Golden section, ratios and phi (nitpicking)

[ndufva]

Dear Cecil,

Love your column, but here is some...

<nitpick>
In the column on phi - http://www.straightdope.com/columns/040618.html - you forgot to do sanity check on the ratios mentioned to get at the golden cut.
If the line AC is divided by the point B, then AB/AC = AC/BC will never happen for the simple reason that if B never coincides with either end (i.e. B is always between A and C, never at A or C), 0<AB/AC<1 and 1<AC/BC.
The correct ratios to get at the golden section would be AC/AB=AB/BC.
In the artwork by Slug Signorino, it seems to be that the ratio is AC/BC=BC/AB, but that is just a matter of agreeing on names for the three points :)
</nitpick>

Keep up the good work!
Nicolai

[CalMeacham]

There's a big section on The Golden Ratio in the educational Disney cartoon Donald in Mathemagicland. I also read articles about it in old Science Digests. Dover has two paperbacks out on the topic.

Phi is fascinating, but I've long held that people see it in more places than it really is. After all the hype about how the Golden Rectangle (having the Golden Ratio as the proportions between its sides) is the "ideal" or "preferred" rectangle, and how people pick it out on tests, I took a skeptical look around. If the Golden Rectangle is the best, why doesn't it appear anywhere we commonly use rectangles? Billboards, movie screens (1940s or wide screen) TV sets, paperback books, postcard, pads of paper (legal or normal), playing cards, credit cards, paper money, doorways -- none of these things are Golden Rectangles. Granted, there are sometimes practical reasoins for this ("I suggest you consider the shape of this characteriastic arch, considered in the light of our functionally designed human doorways"), but at some point an arbitrary rectangle should've presented itself as a candiate for goldenhood.

I'm happy to see that I'm not alone in thinking this was. There was a piece in The Skeptical Inquirer not that long ago arguing the same points.

[Colophon]

When come back, bring φ.




Er, yeah... Cecil shoulda told Slug that point C goes on the line AB, i.e. between A and B.

[Foghead]

From the article:
Your height divided by the distance from your belly button to the floor = phi. Get out. Behold the line segment in the drawing. The only people of height AB with their belly buttons at point C are named Igor.
Either Igor's belly is hanging real low, or the points are reversed in this quote also.

[ftg]

Re: Commonality of Fibonacci numbers.

I've developed a lot of algorithms and proofs in Computer Science, publishing several papers, etc. In at least 4 of them, the Fibonacci numbers/Golden Ratio appear somewhere in the design and/or analysis.

In Computer Science, powers of 2 appear quite commonly via the recursive construction: take two of the next smaller size. Fibonacci numbers are just a small variation of this (take the next smaller and the one before that). So, you can think of phi as the 2nd most common exponent base in Computer Science. (Although a distant second.) Natural, common and practical.

If you have access to a Sophomore Data Structures book, it probably covers AVL trees and Fibonacci numbers appear in the analysis of those. Check it out.

[zut]

I'm surprised Cecil didn't consult Hiyruu (http://boards.straightdope.com/sdmb/showthread.php?t=97714) before writing this column.

[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 toTake line segment AC. Place point B on AC so that AC/BC = BC/AB. (Thank God we have artistic genius Slug Signorino to illustrate these advanced concepts.) AC/BC = BC/AB = phi = the golden section.and in paragraph 6,The only people of height AC with their belly buttons at point B are named Igor.

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.

[Colophon]

As regards the nitpick, I realize that it's impossible to get the Slug diagram changed.

Pah - two minutes with a cheap graphic editor is all that would take.

[C K Dexter Haven]

I've directed this to Ed's attention.

A change on the website will (I presume) occur. In terms of THE STRAIGHT DOPE column that's syndicated in newspapers around the country and the whirled, well... I guess it's there.

[wolf_meister]

I too wish to announce yet another faux pas in the very same Cecil column !!!
http://www.straightdope.com/columns/040618.html

The Master has made a mistake? Yes. (Can Armageddon be far away?)

The Annointed One (Cecil Adams), says that phi and pi are both irrational. While this is true of phi it is not true of pi which is transendental.

An irrational number is one that cannot be put into a simple fraction but can be the result of a simple algebraic equation. Basically, phi equals ( 1 + sqrt(5)) / 2.

A transcendental number is one which can never be the result of a simple algebraic equation.

A nitpick you say? Perhaps, but I think Cecil would approve.

[ultrafilter]

I too wish to announce yet another faux pas in the very same Cecil column !!!
http://www.straightdope.com/columns/040618.html

The Master has made a mistake? Yes. (Can Armageddon be far away?)

The Annointed One (Cecil Adams), says that phi and pi are both irrational. While this is true of phi it is not true of pi which is transendental.

An irrational number is one that cannot be put into a simple fraction but can be the result of a simple algebraic equation. Basically, phi equals ( 1 + sqrt(5)) / 2.

A transcendental number is one which can never be the result of a simple algebraic equation.

A nitpick you say? Perhaps, but I think Cecil would approve.

He might approve if you were right. But alas, it's not the case. pi is most definitely irrational, as is every transcendental number.

[wolf_meister]

Okay then I take it that ALL transcendental numbers are irrational but the converse is not true? If so, I stand corrected and have become a bit more learned in the ways of mathematics.

[ultrafilter]

Okay then I take it that ALL transcendental numbers are irrational but the converse is not true? If so, I stand corrected and have become a bit more learned in the ways of mathematics.

Yes, that's correct.

[Lumpy]

The June 2004 issue of Discover magazine has an article on Phi, and it makes an interesting claim: "of all irrational numbers, Phi is, in a very precise technical sense, the furthest from being representable as a fraction".

Can anyone tell me more about this?

[ultrafilter]

The June 2004 issue of Discover magazine has an article on Phi, and it makes an interesting claim: "of all irrational numbers, Phi is, in a very precise technical sense, the furthest from being representable as a fraction".

Can anyone tell me more about this?

I very vaguely remember something about this, but not even enough to search around.

[wolf_meister]

Okay, until someone else steps in with a more detailed answer, irrational numbers (such as pi) can be approximated by a fraction. One of the best is:
355 / 113 = 3.14159292035...
Ignoring rounding, that is pi, accurate to six decimal places.

Perhaps with the case of phi, the fractions are awkward and not as "neat" as the 355/113 fractional approximation for pi.

[SlowMindThinking]

I'd look in "Mathematical Constants" by Steven Finch. If that book doesn't list it, it might well not be worth a nonmathematician knowing.

With 624 pages about mathematical constants, it must have something about being furthest removed from a fraction. As a complete guess, I'd say it would have something to do with a continued fraction representation.

[ftg]

Re: Phi and representation in fractions.

Yup, check "Concrete Mathematics" by Graham, Knuth and Patashnik. Section 6.6ff. It's a consequence of Fibonacci numbers being the worst case for Euclid's algorithm for GCD.

(This also relates to why Phi is used in the quadratic algebra proof of the Lucas-Lehmer test for Mersenne primes.)

[Chronos]

Another use for phi is in pseudorandom sampling, which incidentally also suggests why its' the "most irrational" number.

To illustrate, suppose that I want to sample points from a circle. At each step, I pick a point, and after some step (but I don't know in advance which one), you'll say "stop". My goal is for my set of points to be as uniformly distributed as possible at the end. It turns out that a very simple way to do this is for each point to be phi times the circumference past the previous one (so the first one would be at zero degrees, the second one at 222.49... degrees, the third one at 84.98... degrees, etc.). If I sample in this way, then every point will always be placed into whatever is the largest gap at each step, and divide that gap so that both of the resulting gaps are smaller than the previous smallest gap.

If you did this with any rational number, you would only hit a finite number of points (the number being the lowest denominator of your fraction), each an infinite number of times, and if you did it with an "approximately rational" number, you would cluster around the points corresponding to the rational number you're approximating. Hence, phi, for which this clustering does not occur, is "most irrational" or "least approximately rational".

[CalMeacham]

The June 2004 issue of Discover magazine has an article on Phi, and it makes an interesting claim: "of all irrational numbers, Phi is, in a very precise technical sense, the furthest from being representable as a fraction".

Can anyone tell me more about this?



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.

Unfortunately, I don't know if I can write numbers in normal notation here, so bear with a lot of parentheses.


Phi can be written as 1/(1+1/(1+1/(1+1/.......))). The more terms you keep in your expansion (the more convergents, to use the technical term), the more closely you'll approximate the real value. There are lots of other continued fractions, but phi is one of the few that uses only ones.


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

[bonzer]

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)2/5. 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]

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. :D

[Sleel]

Here's (http://www.beautyanalysis.com/index2_mba.htm) 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]

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... = phi2
Nifty, ain't it?

[ultrafilter]

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 + y2 = x2, which rearranges to x2 - xy - y2 = 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.
:cool:

[RM Mentock]

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]

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:
:cool:

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]

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]

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]

Because people are stupid.

I'll take "Answers to 90% of questions that begin with 'why'" for $200, Alex.

[Ogre]

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:
:cool: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]

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.