# Golden section, ratios and phi (nitpicking) **URL:** \ **Category:** Cecil's Columns/Staff Reports\ **Created:** [June 18, 2004, 11:20am UTC](https://boards.straightdope.com/t/golden-section-ratios-and-phi-nitpicking/251018 "2004-06-18T11:20:50Z")\ **Posts on this page:** 20\ **Page:** 1
**Author:** ![ndufva](https://avatars.discourse-cdn.com/v4/letter/n/6bbea6/32.png) [@ndufva](https://boards.straightdope.com/u/ndufva)\ **Post date:** [June 18, 2004, 11:20am UTC](https://boards.straightdope.com/t/golden-section-ratios-and-phi-nitpicking/251018/1 "2004-06-18T11:20:50Z")
Dear Cecil, Love your column, but here is some… \ In the column on phi - [http://www.straightdope.com/columns/040618.html](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\ Keep up the good work! Nicolai ---
**Author:** ![CalMeacham](https://sea3.discourse-cdn.com/straightdope/user_avatar/boards.straightdope.com/calmeacham/32/35_2.png) [@CalMeacham](https://boards.straightdope.com/u/CalMeacham)\ **Post date:** [June 18, 2004, 11:33am UTC](https://boards.straightdope.com/t/golden-section-ratios-and-phi-nitpicking/251018/2 "2004-06-18T11:33:08Z")
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. ---
**Author:** ![Colophon](https://avatars.discourse-cdn.com/v4/letter/c/f05b48/32.png) [@Colophon](https://boards.straightdope.com/u/Colophon)\ **Post date:** [June 18, 2004, 12:07pm UTC](https://boards.straightdope.com/t/golden-section-ratios-and-phi-nitpicking/251018/3 "2004-06-18T12:07:07Z")
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. ---
**Author:** ![Foghead](https://avatars.discourse-cdn.com/v4/letter/f/76d3ee/32.png) [@Foghead](https://boards.straightdope.com/u/Foghead)\ **Post date:** [June 18, 2004, 12:20pm UTC](https://boards.straightdope.com/t/golden-section-ratios-and-phi-nitpicking/251018/4 "2004-06-18T12:20:04Z")
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. ---
**Author:** ![ftg](https://sea3.discourse-cdn.com/straightdope/user_avatar/boards.straightdope.com/ftg/32/2801_2.png) [@ftg](https://boards.straightdope.com/u/ftg)\ **Post date:** [June 18, 2004, 3:21pm UTC](https://boards.straightdope.com/t/golden-section-ratios-and-phi-nitpicking/251018/5 "2004-06-18T15:21:10Z")
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. ---
**Author:** ![zut](https://sea3.discourse-cdn.com/straightdope/user_avatar/boards.straightdope.com/zut/32/2875_2.png) [@zut](https://boards.straightdope.com/u/zut)\ **Post date:** [June 18, 2004, 4:30pm UTC](https://boards.straightdope.com/t/golden-section-ratios-and-phi-nitpicking/251018/6 "2004-06-18T16:30:40Z")
I’m surprised **Cecil** didn’t consult [**Hiyruu**](http://boards.straightdope.com/sdmb/showthread.php?t=97714) before writing this column. ---
**Author:** ![Chronos](https://sea3.discourse-cdn.com/straightdope/user_avatar/boards.straightdope.com/chronos/32/134_2.png) [@Chronos](https://boards.straightdope.com/u/Chronos)\ **Post date:** [June 18, 2004, 5:42pm UTC](https://boards.straightdope.com/t/golden-section-ratios-and-phi-nitpicking/251018/7 "2004-06-18T17:42:57Z")
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 > [@](#): > > Take 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. ---
**Author:** ![Colophon](https://avatars.discourse-cdn.com/v4/letter/c/f05b48/32.png) [@Colophon](https://boards.straightdope.com/u/Colophon)\ **Post date:** [June 18, 2004, 6:09pm UTC](https://boards.straightdope.com/t/golden-section-ratios-and-phi-nitpicking/251018/8 "2004-06-18T18:09:54Z")
> [@Chronos](#): > > 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. ---
**Author:** ![C\_K\_Dexter\_Haven](https://avatars.discourse-cdn.com/v4/letter/c/b2d939/32.png) [@C\_K\_Dexter\_Haven](https://boards.straightdope.com/u/C_K_Dexter_Haven)\ **Post date:** [June 18, 2004, 9:00pm UTC](https://boards.straightdope.com/t/golden-section-ratios-and-phi-nitpicking/251018/9 "2004-06-18T21:00:03Z")
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. ---
**Author:** ![wolf\_meister](https://sea3.discourse-cdn.com/straightdope/user_avatar/boards.straightdope.com/wolf_meister/32/15202_2.png) [@wolf\_meister](https://boards.straightdope.com/u/wolf_meister)\ **Post date:** [June 18, 2004, 11:35pm UTC](https://boards.straightdope.com/t/golden-section-ratios-and-phi-nitpicking/251018/10 "2004-06-18T23:35:24Z")
I too wish to announce yet another faux pas in the very same Cecil column !!! [http://www.straightdope.com/columns/040618.html](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. ---
**Author:** ![ultrafilter](https://avatars.discourse-cdn.com/v4/letter/u/3d9bf3/32.png) [@ultrafilter](https://boards.straightdope.com/u/ultrafilter)\ **Post date:** [June 19, 2004, 12:09am UTC](https://boards.straightdope.com/t/golden-section-ratios-and-phi-nitpicking/251018/11 "2004-06-19T00:09:21Z")
> [@wolf\_meister](#): > > I too wish to announce yet another faux pas in the very same Cecil column !!! > [http://www.straightdope.com/columns/040618.html](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. ---
**Author:** ![wolf\_meister](https://sea3.discourse-cdn.com/straightdope/user_avatar/boards.straightdope.com/wolf_meister/32/15202_2.png) [@wolf\_meister](https://boards.straightdope.com/u/wolf_meister)\ **Post date:** [June 19, 2004, 12:54am UTC](https://boards.straightdope.com/t/golden-section-ratios-and-phi-nitpicking/251018/12 "2004-06-19T00:54:10Z")
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. ---
**Author:** ![ultrafilter](https://avatars.discourse-cdn.com/v4/letter/u/3d9bf3/32.png) [@ultrafilter](https://boards.straightdope.com/u/ultrafilter)\ **Post date:** [June 19, 2004, 12:55am UTC](https://boards.straightdope.com/t/golden-section-ratios-and-phi-nitpicking/251018/13 "2004-06-19T00:55:13Z")
> [@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. Yes, that’s correct. ---
**Author:** ![Lumpy](https://sea3.discourse-cdn.com/straightdope/user_avatar/boards.straightdope.com/lumpy/32/446_2.png) [@Lumpy](https://boards.straightdope.com/u/Lumpy)\ **Post date:** [June 19, 2004, 1:06am UTC](https://boards.straightdope.com/t/golden-section-ratios-and-phi-nitpicking/251018/14 "2004-06-19T01:06:15Z")
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? ---
**Author:** ![ultrafilter](https://avatars.discourse-cdn.com/v4/letter/u/3d9bf3/32.png) [@ultrafilter](https://boards.straightdope.com/u/ultrafilter)\ **Post date:** [June 19, 2004, 1:08am UTC](https://boards.straightdope.com/t/golden-section-ratios-and-phi-nitpicking/251018/15 "2004-06-19T01:08:09Z")
> [@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? I very vaguely remember something about this, but not even enough to search around. ---
**Author:** ![wolf\_meister](https://sea3.discourse-cdn.com/straightdope/user_avatar/boards.straightdope.com/wolf_meister/32/15202_2.png) [@wolf\_meister](https://boards.straightdope.com/u/wolf_meister)\ **Post date:** [June 19, 2004, 1:34am UTC](https://boards.straightdope.com/t/golden-section-ratios-and-phi-nitpicking/251018/16 "2004-06-19T01:34:23Z")
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. ---
**Author:** ![SlowMindThinking](https://avatars.discourse-cdn.com/v4/letter/s/dc4da7/32.png) [@SlowMindThinking](https://boards.straightdope.com/u/SlowMindThinking)\ **Post date:** [June 19, 2004, 1:34am UTC](https://boards.straightdope.com/t/golden-section-ratios-and-phi-nitpicking/251018/17 "2004-06-19T01:34:38Z")
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. ---
**Author:** ![ftg](https://sea3.discourse-cdn.com/straightdope/user_avatar/boards.straightdope.com/ftg/32/2801_2.png) [@ftg](https://boards.straightdope.com/u/ftg)\ **Post date:** [June 19, 2004, 2:04am UTC](https://boards.straightdope.com/t/golden-section-ratios-and-phi-nitpicking/251018/18 "2004-06-19T02:04:55Z")
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.) ---
**Author:** ![Chronos](https://sea3.discourse-cdn.com/straightdope/user_avatar/boards.straightdope.com/chronos/32/134_2.png) [@Chronos](https://boards.straightdope.com/u/Chronos)\ **Post date:** [June 19, 2004, 4:17am UTC](https://boards.straightdope.com/t/golden-section-ratios-and-phi-nitpicking/251018/19 "2004-06-19T04:17:35Z")
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”. ---
**Author:** ![CalMeacham](https://sea3.discourse-cdn.com/straightdope/user_avatar/boards.straightdope.com/calmeacham/32/35_2.png) [@CalMeacham](https://boards.straightdope.com/u/CalMeacham)\ **Post date:** [June 19, 2004, 4:51am UTC](https://boards.straightdope.com/t/golden-section-ratios-and-phi-nitpicking/251018/20 "2004-06-19T04:51:55Z")
> [@](#): > > 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. [Next page](https://boards.straightdope.com/t/golden-section-ratios-and-phi-nitpicking/251018.md?page=2)