142857 is the number used to construct the Enneagram, a sacred symbol of the Gurdjieff Work, that is used to explain and visualize the dynamics of the interaction between the two great laws of the Universe (according to Gurdjieff), the Law of Three and the Law of Seven. The movement of the numbers of 142857 divided by 1/7,2/7. etc., and the subsequent movement of the Enneagram, are portrayed in Gurdjieffs Sacred Dances, known as the Movements.
Law of Three [source]
The Law of Three, in a short description, means that three forces enter into every manifestation, into every phenomenon, and every event. They are called (but these are only words, because they do not express their qualities) positive, negative and neutralising, or active, passive and neutralising; or, still more simply, they may be called first force, second force and third force.
These three forces enter into everything. In many cases we understand the need for two forces — that one force cannot create an action, that there is action and resistance. But generally we are not aware of the third force. This is connected with the state of our being, the state of our consciousness. In another state we would be aware of it in many cases where we do not see it now. Sometimes we can find examples of the third force in ordinary scientific study — for example, in chemistry and in biology we can find the necessity for a third force in the creation of events and phenomena.
We begin with the study of psychology. Later we shall talk more about three forces and we may find some examples of their interaction. But it is better to be prepared and get accustomed now to the idea of the need to study these three forces.
Law of Seven
The Law of Seven must also be described briefly. It means that no process in the world goes without interruptions.
In order to understand the meaning of this law it is necessary to regard the Universe as consisting of vibrations. Suppose something begins to vibrate at 1000 vibrations a second and the frequency increases to 2000 vibrations a second. This period is called an octave, because this law was applied to music and the period was divided into seven notes and a repetition of the first note. The octave, particularly the major octave, is really a picture or formula of a cosmic law because, in common arrangements, within one octave there are two moments when octaves slow down by themselves. Vibrations do not develop regularly. In the major octave this is shown by the missing semi-tones; that is why we are told that it is a picture of a cosmic law; but this law is not restricted to music.
The reason why it is necessary to understand the Law of Seven is that it plays a very important part in all events. If there were no Law of Seven, everything in the world would go directly to its final conclusion; but because of this law, everything deviates. For instance, if rain began it would go on without stopping; if floods began, they would cover everything; if an earthquake began it would go on indefinitely. But they stop because of the Law of Seven, because at every missing semi-tone things deviate; they do not go by straight lines.
The Law of Seven also explains why there are no straight lines in Nature. Everything in Life and in our machine is based on this law. So we shall study it in the work of our organism, because we have to study ourselves not only psychologically, not only in connection with our mental life, but also in connection with our physical life. In our physical processes we find many examples of the working of this law.
At the same time, the Law of Seven explains that, if you know how and at what moment to do it, you can give an additional shock to an octave and keep the line straight. We can observe in human activity how people start to do one thing and after some time do a quite different thing, still calling it by the first name without noticing that things have completely changed. But in personal work, particularly in work connected with this system, we must learn how to keep these octaves from deviating, how to keep a straight line. Otherwise we shall not find anything.
Misc
The Fourth Way enneagram is a mystic figure, believed to have been first published in 1947 in In Search of the Miraculous by P.D. Ouspensky. The term “enneagram” derives from two Greek words, ennea (nine) and grammos (something written or drawn). The enneagram is a nine-pointed figure usually inscribed within a circle.
Ouspensky claimed that the enneagram was part of the teachings originally presented by G.I. Gurdjieff in Russia during the First World War. Gurdjieff is quoted by Ouspensky as claiming that this form of enneagram was an ancient secret and was now being partly revealed for the first time.[1] Although no earlier publication of the Fourth Way version of the enneagram can be cited, it has been proposed that it may derive from, or be cognate to, the Jewish Tree of Life (Kabbalah) as used in Renaissance Hermeticism (which used an enneagram of three interlocking triangles, also called a nonagram)[2] or a nine-pointed figure used by the Christian medieval philosopher Ramon Llull.[2] Idries Shah, a populariser of Sufism, has claimed that the enneagram has a Sufi provenance and that it has also been long known in coded form disguised as an octagram.[3] Another claim to a Sufi provenance is offered by the Sufi Enneagram website. Robin Amis claims an Orthodox Christian origin, claiming that both Gurdijeff and Ouspensky developed their teaching with insights gained from visits to Mount Athos.[4]
This ancient approximation to Pi has been associated with Jewish mysticism. In particular, it is known in certain circles of Kabbalists such as the “Mediogegnians”, that the twenty two characters of the Hebrew language represent a complete circumference that when divided by seven (the sacred number of cycles) produces the Kabbalistic Pi , also known within this circle of practitioners as the “perfect ” Pi.
142857 is the six repeating digits of 1/7, 0.142857, and is the best-known cyclic number in base 10.[1][2][3][4] If you multiply the number by 2, 3, 4, 5, or 6, the answer will be a cyclic permutation of itself, and equivalent to 2/7, 3/7, 4/7, 5/7, or 6/7, respectively.
- 1 × 142,857 = 142,857
- 2 × 142,857 = 285,714
- 3 × 142,857 = 428,571
- 4 × 142,857 = 571,428
- 5 × 142,857 = 714,285
- 6 × 142,857 = 857,142
- 7 × 142,857 = 999,999
If you multiply by an integer bigger than 7, there is a simple process to get to a cyclic permutation of 142857. By adding the first six digits (ones through hundred thousands) to the remaining digits and repeating this process until you have only the six digits left, it will result in a cyclic permutation of 142857
- 142857 × 8 = 1142856
- 1 + 142856 = 142857
- 142857 × 815 = 116428455
- 116 + 428455 = 428571
Multiplying by a multiple of 7 will result in 999999 through this process
- 142857 × 74 = 342999657
- 342 + 999657 = 999999
If you square the last three digits and subtract the square of the first three digits, you also get back a cyclic permutation of the number.
- 8572 = 734449
- 1422 = 20164
- 734449 – 20164 = 714285
It is the repeating part in the decimal expansion of the rational number 1/7 = 0.142857. Thus, multiples of 1/7 are simply repeated copies of the corresponding multiples of 142857:
- 1 ÷ 7 = 0.142857
- 2 ÷ 7 = 0.285714
- 3 ÷ 7 = 0.428571
- 4 ÷ 7 = 0.571428
- 5 ÷ 7 = 0.714285
- 6 ÷ 7 = 0.857142
- 7 ÷ 7 = 0.999999
- 8 ÷ 7 = 1.142857
- 9 ÷ 7 = 1.285714
- 22 ÷ 7 = 3.142857, an ancient approximation to Pi.
Simply combining the digits in different ways and dividing by 7 or a multiple of 7
- 142 + 857 = 999
- 999 ÷ 7 = 142.714285
- 14 + 28 + 57 = 99
- 99 ÷ 7 = 14.142857
- 1 + 4 + 2 + 8 + 5 + 7 = 27
- 27 ÷ 7 = 3.857142
- 857 – 142 = 715
- 715 ÷ 7 = 102.142857
- 1 × 4 × 2 × 8 × 5 × 7 = 2240
- 2240 ÷ 49 = 45.714285
- 1 + 42 + 857 = 900
- 900 ÷ 7 = 128.571428
- 1 × 4 ÷ 2 × 8 ÷ 5 × 7 = 22.4
- 22.4 ÷ 49 = 0.4571428
In base 10, 142,857 is a Harshad number and a Kaprekar number.
