Golden section, ratios and phi (nitpicking)

[bonzer]

@CalMeacham

It could be that what they have in mind here is that phi canot be represented as a fractional ratio, but it can be written as a continued fraction. It is, in fact, arguably the simplest of the continued fractions.

It’s also the slowest converging continued fraction.

For more info on this and on the Golden Ratio, see the book The Divine Proportion by Hinton, published by Dover Books.

You’ve got the title and publishers correct, but it’s actually by H.E. Huntley.

Incidentally, while Huntley doesn’t give a specific reference, he notes (p62) that the study measuring navels was conducted by a crank who also believed that pi = 6(phi)[sup]2[/sup]/5. Though I doubt they’re the only person to have proposed the latter, it does suggest this may be the “R.G.” who’s the example Underwood Dudley uses in the section on phi in his Mathematical Cranks (MAA, 1992, p245-50) and who had a track record of pestering the likes of Martin Gardner.

[Ed_Zotti]

@Chronos

As regards the nitpick, I realize that it’s impossible to get the Slug diagram changed. Or, for that matter, to get Slug to do anything he doesn’t want to do. But the problem could also be fixed by editing the column, which is within the realm of possibility. As follows: In the fourth paragraph, change toand in paragraph 6,

Note, of course, that I’m not saying that Cecil is wrong: Absent the illustration, the column is self-consistent. And since Slug’s illustration is drawn to match the column, and not the other way around, the fault must be considered to lie with Slug. However, Cecil is much more open to editing than is Slug, so it might be prudent to change the text of the column to be consistent with the drawing.

Don’t blame Slug. Cecil relies on me to convey his instructions to the art department, and I got the letters mixed up. It’s fixed now. I hope. My apologies.

[wolf_meister]

In my first posting, I gave the value of PHI as (1 + sqrt(5)) / 2
Although no one asked, here is how that value is derived:

Deriving Phi

A_____________________ B__________C
|-------------------------·-----------|

Line AC is of length PHI
Line AB = 1
Therefore, Line BC = PHI -1

By definition, if AC is length PHI, then the ratio of Line AC to line AB equals the ratio of Line AB / Line BC

(AC / AB) = (AB / BC)

Substituting:
(PHI / 1) = (1 / (PHI-1))

PHI² -PHI = 1

PHI² -PHI -1 = 0

Solving by the Quadratic Formula:

PHI = (1 ± sqrt( 1 - (4 · 1 · -1)) / (2 · 1)

PHI = (1 + sqrt(5)) / 2

It was a story that had to be told.

[Sleel]

Here’s an interesting site concerning Phi’s relationship to our perception of beauty. They don’t say that Phi is universal, just that the closer faces conform to Phi proportions, the more likely we are to perceive them as beautiful.

[ultrafilter]

@wolf_meister

In my first posting, I gave the value of PHI as (1 + sqrt(5)) / 2
Although no one asked, here is how that value is derived:

Deriving Phi

A_____________________ B__________C
|-------------------------·-----------|

Line AC is of length PHI
Line AB = 1
Therefore, Line BC = PHI -1

Doesn’t this assume that phi > 1?

[Chronos]

Phi is a ratio, so you can consider it to be more than 1 or less than 1 (depending on which way you’re comparing). I’ve usually seen the convention that 1.618… is phi.

Incidentally, more phi fun:
0.61803398875… = 1/phi
1.61803398875… = phi
2.61803398875… = phi[sup]2[/sup]
Nifty, ain’t it?

[ultrafilter]

@Chronos

Phi is a ratio, so you can consider it to be more than 1 or less than 1 (depending on which way you’re comparing). I’ve usually seen the convention that 1.618… is phi.

I don’t see how to derive 1/phi from what we start with.

Anyway, here’s a derivation that uses no assumptions about phi. Take AB = x and BC = y. So AC = x + y.

We want (AC / AB) = (AB / BC), which means that (x + y)/x = x/y. Cross-multiply to get xy + y[sup]2[/sup] = x[sup]2[/sup], which rearranges to x[sup]2[/sup] - xy - y[sup]2[/sup] = 0.

By the quadratic formula, x = (y + y*sqrt(5))/2. Since one of those roots is negative, we can throw it out. Therefore, x/y = (1 + sqrt(5))/2.

[Mariemarie]

I thoroughly enjoyed Cecil’s column on phi. I hope Cecil will comment further on The Da Vinci Code. I thought the book was a steamy lump of doodoo, and it puzzles me that it’s been such a best seller. I love Cecil’s delicious little jabs at Brown:

So what I’ll do here is go through Brown’s often loopy assertions and follow each with the facts.

[RM_Mentock]

@ultrafilter

Since one of those roots is negative, we can throw it out.

Hey, don’t through that out, I can eat that.

The negative root just means that A is between B and C, and the number is the reciprocal phi.

Phi? I thought he said pi.

[RiverRunner]

@Mariemarie

I thoroughly enjoyed Cecil’s column on phi. I hope Cecil will comment further on The Da Vinci Code. I thought the book was a steamy lump of doodoo, and it puzzles me that it’s been such a best seller. I love Cecil’s delicious little jabs at Brown:

Yeah, I got two(!) copies of it for Christmas but I have only been able to make it through the first couple of chapters. I might try it again on the trip I’ll be taking soon, but I haven’t been able to work up any enthusiasm for the idea. It’s like trying to look forward to a prostate exam.
RR

[ultrafilter]

@RM Mentock

The negative root just means that A is between B and C, and the number is the reciprocal phi.

Are you thinking of a directed distance here?

[John_W.Kennedy]

@Mariemarie

I thoroughly enjoyed Cecil’s column on phi. I hope Cecil will comment further on The Da Vinci Code. I thought the book was a steamy lump of doodoo, and it puzzles me that it’s been such a best seller.

Because people are stupid.

[ultrafilter]

@John W. Kennedy

Because people are stupid.

I’ll take “Answers to 90% of questions that begin with ‘why’” for $200, Alex.

[Ogre]

@Mariemarie

I thoroughly enjoyed Cecil’s column on phi. I hope Cecil will comment further on The Da Vinci Code. I thought the book was a steamy lump of doodoo, and it puzzles me that it’s been such a best seller. I love Cecil’s delicious little jabs at Brown:

You may then enjoy reading, oh, any one of the threads about Da Vinci code in Cafe Society. Some folks tagged it as a “mindless, fun read,” but the rest of us pretty much tore it to quivering shreds, and then peed on the shreds.

[RM_Mentock]

I stopped by Cafe Society, but didn’t see any da Vinci Code threads on the first page. They were talking about Michael Moore, I think.

[Derleth]

@RM Mentock

I stopped by Cafe Society, but didn’t see any da Vinci Code threads on the first page. They were talking about Michael Moore, I think.

Hopefully, that means people who post to the SDMB are either smart enough to reject it out of hand, or smart enough to do the research needed to indicate it is pretty much junk.

Aside from that, I don’t know. It could be a confounding factor we’re not accounting for, a small statistical glitch that makes Dopers less likely to fully track best-seller lists with regards to Cafe Society posts. Maybe we largely conform to the geek stereotype and follow science fiction, fantasy, and horror more closely than `mainstream’ fiction. With the democritization of Internet access this is less likely, but the makeup of the SDMB’s core clientele is skewed for other reasons.

Of course, spinning statistical hypotheses like this is pretty much pointless. But I think one or more of my factors could easily account for this lack.