John Grieve: Phi, the Pentagon and Self-Similarity    
 Phi, the Pentagon and Self-Similarity2 comments
3 Jun 2008 @ 09:12, by John Grieve

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.

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13 Aug 2008 @ 08:12 by erlefrayne : Synthetic Treatment Appreciated
A synthesis of Eastern and Western, exoteric science and mystical science, like 'tao of physics'. A Oneness of Mathematics, not dichotomized mathematics. Well appreciated, Partner.  

17 Nov 2008 @ 09:11 by Vincent @ : Strong points of mysticism...
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