Thursday, June 15, 2006

The Pearl Poet and Condren's Golden Section

Thanks to Wikipedia, I now have a greater understanding of the famous Golden Ratio and some questions for Professor Condren (not that he's reading my blog).

On the left above, you can see Wikipedia's image of a rectangle containing (almost) the Golden Ratio:
The dimensions of the gray rectangle are in the ratio 144/89 = 1.6180, very nearly the golden ratio, thought by many to be a most pleasing ratio for a rectangle.
On the right above, you can see Michael Lahanas's image of a line whose segments show the Golden Ratio, which is explained as follows:
A line is divided into two segments AG and GB. The entire line AB is to the AG segment as AG is to the GB segment.
The Golden Ratio is an irrational number that is today expressed by the following formula:

I hope that this is visible, but if not, go to this image on Wikipedia if you haven't already done so.

As an irrational number, the Golden Ratio can be given a decimal expansion for as far as one wishes: 1.61803398874989484820....

We now approach my question. Condren states the following in his post on Drexel University's Math Forum (June 1999):

The line counts of the four poems [in the Pearl Poet manuscript] are, respectively, 1212, 1812, 531, and 2531. This highly artful arrangement, minus the signature twelves in the first half and signature 31s (the tenth prime as the Middle Ages reckoned primes) in the second half, gives us two halves of 3000 lines each. More intriguing still, the two outer poems divided by the two medial poems give the Golden Section.
Okay, if we add 1212 and 2531, we get 3743, and if we add 1812 and 531, we get 2343. If we then divide 3743 by 2343, we obtain 1.59752454118.... That's not especially close to the Golden Ratio. So, let's try again without the "signature twelves" and "signature 31s" -- a move that already raises some danger signals for me, but okay. If we add 1200 and 2500, we get 3700, and if we add 1800 and 500, we get 2300. If we then divide 3700 by 2300, we obtain 1.60869565217....

That's closer but still no cigar. We can round off Condren's mean and the actual Golden Mean to 1.609 and 1.618, respectively. That's pretty close, but not the same. If we round off a bit more, we get 1.61 and 1.62. Closer yet, but still not the same. If we round off one more time, we get 1.6 and 1.6. The same ... finally.

At this point, I'd need to know two things:
1. What was the Medieval approximation for the Golden Ratio?
2. Why does Condren drop the "signature twelves" and the "signature 31s"?
I presume that Condren has some answers, but for now, I'm even more skeptical than I already was.



At 9:27 AM, Anonymous dilys said...

There may have been some slippage, but I have for awhile tried to make anything I design congruent at least in outline with the proportions of the Golden Section.

I have not noticed either visually or kinesthetically (that "body feel" when you look at something) that it was the high road to aesthetic satisfaction.

I remember reading that in ancient China that a public building not to those proportions resulted in execution of the architect. That, I suppose, introduces a practical satisfaction measure.

At 12:27 PM, Blogger Horace Jeffery Hodges said...

Dilys, I clicked on the link and posted a comment there on your blog.

I suspect that the aesthetic satisfaction from the Golden Section would be primarily intellectual rather.

I've never heard this about ancient Chinese architecture. I'll have to look into it.

Jeffery Hodges

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