23 Apr 2008 @ 09:25, by John Grieve
Pascal’s 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 Pascal’s 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. Pascal’s 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 can’t 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.
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