The Myth:

Author Dan Brown in his best selling novel The Da Vinci Code makes special mention of the golden ratio, also known as . The main character Robert Langdon has a flashback starting on page 93 to a lecture he gave to his students about the golden ratio and many amazing and interesting facts involving the ratio. They include:

1.  =1.618.
2.   is derived from the Fibonacci sequence.
3. It is a fundamental building block in nature for plants, animals and even human beings.
4. Early scientists heralded it as the Divine Proportion.
5. The number of female bees divided by the number of male bees is always the golden ratio.
6. The nautilus forms a spiral in which the ratio of each spiral's diameter to the next is the golden ratio.

1. Sunflower seeds grow in opposing spirals, and the ratio of each rotation's diameter to the next is the golden ratio.
2. Pine cone petals, leaf arrangements on plant stalks, insect segmentation display astonishing obedience to the divine proportion.
3. The golden ratio appears in Da Vinci's The Vitruvian Man (albeit, this is implicit in the text, not outright stated).
4. Marcus Vitruvius praised the divine proportion in De Architectura.
5. Measure the distance from the tip of your head to the floor and then divide that by the distance from  your belly button to the floor, and you get .
6. The distance from your shoulder to your fingertips divided by the distance from your elbow to your fingertips is the golden ratio.
7. The distance from your hip to the floor divided by the distance from your knee to the floor is the golden ratio.
8. Finger joints, toes, spinal divisions all have something to do with the golden ratio.
9. Artwork by Michelangelo, Albrecht Dürer, Da Vinci and many others demonstrate intentional and rigorous adherence to the golden ratio.
10. The Greek Parthenon, the Pyramids of Egypt and the UN Building exhibit use of the golden ratio.
11. In music the golden ratio appears in the organizational structures of the sonatas of Mozart, Beethoven's Fifth symphony, works of Bartók, Debussy, and Schubert.
12. Stradivarius used the golden ratio to calculate the placement of the f-holes on his violins.
13. The lines of a pentagram divide themselves into segments according to the Divine Proportion.

The Reality:

I'll take each of the claims above in turn briefly here that don't already have a page devoted to them.

(1) This is an approximation, and given Brown's emphasis on ancient knowledge I think it is a little sad that he doesn't note that the number is actually irrational. Moreover, being an irrational number, and measurement being an approximate process, it is impossible for one to make the claim that any ratio of measured numbers is exactly the golden ratio. The irrationality of the golden ratio apparently caused some distress to the Pythagoreans, one of the worlds oldest secret societies, so you think it would have been right up his alley. 

(2) The golden ratio predates the Fibonacci sequence by almost two millennia. It is true however that the limit of successive terms of the Fibonacci sequence is the golden ratio.

(3) This is a muddled claim. More commonly what we see is that the Fibonacci numbers occur to a surprising degree in nature in a phenomenon known as phyllotaxis. The golden ratio does provide an explanation for the appearance of some of these numbers, especially in the numbers of spirals of each orientation in the heads of flowers and on pine cones and such, though admittedly, some of the best work done on this is not very accessible to the layman.

(4) The first person to herald this ratio as the Divine Proportion was Luca Pacioli. His reasons were hardly scientific, and his interests were largely theological and mathematical.

(5) This is, to the best of my ability to verify, a confusion on Dan Brown's part. The number of male to female bees in a hive can vary a great deal, but given that the number of male bees can be as little as zero, and that their number rarely is higher than about 200 while the number of female bees numbers in the thousands this is a wildly inaccurate statement. What does appear to be true, due to some peculiarities of the breeding processes of bees is that the numbers of male to female bees in the family history of a male bee at any given generation is a ratio of adjacent Fibonacci numbers [Liv02].

(6) This appears to be a variant of the nautilus shell is a golden spiral myth.

(7,8) The diameter of a spiral of either orientation on a flower should be the same, since it is just the diameter of the flower head. What he is probably referring to here is phyllotaxis occurring in the count of spirals in each direction which in many species of plants is always a pair of successive Fibonacci numbers.

(9,10) I have never been able to find anything in De Architectura mentioning the golden ratio (which Vitruvius would have referred to as the extreme and mean ratio). Places where many authors point when making this kind of claim seem to have to do with his commentary on the pleasing proportions of floor plans, being 2:3, 3:5 and 1: . Vitruvius can't have been aware of the occurrence of the first two ratios as being instances of Fibonacci numbers because he died over one thousand years before they were studied, and none of them falls inside the acceptable range for claims of use of the golden ratio used on this site. Being a roman architect, and being a practical man, he probably recognized these as easily constructed ratios, and that because they are in the 1.4-1.67 range they are not too much like squares but not too narrow for most architectural purposes.

As to The Vitruvian Man, the proportions Vitrivius describes for the human body are self referential (four fingers to a palm, a man's height is six times the length of his foot and so on). Mostly Vitruvius appears to be making the point that architecture is designed in terms relative to a human scale, and that it pays to know the proportions of the human body to know that scale. To the extent that The Vitruvian Man actually encompasses the proportions of Vitruvius, it doesn't involve the golden ratio. I am not aware of any references of any quality that would lead one to believe that this sketch by Da Vinci has anything to do with the golden ratio and instead is a very nice illustration of the proportions of the human body as described by Vitruvius. On the other hand I am not an art historian, don't read italian, haven't got a clue what was in Da Vinci's head when he drew this, and haven't read a translation of what else appears on the page The Vitruvian Man appears on. Clearly I have a little more poking to do on this one.

(11-14) These claims suffer from two faults in my view. Not only is there considerable human variation, one must also ask the question, where exactly are you measuring from? Take the example of the arm measurements; where does your shoulder end? Your elbow? Take the belly button measurement, should you measure from the top of the belly button or the bottom or try to find the center? The second major fault is that all of these rely on the ratio of the longer to the whole, as opposed to the shorter to the longer, and as is mentioned in our discussion of the golden ratio, the former is always closer to the golden ratio.

(15) I'm continuing to try to find good references on this. My best reference on Dürer is [Liv02], who subscribes to this point of view, but I don't find his presentation convincing. What Livio does lay out is that Dürer was greatly interested in, and even wrote on, geometric topics including figures that use the golden ratio such as the dodecahedron, but this is a far cry from saying Dürer used it as a compositional element. Claims about the works of Da Vinci involving use of the golden ratio are generally made about works made significantly prior to his association with Pacioli, and suffer from a kind of viewer selectivity bias rampant in claims about the golden ratio and art. Markowsky [Mar 92] has an excellent discussion of the claims made about Da Vinci's St. Jerome,  in which he points out that amongst other flaws, one needs to chop off Jerome's arm to achieve the desired result. In fact, Markowsky points to a great deal of literature addressing these kinds of issues, and as time permits I will get copies of them and summarize them on this site.

(17) The Mozart sonata myth is addressed elsewhere, and as time permits I will address the others.

(18) This is another example of something that Livio seems to subscribe to, but for which I have yet to find good evidence and the evidence presented in [Liv02] I don't find convincing. The evidence given in [Liv02] consists of a single sketch on page 184, and I don't see enough in this sketch to make it remotely clear where and how the golden ratio comes into it. That being said, Livio does point out that hardly anyone attributes the sound of a Stradivarius to the placement of these holes, instead most violin makers point to his choices of wood and varnish.

(19) This is correct as long as the pentagram is a regular pentagram, which I'm sure is what Dan Brown intended.

References:

[Bro03] Dan Brown, The da vinci code, Doubleday, 2003.

[Liv02] Mario Livio, The golden ratio, Broadway Books, New York, 2002, The story of phi, the world’s most astonishing number. MR MR1938220 (2003k:11025)

[Mar92] George Markowsky, Misconceptions about the golden ratio, The College Mathematics Journal 23 (1992), no. 1, 2–19.

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