New Civilization News: Phi is the constant of Self-Similarity    
 Phi is the constant of Self-Similarity2 comments
13 Apr 2008 @ 09:47, by John Grieve

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.

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