Below... Regards, Bob.... -------------------------------------------------------------------- "Beer is proof that God loves us and wants us to be happy!" - Benjamin Franklin
From: "Dr E D F Williams" <[EMAIL PROTECTED]> > It only leads to the 'Golden Section' because you want it to Herb. Nature > does not obey numbers! > There is nothing special about those numbers at all. This is simply not true. Nature obeys all sorts of numbers. All nature is subject to the basic constants of the universe. The numbers may be considered special in that any variation in them would result in a very different universe. Everything, you me, nature are subject to and constrained by these numbers. If any item in nature grows and increases in complexity as it grows, (say a tree) and the mature looks like the juvinal, the branching must on average follow the "Golden Section". The number of seed spirals in a sunflower will always be a Fibonacci number. Living nature picks or obeys certain mathematical formations because evolution has discarded others through competition, lower effeciency of propagation or lack of robustness regarding survival. All DNA is subject to mathematical constraints resulting from geometries of the molecules making up the DNA which are in turn dictated by the mathematics of the geometry governing their individual atoms which is inturn the result of several of these universal constants. This results in a spiral of a spiral that compacts an amazing amount of information in an extremely small space and which can still be unzipped like a zipper to replicate a gene or code a protein. This paticular pattern exists in all living things because this geometry, resulting from fundamental constants is the only one that nature here on earth has found to work. All undamaged snowflakes are hexagonal for a reason. There is a "magic" number in a water molecule, 2/3pi. This does not mean that the "Golden Ratio" is some most pleasing form to humans and I've offered no opinion on this. It presumes connections we cannot prove. Nevertheless, Fibonacci numbers do show up in nature and there is a reason why they do. To say that there is nothing naturally or especially "pleasing" about the "Golden Ratio" to at least some humans is probably arrogant. Keep in mind that the "Golden ratio" is an unique geometric construction like pi, not some number picked from thin air. > But there may well be something very special about a thing they may have > been used to describe. > > There are many ways in which a picture may be presented in a pleasing way. > The 'golden' way is only one example. We all know that numbers are among the > symbols of a special universal language called Mathematics. The numbers > themselves have no special quality. Pi is an exceptionally special number, and without knowledge of it our entire civilation would be back to flaking rocks, attaching them to sticks and spearing animals for dinner. There are a whole host of "special numbers" that lie behind who we are, and without knowledge and use of them you would not be taking photos or typing on your keyboard. The discovery of each of these numbers has been as much a milestone of civilation as the invention of the wheel. > Games have been played with these > symbols for a long time. Thousands of books have been written, and read too, > on how numbers affect our lives. I saw, recently, a book about the secret > code of the bible - number nonsense taken to an extreme. The only valid > statements that can be made about it is that it is a book, printed on paper > and seems to have made money for the author and publisher. But it is only > one of a long string of them going back for decades. No one is trying to "divine" secrets here. Folks have been making observances here and also discussing the observances of those that came before them. Sounds like scientific endeavor to me. > When you say that these special numbers occur in nature what you're actually > saying is that they have some kind of magical or special aesthetic quality. No, so far as I can tell, folks have made observations of their own and referred to those who came before them who made observations. No one here is referring to magic. > Yes? Its the other way around. The numbers derive from the way nature is > arranged. Looking at an X-ray diffraction pattern, or the arrangement of > atoms in an electron micrograph, or the number of electrons in the shells > about an atom, virus particles, or the incredible DNA molecule. Or even > counting the number of coils in a sea-snail shell, or measuring snow > crystals, and finally dividing, multiplying, solving quadratics, > differentiating, integrating, and ending with some numbers that you conclude > represent some kind of a 'golden rule of nature' is comparable to the secret > worship of numbers by the Pythagoreans. Nonsense! The numbers do NOT derive from looking at an X-ray diffraction pattern, or from any other diffraction pattern for that matter. Diffraction patterns are described by Maxwell's Equasions for propagation of electromagetic waves. These are Law. The only special numbers used are pi, e, and c (speed of light). None of these numbers were derived by observing diffraction patterns. In other words, nature IS constrained by these special numbers. In the first paragraph I described how your other examples discussed here exist as they do because of "special numbers". The 'golden ratio' is a geometric definition in much the same way as pi is defined as the ratio of the circumfrence of a circle to it's diameter. Phi, the 'golden ratio' is defined as "the ratio obtained if a line is divided so that the length of the shorter segment is in the same proportion to that of the longer segment as the length of the longer segment is to the entire line." This is why it pertains to growth in nature. or the arrangement of atoms in an electron micrograph > There is no example of a golden section in nature. The golden ratio exists all through nature selected by evolution for reasons of efficiency and conservation energy. > For every one you can > make fit the rule by manipulating it into the shape of a rectangle, The "Golden rectangle" is not the golden ratio. The golden rectangle is a rectangle where the ratio of two adjacent sides is the golden ratio. > millions > can be found that simply don't match. And I add, although this might be a > little out of place here, during the last 45 years or so pictorial > 'fractals' have been appended to the 'number magic' quiver. The 'Golden > Section' has no more aesthetic validity than the universal magic word > abracadabra. Perhaps, but the pronouncement is arrogant. The most telling evidence of a tendency for selection of pleasing photographic format within our society would be to develop the probably density function of height to width of acclaimed art, especially that which involves cropping without regard to paper format. It would be interesting to see if it has two peaks, one around 0.62 and the other around 1.62. One might then still argue that this is due to some sort of conditioning in society, but then that's irrelevant isn't? It's still what they prefer and what they found pleasing. [further anecdotal evidence deleted] > But as I said, this is really quite a load and will lead nowhere. Some > members will go so far as to post with ~other member's names~ in the subject > line. You are correct to complain about this.
