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Date: | Thu, 3 Feb 2005 14:24:33 -0500 |
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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
-----Original Message-----
From: Pre-patinated plastic gumby block w/ coin slot
[mailto:[log in to unmask]]On Behalf Of John
Callan
Sent: Thursday, February 03, 2005 9:21 AM
To: [log in to unmask]
Subject: [BP] Golden
Cuyler,
I may be doing something wrong, but I keep coming up with 1-1/4". When
I went through the A/C=B/A routine I came up with .61538xxx and
.625xxx. We are talking about a difference of 0.01+/-. At this scale,
3"+/-, its a difference a craftsperson, can deal with it, probably with
sandpaper. If I needed to work at microscopic level, I might want to
figure out how to increase the accuracy. If I'm working on the
preliminary design for an airport this might be too much accuracy.
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