John Grieve

Phi, the Pentagon and Self-Similarity

Tue, 3 Jun 2008 09:12:31 GMT

Phi, The Pentagon and Self-Similarity

It is my contention that there is just one unified knowledge, there is just one mathematics, not straight mathematics versus mystical mathematics, or straight geometry versus sacred geometry, just one science which is the union of the strong points of mysticism with the strong points of quantitative science.

To this end I am putting forward the assertion that Phi (1.61803.) is the constant of Self-similarity and that they are always found together.

I have previously shown this to be the case where we have the Fibonacci sequence and also the Pythagorean theorem ( through the use of self-similar right-angled triangles). Now I will add to this by demonstrating that the so-called mystical pentagon exhibits self-similarity, and that the well-known prevalence of Phi in anything connected with pentagons, pentacles or pentagrams is connected with this fact and not anything other-worldly.

Phi (1.61803..) is a very unusual ratio and is demonstrated by Euclid in his Elements where he calls it division in the extreme and mean ratio. What this means is that in any straight line, if you cut it at a certain point where the ratio of the whole to the greater part is exactly the same as the ratio of the greater part to the smaller part, that ratio will be the irrational number Phi, approximately 1.61803989. Then Euclid goes on to use this line to construct a regular pentagon and the regular solids based on it. All the remarkable properties of pentagons are based on the fact that the diagonals of a pentagon cut each other in this ratio of Phi. What has been overlooked in all this is that if you draw all the diagonals of such a pentagon, then they cut each other in this ratio and they create in the middle of the previous pentagon an exactly self-similar pentagon, rotated 180 degrees, whose sides are smaller than the larger pentagon by a factor of Phi squared. This is well-known but its significance seems to be ignored. Clearly if we construct diagonals of this smaller pentagon we create a smaller one and so on ad infinitum. This is clearly a case of self-similarity as I have sought to demonstrate as the basis of Phi. I would also speculate, though this is unproven, that the expression for Phi =(1 + SquareRoot(5)) / 2 is but a specific example of a general formula which links the number of sides of a polygon, in this case 5, to the ratio of the intersecting diagonals of that polygon, whether it has 5, 6, 7 or more sides. In other words this remarkable ratio is produced by ordinary mathematical processes, which need to be explained and clarified and not mystified. The real deep mysteries in all this are self-similarity,fractals, chaos and complexity theory which as yet are imperfectly understood.

Pascal's Triangle, Self-similarity and Phi

Wed, 23 Apr 2008 09:25:52 GMT

Pascals Triangle, Self-Similarity and Phi

In maths the simple operation of adding two consecutive elements in a sequence and then iterating, which process is well-known to us in the Fibonacci sequence, leads to many of the more remarkable properties we come across in nature and mathematics. The Fibonacci sequence, as I pointed out in my last article, is based on self-similarity and exhibits the mystical number and proportion called the Sacred Ratio, approximately 1.618 which is an irrational number.

Now something very similar occurs in that very famous table of numbers, known to the ancient Chinese, but known to us as Pascals Triangle. This is a symmetrical table, with ones at the apex and at each edge, with the intervening numbers created by adding together the two numbers directly above and to either side. Pascals triangle has a multitude, maybe indeed an infinite number of remarkable properties. Every interesting thing in mathematics more or less, can be found in different ways in this pyramid. Fibonacci itself, can be found in sequence if you add the short diagonals. This of course yields Phi, the Golden Proportion. However, this is a bit misleading, because Phi is conspicuously absent from the other patterns you will find in this triangle. This is because if you divide any two of the numbers in the table they will be a rational fraction, not irrational. Adding different numbers together, as in the Fibonacci example, is the only way to get a sequence which gives Phi. The whole structure is based on the iterative technique mentioned above, and I suspect that this technique is a cornerstone of self-similarity, though I cant demonstrate it as convincingly here as I did in my previous article on Fibonacci.

I believe, since Nature produces Phi all over the place, and Fibonacci sequences in the number of petals of flowers and the spirals of shells, that at an early stage in the evolution of life, in plant RNA and animal DNA, the simple iterative technique I refer to, was encoded and passed down to following generations. Thus we find Phi everywhere in Nature. Why it leads to very remarkable properties in mathematics is another issue and one I will address in a later article.

Phi is the constant of Self-Similarity

Sun, 13 Apr 2008 09:47:07 GMT

Mysticism and Science: A new Union

1.Phi is the constant of self-similarity

It is my belief that the way forward involves the coming together of mysticism and science, to give a new holistic discipline which will combine the quantitative strengths of science with the holistic and qualitative strengths of mysticism.

What I am writing is not sacred geometry or sacred mathematics, but just plain true knowledge. My first assertion is that the number or ratio Phi, known from antiquity, is the constant of self-similarity. Let me illustrate this with a simple numerical example. Take the Fibonacci sequence

1 1 2 3 5 8 13 21 34 55 89 144

each number is the sum of the previous two terms, thus 8 = 5 + 3. It is very significant that if you divide any term by the immediately preceding term you derive a fraction which is alternately greater then less than Phi, and which quickly closely approximates its value of 1.618. For example 89 divided by 55 is 1.61818, whereas 55 divided by 34 ( the preceding number) is 1.617647058

Now, the next thing to observe is that this sequence is self-similar:

1 1 2 3 5 8 13 21 34 55 89 144

0 1 1 2 3 5 8 13 21
taking away each previous term, in sequence leaves a sequence which is identical to the original one. The whole thing appears to be nested and self-similar, and this process can be repeated ad infinitum.

Now let us look at the famous right-angled triangle and Pythagoras theorem. There are, it seems, hundreds of valid proofs of the theorem that the square on the hypoteneuse is equal to the sum of the squares on the other two sides. One of the least well-known of these proofs is the one which uses the fact that the two small right-angled triangles formed when you drop a perpendicular from the original right-angle to the opposite hypoteneuse, are similar to each other and to the larger triangle. In other words this is another example of self-similarity. It is possible to construct spirals ad infinitum around the vertices of the ensuing smaller and smaller right-angled triangles which you can construct within these two triangles. And where you find equiangular spirals you will always find the ratio Phi, approximately 1.618

There are many other sequences e.g. the Lucas sequence, which like the Fibonacci show the Phi ratio, and they always display a form of self-similarity. I will leave it to you to investigate.

Phi is indeed the constant of self-similarity.

CyberEnlightenment, "Political Correctness" and Fascism

Wed, 9 Apr 2008 09:25:18 GMT

CyberEnlightenment, Political Correctness and Fascism

The Enlightenment which issued from the declining feudal society of the mid-18th century and which culminated in the French Revolution and the transformation of Europe is with us again. It has returned in a new different form.
Anything which is partial, incomplete and fragmentary, if it is raised up and made into a totality, a false totality and a false whole, becomes demonic and very dangerous. This is what happened to the Enlightenment. Here, the rational faculty, at the expense of feeling, emotions and intuition, was raised up to the heights and made into Everything; which it of course is not. The result was very predictable. What is thrown out by the front door comes back unannounced through the back door. All the dark, irrational forces of emotion, Unconscious etc. exerted themselves and burst forth in the French Terror, Nazi Germany , Stalinism and all the other horrors of recent times. So much for the past.
But I assert the same thing is happening now. New technological developments in communications and information technology over the past 50 years have created, and still create, a new hyper-rationality, hyper-logicality which I call the Cyber Enlightenment. Evidence of this is all around us. Apart from personal things like obsession with computers and programming and logical ways of doing things, there are disturbing trends at the social and political levels. I am talking about the situation in England but I daresay the same thing is happening in other parts of the world, most notably the USA. The biggest example of what I am referring to is Political Correctness. This has gone from a tiny seed of 15 to 20 years ago, flourishing in the last ten years of the present Labour government and threatens to become an ideology as entrenched in our society as Stalinism was in the Soviet Union. A certain amount of Political Correctness is fine, conducive to showing respect to minorities and other groups, but it has become an overweening power in the land and threatens to become a major plank of some sort of totalitarian ideology which is a stealthy restraint on our liberty. It is now impossible to mention publicly or in private very many pressing problems and iniquities in our society without being accused of political incorrectness and worse. I repeat, when the truth cannot be expressed in this way, it is a restraint on our liberty.
Other planks of this emerging soft totalitarianism are the nanny state, Health and Safety, omnipresent bureaucracy and stealth taxes. We are all too familiar with all these things to need any detailed analysis and elaboration. These are all examples of Hyper-Rationality.
I believe that just as the 18th century Enlightenment was followed by great turbulence and a descent into darkness, similarly we are heading on the same trajectory now unless something is done to change course.

When someone criticises a situation they are often asked if they have anything better to offer themselves, with the implication that if they havent then they should shut up. Well, I do have something to offer which I believe is superior to this CyberEnlightenment and CyberSavagery which are closing in on us.

The only thing which can be raised up as a totality is a whole; is a totality. A real totality and not a false one, which is really only half of something, as the Reason is only half of us.

God/dess is the only real totality but on the human level a whole person is a real totality. A person who combines Yin and Yang, male and female, good and bad, intellect and feeling, heart and mind, Body and Soul and so on. Such a person blows away the cobwebs of political correctness like a whirlwind. That is the only answer to the current ills which threaten us, and such a person will say a lot of things, different aspects of the truth, that are unwelcome to more conventional people. I am talking here about self-realisation or Enlightenment. Real Enlightenment, not just of the intellect. And not just of one person but of the whole of society. That is the solution to all the problems that are facing the world today.

Dialectical Analysis of the Sudoku Puzzle

Mon, 7 Apr 2008 10:59:08 GMT

Dialectical analysis of the Sudoku puzzle

A sudoku is a puzzle composed of 81 individual squares or cells, which join together into 9 larger squares of nine cells each. It can be looked at also as an array of nine columns and nine rows, each composed of nine cells.

The rules are what defines a game or puzzle and they create the contradictions in it. The basic rule of sudoku is that all the numbers 1 to 9, in any order, have to be in every row, column and square. But the defining rule or characteristic is that no number can recur in any row, column or square. Thus the main opposites here are Identity and Difference

This puzzle is constructed as a logical deduction puzzle whereby it should be possible to deduce all the absent numbers from the ones that are given, one by one, serially and piecemeal.

Now, just as the logical deductive process focuses on the rule that the numbers must be different, and uses that fact to deduce the outstanding numbers in the grid, so contrariwise, the dialectical method uses the opposite fact that in some parts of the grid, not subject to the rules, the numbers will be the same, will recur. This will give rise to certain characteristic patterns which can be used to intuit, guess and deduce the complete picture, holistically.

There are many patterns that arise in this way, but I will only mention three to illustrate my point.

Firstly, numbers cannot recur on rows, columns or in larger squares because of the rules. But they can and do recur when counted along diagonal lines. Thus on a diagonal across the whole grid it is sometimes possible to count three or four recurrences of the same number for example 4 or 7. Every diagonal is of a different length and there are sub-patterns within them that will reward study.

Secondly, what I call knights-move pattern. In many cases this pattern, created directly by the rules as a natural consequence of them, follows exactly the move of a knight in chess, and the same number recurs in the next column or row in this position. However, the rule that no number can recur in a larger square sometimes affects this pattern, and the same number recurs in a slight variation of the true knights-move, in an extended version.

Thirdly, it is noticeable that when a number is in a small square within a larger square, as we move from one column to the next, or one row to the next, the position of the same number is usually different, relatively, but not always. Thus, for example, in the top large square a 3 may be in the top left-most small square. In the next large square underneath it may be in the middle small square, while in the bottom large square it may be that it is in the right-most small square.

These and other patterns can be observed and learnt. When doing a sudoku you can use knowledge of these patterns to intuit, guess and deduce the overall distribution of the numbers throughout the whole puzzle, rather than just using a mechanical, solely deductive process to solve the puzzle piecemeal. These are examples of the Taoist and Confucianist methods of solving problems that I have written about elsewhere. Of course, it is possible to combine the two methods to get the best result.

The Social Unconscious,Civilization and Sexuality

Sat, 29 Mar 2008 09:32:05 GMT

Social Unconscious, Civilization and Sexuality

The rational mind was the great celebrated faculty of the Enlightenment era, after which came the French revolution and Terror, and even worse things in the 20th century. All this was followed by the scientific elaboration of the notion of the Unconscious mind by Freud, just over one hundred years later which threw rationalism into turmoil.
Just as the individual has a rational or conscious mind, and also an Unconscious, so too, I now propose, does society as a whole. The analogy of society/individual is an old one, but I think this comparison of a social conscious and a Social Unconscious is new. Certainly society tries to be very rational with its theories, sciences, laws, rules and regulations and so on, and this corresponds naturally to the rational mind of the individual. What of the Social Unconscious, what does this consist of ? We are at an introductory stage in the study of this subject and I now propose three areas to look at, though I am sure that in due course more will come to light. These areas are Civilization, Sexuality and a structure in society which I call the Pyramid or triangle, which I will not be elaborating in this piece but will cover at length in an essay dedicated solely to it.

Civilization is a very strange subject and area of study. In one way it is a specific form of society as delineated by anthropologists, Marxists etc. but in another way it is a thing about which people have very personal and emotional, even irrational views and they feel very threatened by any criticisms of it to the extent of accusing you of wanting to destroy it if you make anything in the nature of a critique of it. Furthermore, there is no proper study of this subject. It is very broad and considered too much so for any one academic discipline. It is the area of dilettantes like Freud who wrote Civilization and its Discontents and many other writers who werent specialists in this field. Every subject under the sun has an academic discipline devoted to it, except this one. Even though civilization is presently under great threat from climate change and other dangers, and every civilization that has ever existed has ultimately collapsed, there is still an incredible reluctance on the part of most people to talk about the subject, discuss it rationally And make plans to save it from destruction. All of these things, and I could say more, suggest to me that this subject of civilization is part of the aforementioned Social Unconscious. To say that society and people have a total blindspot to this subject is a vast understatement. Like many blindspots and neuroses and states of denial it can prove fatal if not brought to light and subjected to therapeutic treatment.

Much the same can be said about the subject of sexuality. One can discuss it rationally but many people cant, and seem very ill at ease in any proximity to the subject. I have sometimes tried to discuss the ideas of that genius Oscar Wilde with seemingly intelligent people, only to be told I dont want to talk about that homosexual. Obviously their own problems with their own sexuality makes it impossible for them to even talk about any subject even approximately close to their difficult feelings. Its not just individuals who have these problems with sexuality, society as a whole has taboos, no-go areas, evasions, distortions, denials, hypocrisies and double-standards of all sorts and that too suggests that sexuality is part of the Social Unconscious.

In a later writing I will look at a structure in society which seems to be unknown to social and political scientists, which in itself suggests that it may be part of this Social Unconscious, and which I have termed the Pyramid or Triangle.

Just as neurotic individuals need to do therapy to become healthier people, so our society needs to do some social therapy. It needs to look at the subject of civilization closely and discuss it widely. The same with the taboo areas of sexuality and all the connected issues. This is not an option but a necessity.

Book of Aphorisms and Commonplaces

Tue, 25 Mar 2008 10:01:20 GMT

Commonplaces

#1) Men make counterfeit money;
in many more cases, money makes counterfeit men

Sydney J. Harris

Aphorisms

#1) Mind over Matter
Matter UnderMined

#2) There is an art of Politics but politics is not Art
Politicians tell the truth or half-truth in order to lie
Art tells lies in order to speak the truth

#3) The evolution of the Butterfly shows the survival of the flittest

#4) Man (or woman) may be "the measure of all things", but only of those things that are measurable. They are not the measure of those other things which are beyond Quantity and Measure.

#5) The Best things in life are free. The Good things in life are quite expensive.

Taoism and Confucianism : Part II

Mon, 3 Mar 2008 12:50:48 GMT

Problem 3: How to bring fresh water from a nearby river to a small Chinese town.
This example is given by Joseph Needham in his book Science and Civilization in China as illustrating the main differences between Confucianist and Taoist approaches to doing things. The problem is how to provide fresh water to a small town from a nearby river. The Confucianist approach would be to divert water from the river at a convenient point well below the town and then use much man-power, mechanical devices and expense to lift it up again to the level of the town and so distribute it. This will solve the problem while generating employment, circulating money, aiding the invention and perfection of mechanical devices and generally keeping everybody busy. The Taoist approach would be to divert the water at a convenient point above the level of the town and using the natural tendency of water to find its own level, that is to run downwards, create an aqueduct to lead the water to the village or town and so distribute it. This method uses natures properties to move the water rather than human effort, creates much less employment (a one-off aqueduct rather than constant lifting of water), circulates less money and does not need inventions or mechanical devices.

This example clearly reveals that Confucianism is a social philosophy, and its solutions to problems are designed to benefit society in creating wealth, employment and invention. The Taoist method is one which is based on the individual, addresses the essence of the problem rather than appearances, and is in tune with nature. Instead of expending effort in lifting water upwards after we have allowed it to run downwards, we just divert it at the right point to run by its own momentum (with gravitys help) into the town.

Conclusion
Clearly, there are simple, easy ways to solve problems and there are difficult, tedious ways. It seems very much the case that civilized society, with its obsession with externals, appearances and irrelevant details prefers the difficult way to do things. This is in keeping with its greatly over-yang nature. Over-Yang means giant, mechanical, crude, external, superficial and so on but most importantly, over-masculine. There is a clear link between the problems in our society and problems in our psyches concerning sexuality and gender. The solution must be to redress the balance and level-off with an equal emphasis on Yang and Yin values. Yin, after all, represents the small, the inner, the subtle, the essence and naturally, the feminine. If we are in harmony then society is in harmony.

It is my belief that every problem, whether it is Fermats Last Theorem, or CERNs accelerator, or getting cheap energy through FUSION, has a simple solution as well as a difficult, complicated one. But our societys obsession with doing things the Confucianist way, in order to create wealth and employment and inventions, means that people have forgotten, to a large extent, the ancient Taoist (and universal) approach that seeks simple, easy and cost-effective solutions to difficult problems. People just dont believe that there are simple solutions to many of these problems. If the experts cant solve them then they must be impossible, they think. But maybe the experts are looking in the wrong place, and the wrong way, and from the wrong perspective.

I am not asking people to abandon completely the Confucianist approach, which is so engrained in all of us, particularly men, by our upbringing and education. All I ask is for the imbalance to be less completely one-sided and total.

Ideally we should use both techniques to solve difficult problems, both the Yin and the Yang, both the left brain and the right brain, both Taoist and Confucianist.

Taoism And Confucianism: Part I

Mon, 3 Mar 2008 12:47:15 GMT

Taoism and Confucianism

Confucianism and Taoism are seemingly opposite but also complementary ways of approaching life and solving its problems. They are world-views which originated in China but that are both universal. Confucianism is named after the Chinese Sage Confucius (Kung Fu Tse) who incidentally was not a Confucianist, and developed a moral and ethical philosophy, six centuries before Christ. Taoism is the nature mysticism local to China and is far more aimed at individual development and enlightenment than social. Both are universal in that Confucianism is a way of doing things which comes naturally to civilized, organised, bureaucratised and conformist (official) societies world-wide, while Taoism is part of the perennial philosophy and is the personal ideology of the individual, the lover, the eccentric, the spiritual searcher after truth, the rebel and the seeker of simple solutions to lifes problems and union with Nature.

In this paper I will briefly outline a few problems of a simple kind and present Confucianist and Taoist treatments of their solutions.

Problem 1: Opening a vacuum-held jar lid.
The Confucianist approach to getting a top off from a jar, held tight by a vacuum, is primarily to apply brute force externally to the lid and force it off. This is done with either the hands or a mechanical device, a lever. With small jars this is a practical technique but with larger and larger jars and lids, it becomes exceedingly difficult and tedious.

The Taoist technique is to release the vacuum. This addresses the essence, not the externals of the problem because inside, the lid is held on by the power of vacuum in the top of the jar. The vacuum is released by inserting a wedge sort of object, it could be a spoon or other strong object, between the side edge of the lid and the jar top. Applying a small amount of pressure causes the two to separate allowing air into the top of the jar. The lid then unscrews with the minimum of effort.

Comparing the two techniques, it is clear that the Confucianist method uses a cumbersome, mechanical approach which does not address the essence of the problem. With large jars it would be completely inappropriate and impractical. On the other hand the Taoist method is quick, simple, costs nothing and is exactly the same whether you are dealing with a tiny jar or a large one. The Taoist technique is geared to the inner essence of the problem which the Confucianist method ignores, looking only at external superficial appearances in trying to deal with the problem.

The second problem 2: Calculating the number of games played in a knock-out tournament.
This simple mathematical problem is given by Edward de Bono in his book on lateral thinking as an illustration not of Confucianism and Taoism but of his technique at lateral thinking. A teacher wants to keep a class busy for half-an-hour with a simple, mechanical but tedious sequence of calculations that lead to the desired answer. However, one bright student, using lateral thinking solves the problem in a matter of seconds. What is the problem? Imagine a football or chess tournament which has 16 teams or players in it. How many individual games need to be played before a winner emerges? The Confucianist technique is to calculate the number of games piecemeal. First there are 8 pairs of teams, then when the winners go through this becomes 4 pairs, then 2 pairs and finally one pair in the final. The total number of games then becomes (for a tournament comprising 16 teams), 8 + 4 + 2 + 1 = 15. So the answer is 15.

The lateral thinker or Taoist uses a different approach. Instead of concentrating on games won he or she looks at the opposite, that is games lost. With clear intuition he realises that in the tournament all teams or players will lose a game except the final winner. So the number of games lost is the same as the number of games won, except it is much easier to calculate. If there are 16 teams in the tournament then 15 losses occur = 16 1. Both techniques arrive at the same answer but one is much more simple and quick than the other. The Taoist technique takes seconds and it doesnt matter whether there are 16 teams in the tournament or 256. The method is essentially the same. If there were 256 teams in the tournament the number of games played (lost) would be 256 1 = 255. Using the Confucianist technique to calculate the numbers at each stage of the tournament piecemeal would take hours in this case.

So in this example, we see slightly different aspects of the difference between the two techniques. The Confucianist method is slow, laborious and we suspect, primarily designed, in this case, to keep a class busy and out of mischief for half an hour. The Taoist method goes straight to the heart of the problem and instead of being misled by societys obsession with winning it solves the riddle by looking at the key to the problem which is that the number of games lost is equal to the number of games played. All that can be said in favour of the Confucianist approach is that it is useful to society and in generating the result generates additional statistics and data about the different stages of the tournament and finally suggests a mathematical relationship between the sums at the powers of 2 (8 + 4 + 2 + 1 = 2^4 1).

Fermat's Last Theorem

Sun, 24 Feb 2008 09:25:57 GMT

Two Conjectures concerning Fermats Last Theorem

For most of my life, I have been an amateur mathematician and interested in the problem called Fermats Last Theorem. Like most other people I was delighted when Prof. Wiles came up with his proof in 1994 and I only regretted that I was not able to understand the advanced mathematics which it contained. I have always believed that Fermat had a proof of his own and that it has to be based on the mathematics available in his time. So I continued to struggle with this problem.

My recent researches have led me to two interconnected hypotheses which, if proven, lead to a solution of this problem and which arguably could be the same approach which Fermat used.
My first conjecture concerns the equation:

X^3 + Y^3 = Z^3

If we assume that there are indeed two cubes which when added together equal a third cube then we realize that since a cube has six sides, each of which is a square, then we have the interesting fact that the first cube has a square equal to X^2, the second cube has a square equal to Y^2 and the third cube has a square equal to Z^2. This obvious fact leads me to a question which I believe is original, as I have never come across it in conversation with other mathematicians or seen it in the literature. Is it possible, I ask myself, that given the hypothetical existence of these sides x, y, and z and these squares X^2, Y^2 and Z^2, that there is a definite relationship between these three squares.
The surface area of the first cube is 6x^2, that of the second cube is 6y^2 and that of the third cube 6z^2.It is clear to me that since cubes are based on squares and squares are based on Pythagorean triangles, that it is the underlying Pythagorean triangles which make it impossible for two cubes to add to another cube or two fourth powers to add to another fourth power and so on. In fact I would say that here X^2 + Y^2 = Z^2 though this implies a contradiction.

This is paradoxical and clearly impossible. Prof. Wiles, whatever the merits or demerits of his proposed solution, does nothing to clarify the amazing contradiction at the heart of the Fermat problem.

If this is indeed so, the fact that X, Y and Z must be Pythagorean triples leads me to my next conjecture. I believe, though I cannot prove, that the same relationship between N=2 and N=3 in the Fermat equation
X^N + Y^N = Z^N

Exists between N=3 and N=4 and in general between any two consecutive powers. In other words, the values which satisfy the equation for any N must also satisfy the equation for N+1. So the solutions are like a set of Russian dolls or nested boxes in that each is contained in the previous one. Since we know that there are no solutions for N=3 then establishing these two conjectures would immediately prove Fermats Last Theorem.