I just finished Dan Brown's best-seller The Da Vinci Code, about a thousand-year-old conspiracy involving the Catholic church. While the modern murder mystery part of the book is obviously fictional, you're encouraged to believe that the historical background is authentic. As a Straight Dope reader I know otherwise, so I'm not going to ask about hanky-panky between Jesus and Mary Magdalene (unlikely), the Holy Grail (a literary invention), or the Priory of Sion (too much like the Illuminati). What intrigued me was phi, also known as the Divine Proportion, a mystical ratio the book claims shows up everywhere in nature and art. I remember enough about the golden section and Fibonacci numbers, which also figure in the book and are related to phi somehow, to know that some of this is legit. But phi itself is new to me. What's the straight dope on this magic number?
Illustration by Slug Signorino
Brown’s exegesis of phi — for that matter, his whole book — is so cartoonish that you’re inclined to dismiss it out of hand. (To cite one egregious example of his imprecision, he continually refers to the painter of the Mona Lisa as “da Vinci.” As anyone with a semester of art history knows, the man’s name was Leonardo; da Vinci merely refers to his birthplace.) Phi is a cool concept, though, and Lord knows I don’t get many chances to expound on higher mathematics and dump on a fellow scribbler at the same time. So what I’ll do here is go through Brown’s often loopy assertions and follow each with the facts.
There’s this number known as phi. Yeah, although the term wasn’t invented till the early 1900s, by American mathematician Mark Barr. For that matter, synonyms like golden section and golden ratio, notwithstanding their air of antiquity, may date back only to the 19th century. The basic concept, however, was first advanced by the Greek geometer Euclid.
Phi = 1.618. Not quite. Phi is the infinite nonrepeating decimal 1.6180339887 … This may seem like a trivial difference, but it’s the whole point — phi, like pi, is an irrational number that can’t be expressed as the ratio of whole numbers. However, you can see where an explanation like that might not fly in a beach book.
Phi is derived from the Fibonacci sequence 1, 1, 2, 3, 5, 8, etc. It can be, but the original, simpler explanation is this: 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. Transfixed by the divinity of this proportion, are we? Maybe not, but some claim the ratio is uniquely pleasing in art.
Back to the Fibonacci sequence. Each number in it is the sum of the preceding two numbers (1 + 1 = 2, 2 + 1 = 3, etc). As Brown rightly notes, the quotients of successive adjacent terms (2/1, 3/2, 5/3, etc) converge on phi as you get further out. This is not the miracle some think, but it’s still cool.
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 AC with their belly buttons at point B are named Igor. On me the ratio is about 1.7, not 1.618+. A huge difference? No, but Brown’s hero observes, “Plants, animals, and even human beings all possessed dimensional properties that adhered with eerie exactitude to the ratio of phi to 1.” That’s just not so. I venture to say there’s wide variation among individuals not only for navel placement but also for the other supposed anatomical occurrences of phi Brown cites.
Many natural phenomena exhibit some relation to phi. Up to a point, true. The spiral in the illustration was generated by assembling rectangles out of squares whose sides correspond in length to successive numbers in the Fibonacci series, then drawing arc segments in each square. This gives us a reasonable approximation of the shell of a chambered nautilus, seen in cross section. You can also find Fibonacci numbers in leaves spiraling around a plant stem, in the arrangement of seeds in flower heads, and so forth. Surprising? Not really. The Fibonacci series is a simple algorithm enabling proportions to remain roughly constant (i.e., approximately phi to one) with successive accretions of growth. If it works for a number series, it’ll work for plants, shells, etc.
Phi governs the proportions of the pyramids, the Parthenon, and so on. There’s little evidence that ancient architects used the golden section, but even if they did, so what? Euclid figured it out, and others could have too. Later creative types have certainly shown an interest — Dan Brown, for one. We know-it-alls may scoff, but how often do you see someone build a popular thriller around obscure concepts in math?
Send questions to Cecil via firstname.lastname@example.org.