Hi Bruce,
Nice chart. You are using the number series known as the Fibonacci Series,
discovered by Middle Aged monk who loved matematics. The Fibonacci Series
features numbers in which each is the sum of the preceding two. Therefore,
you have to add another two units at the beginning, a "0" and a "1" to make
the series perfect. After all, the Universe had to begin with the big
yawn, and then there was One, who got lonely, so then there were Two, and
then .........
The Fibonacci Series and the Golden Ration are not the same thing. The
Ratio is pure in its geometric form. What iz interezting is how they
describe each other, becoming more precise the higher the plane of
reference. There are other places where the Golden proportion has
parallels too, but Fibonacci's numbers are the easiest to see.
An interesting thing about the ratio is that it is reciprocal. If you
divide one number into the next, you get 1.618.... If you divide it into
the previous, you get 0.618....
To really make Ralph steam, go count the spiral rows of seeds in a
sunflower. Count both left and right spirals. One will be 34 and the
other will be 55. Been there, done that. It works. Obviously you don't
believe anything you can't do yourself.
cp in bc
----- Original Message -----
From: "Bruce Marcham" <[log in to unmask]>
To: <[log in to unmask]>
Sent: Thursday, February 03, 2005 11:24 AM
Subject: Re: [BP] Golden
jc:
Have a look at this table below (somewhat corrupted when viewed in plain
text) I did on what I understand the series to be:
Golden Ratio
Ratio Angle
(degrees)
Iteration Side A Side B (A/B) (arctan A/B)
1 1 1 1 45
2 1 2 0.5 26.56505118
3 2 3 0.666666667
33.69006753
4 3 5 0.6 30.96375653
5 5 8 0.625 32.00538321
6 8 13 0.615384615
31.60750225
7 13 21 0.619047619
31.75948008
8 21 34 0.617647059
31.70142967
9 34 55 0.618181818
31.72360296
10 55 89 0.617977528
31.71513352
11 89 144 0.618055556
31.71836855
12 144 233 0.618025751
31.71713288
13 233 377 0.618037135
31.71760487
14 377 610 0.618032787
31.71742458
15 610 987 0.618034448
31.71749344
16 987 1597 0.618033813
31.71746714
17 1597 2584 0.618034056
31.71747719
18 2584 4181 0.618033963
31.71747335
19 4181 6765 0.618033999
31.71747482
20 6765 10946 0.618033985
31.71747426
21 10946 17711 0.61803399
31.71747447
22 17711 28657 0.618033988
31.71747439
23 28657 46368 0.618033989
31.71747442
24 46368 75025 0.618033989
31.71747441
25 75025 121393 0.618033989
31.71747441
26 121393 196418 0.618033989
31.71747441
27 196418 317811 0.618033989
31.71747441
28 317811 514229 0.618033989
31.71747441
29 514229 832040 0.618033989
31.71747441
30 832040 1346269 0.618033989
31.71747441
31 1346269 2178309 0.618033989
31.71747441
32 2178309 3524578 0.618033989
31.71747441
Note that it pretty quickly settles out on a ratio of 0.618 (though it does
show up as 0.61835 and 0.625 early on) and the angle is 31.72 degrees (or
31.7174744...).
I think someone made mention of the idea that maybe this series is one that
approaches a given ration when the calculation is carried out a large number
of times--the table above shows that it does.
I've also attached the calculations as an Excel spreadsheet. Excel is a
favorite tool of mine for repetitive calculations.
I note that the ratio of 1/1.618 is 0.618...
Interesting, but my mind is getting fried on this whole business.
Do you want Golden Fries with that?
BM
P.S. I again refer you to the item I googled which has some high fallutin'
derivations and constructions on the subject:
http://jwilson.coe.uga.edu/emt669/Student.Folders/Frietag.Mark/Homepage/Goldenratio/goldenratio.html
--
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uncoffee-ed, or to change your settings, go to:
<http://maelstrom.stjohns.edu/archives/bullamanka-pinheads.html>
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