******Dead Greek Guys, etc.******
Hi Pat,
After checking a few encyclopedias and only finding the Philosohy of The Golden Mean espoused bt Aristotle aI was abhout to give up when I saw a copy of "Sacred Geometry", an Astro book of my wife's which happened to be on my desk.
Naturally, it is in there but the only reference that I can find as to the Origin of the School of Thought is a vague reference to the beliefs of the Pythagoreans so in one way that supports my thinking that Fibonacci built on the work of Pythagoras.
You are right about the origin of the golden mean being officially traced back to the Pythagoreans. Some might say that the idea was well understood before them by the Egyptians, Babylonians, and Sumerians, as is demonstrated in some of their architecture..... but the direct connection is sort of vague, imo. Moreover, if the earlier guys did know of it, they did a poor job of conveying that in their records, at least the extant records. My guess would be that the guys before the Pythagoreans knew it as a good aesthetic building ratio to use, and maybe even for use in art too, but did not delve much into the other natural occurrences of the #'s.... BWDIK. Back at home, I have some books which talk about this.... although some have been lent out, and I forget if I ever got them back.... I will check when I can. The thing is I doubt the Greeks ever knew of it as the "the golden mean" so to speak. Euclid and others mention it as the mean-extreme ratio..... and it's probably safe to assume that the divine proportion term was used much later too.
The Greeks had no symbols for #'s, as we think of them (e.g 1,2,3, etc.). That was fine with respect to rational numbers. Instead of Greek mathematicians representing #'s as symbols (as the Egyptians and Babylonians did), they thought of # as a ratio of lengths. This is slightly similar to the Egyptian system of counting, which revolved around 1. To the Egyptians, unity was the largest #.... therefore, what we would think of as 1000, would be thought of by Egyptians as 1/1000 since unity is derived from one, and unity is supreme. Counting was done in ratios.... to the Egyptians, it was ratios from unity.... to the Greeks, it was ratios of lengths.
Concomitant to the Greeks' notion of #'s being thought of as ratios of lengths was their assumption that these ratios were commensurable. For example, with a basic monad, a group of three units to two units could be thought of as 3:2 or 2:3 since three groups of two units equals two groups of three units. Put simply, the sh*t hit the fan when they figured out that some pairs of lengths were not commensurable (i.e. they found some irrational #'s). In particular, the Pythagoreans found that the ratio of a diagonal of a square and the diagonal of a pentagon, to their respective sides were incommensurable ratios: (sqroot 2):1 and -phi- (1.618...):1. Using similar geometry, -phi- is figured to be the {(sqroot 5)+1}/2. This discovery called into question the foundation of the Greek counting system, so the Pythagoreans resolved this issue by banishing anyone who revealed the secret. Being a rather religious bunch (they would probably be considered a cult by today's standards<g>), they deduced that this "irregularity" was the work of the gods.
The basic formula later derived was A:B = B:(A+B) a reciprocal relationship between two unequal parts of the whole, where the small part is in the same proportion to the large part, as the large part is to the whole. The golden mean term sounds like it came from the Renaissance, but that's just a guess on my part.... ditto for the divine proportion.
Leonardo de Pisa did a lot of math stuff in his day too, not the least of which was popularizing the Arabic (our modern) counting system within Europe -I won't ramble on about it <ggg>, but he also had the proverbial rabbit reproduction problem, from which we get the Fibonacci series. The relationship between the golden mean and the fibonacci series is the following..... any # in the fibonacci series divided by the following one approximates (i.e. converges on) 0.618... and any # in the fibonacci series divided by the previous one approximates 1.618..., these being the proportional rates between minor and major parts of the golden section (a.k.a. phi).
Put another way, if we are to assume that mathematics is the language of nature, then I would make the analogy that the golden section and the fibonacci series, are sort of like snow and sleet. Both are water in the forms of precipitation, beyond that, the rest is details and semantics. But they represent the same thing.... in essence. I am basing my ramblings on some books I've read, as well as vague memories of tangential discussions by one of my greek professors in high school on greek mathematical history. The names of the books that immediately come to mind are "connections", "the power of limits" and "the geometry of art and life", all of which I forget the authors' names..... there might be a few other books too.
edit: in doing a spell check, i called up a metacrawler window, and did a quick search.... these links sort of rhyme with the version that I remember.... for all I know, I may have overlooked something. FWIW.... most people may well already know these things.
fibonacci links-
inil.com
lib.virginia.edu
pythagoras- golden mean link
perseus.tufts.edu
GEEZ, I bored myself with this post..... sorry about that. <ggggggg> That's about all for this ¿ñºå¢æºçÑ¿¿ addled brain of mine. <g> TTYL, take care
Regards,
Frank |
