong similarity to properties of S-matrix describing fundamental processes of microcosm. Similar concurrence on set of parameters can not be absolutely casual... If we take into account interrelation of microcosm with processes of macrocosm we can tell that "Book of Changes " is not simply "a board for guessing " but a certain symbolical display of "code of the Universe".
And the problem about sources of "Book of Changes" rises with an extraordinary acuteness... From whence many thousand years back ancient Chinese have received knowledge to which the modern newest science steal up only now?..
It is a riddle of riddles!!!