<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>
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.
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.
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 to
and 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.
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.
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.
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.
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”.
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.
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.
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.)
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”.
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.
