13 – 3 – 2 – 21 – 1 – 1 – 8 – 5
O, Draconian devil!
Oh, lame saint!
Langdon read the message again and looked up at Fache.
“What the hell does this mean?”
Harvard University Professor Robert Langdon, the hero of Dan Brown’s best-selling novel The Da Vinci Code, is initially baffled by the message, scrawled in invisible ink on the floor of the Louvre in Paris by a dying man with a passion for secret codes.
Langdon, whose specialty is religious symbology, soon figures out that the words are a pair of anagrams for “Leonardo da Vinci” and “the Mona Lisa.” But what about those numbers? They may puzzle Langdon for a while, but any mathematician will recognize them at once. They are the first eight members of the Fibonacci sequence, written in a jumbled order. A young French code breaker named Sophie Neveu makes the same observation and explains that the Fibonacci sequence is one of the most famous mathematical progressions in history.
Having cracked the first two of what turn out to be a whole sequence of secret codes, Langdon and Neveu find themselves on a fast-paced adventure that eventually threatens their lives as they uncover a sinister conspiracy within the Roman Catholic Church. It’s a fantastic plot that intertwines art history and 2,000 years of church politics.
But what of the mathematical clue? In Chapter 20, Langdon recalls a lecture he gave at Harvard on the Fibonacci numbers and the closely related constant that is his favorite number: the golden ratio, also known as the divine proportion. In his lecture, Langdon makes a series of amazing claims about the prevalence of the divine proportion in life and nature, and I suspect many readers tacitly assume most of it is fiction. That is not the case. As with the novel’s many religious, historical, and art references, some of the things Langdon says about the golden ratio are false — or at least stretch the truth. But some are correct.
The divine proportion — which is sometimes represented by the Greek letter φ, generally written in English as phi and pronounced “fie” — is one of nature’s own mysteries, a mystery that was fully unraveled only 10 years ago. The quest to uncover the φ Code, as I’ll call it, provides a story with almost as many surprising turns, puzzles, and false leads as The Da Vinci Code.
The story of φ begins, like so many mathematical tales, in ancient Greece. The Greeks, with their love for symmetry and geometric order, searched for what they felt was the most pleasing rectangle. Believing that the purest and most aesthetically pleasing form of thought was mathematics, they used math to come up with an answer (see “How the Greeks Found φ,” page 69).
When Langdon begins his Harvard lecture on the divine proportion, he begins by writing the number 1.618 on the chalkboard. Strictly speaking, this is not exactly the golden ratio. The true value is given by the formula
φ = 1 + √5
Unlike authors of best-selling novels, when Mother Nature writes a mystery, she often keeps us from finding the whole answer. Like the ancient Hebrews who could never know the true name of God, we will never know the true numerical value of φ. If you try to use the formula to calculate its value, you will discover that the decimals keep on appearing. The process never stops. In mathematician’s language, the number φ is “irrational.”
As an irrational number, φ is like that other mathematical constant π, whose infinite decimal expansion begins 3.14159 ... Of the two numbers, mathematicians would say that π is more important than φ. But I have a lot of sympathy with the math major in Langdon’s class who raises his hand and says, “Phi is one H of a lot cooler than pi.” π is hot, but φ is cool.
The story continues. Having found their golden ratio, the Greeks incorporated it into their architecture, ensuring that wherever they went in their cities, their eyes would be met with glorious rectangles. This may be true, but modern historians question the claim. Certainly, the oft-repeated assertion that the Parthenon is based on the golden ratio is not supported by actual measurements.
In fact, the entire story about the Greeks and the golden ratio seems to be without foundation. The one thing we know for sure is that Euclid, in his famous textbook Elements, written around 300 B.C., showed how to calculate its value. But he seemed more interested in the mathematics than the architecture, for he gave the golden ratio a decidedly unromantic label: extreme and mean ratio. The term divine proportion first appeared with the publication of the three-volume work of that name by the 15th-century mathematician Luca Pacioli. Calling φ “golden” is even more recent: In 1835 it appeared in a book written by the mathematician Martin Ohm.
Like most successful thrillers, The Da Vinci Code switches between different plotlines, and so does our φ Code. For the next episode, fast-forward from ancient Greece to Pisa, Italy, in 1202. Young Italian mathematician Leonardo Pisano (Leonardo of Pisa) has just completed a book called Liber Abaci, or Book of Calculation. Although there is no way Leonardo can possibly know it, his book is going to change the course of human civilization by bringing to the West the method of writing numbers and doing arithmetic that we use today. That system, completed in India 500 years earlier, was much more efficient than the Roman numerals in use at the time. The new number system eventually provided the basis for the European emergence of modern science and engineering in the 16th and 17th centuries.
One of the many exercises in Liber Abaci is a whimsical problem about rabbits: A man puts a pair of baby rabbits into an enclosed garden. Assuming that each pair of rabbits in the garden bears a new pair every month, which from the second month on itself becomes productive, how many pairs of rabbits will there be in the garden after one year?
It is not hard to see that the number of pairs of rabbits in the garden in each month is given by the numbers in the sequence 1, 1, 2, 3, 5, 8, 13, ... , which is the Fibonacci sequence Langdon saw scrawled in scrambled form on the floor of the Louvre. Fibonacci is the name that mathematical historians later gave to Leonardo Pisano. It is derived from the Latin filius Bonacci, or “son of Bonacci” — Leonardo being a member of the Bonacci family. (Now there are two Leonardos in our story: Leonardo da Vinci and Leonardo da Pisa.)
The general rule that produces the Fibonacci sequence is that each number after the second 1 is equal to the sum of the two previous numbers. So 1+1 = 2, 1+2 = 3, 2+3 = 5, and so on. This corresponds to the fact that each month, the new rabbit births in Liber Abaci consist of one pair born to each of the newly adult pairs plus one pair for each of the earlier adult pairs. Once you have figured out how to generate the sequence, you can solve the rabbit problem by simply reading off the 12th number: 144 pairs.
As people solved the problem in Leonardo’s book, they began to notice that this frivolous sequence of numbers occurred so frequently in nature that not even the bungling French police captain Bezu Fache in The Da Vinci Code could fail to become suspicious. Here are some exhibits suggesting that the Fibonacci sequence is hiding a secret of nature:
Exhibit A: If you count the number of petals in most flowers, you will find that the total is a Fibonacci number. For instance, an iris has 3 petals, a buttercup 5, a delphinium 8, a ragwort 13, an aster 21, a daisy 13, 21, or 34, and Michaelmas daisies 55 or 89 petals.
Exhibit B: If you look at a sunflower, you will see a beautiful pattern of two spirals, one running clockwise and the other counterclockwise. Count those spirals and for most sunflowers you will find that there are 21 or 34 running clockwise and 34 or 55 running counterclockwise, respectively — all Fibonacci numbers. Other flowers exhibit the same phenomenon; the purple coneflower is a good example. Similarly, pinecones often have 5 clockwise spirals and 8 counterclockwise spirals, and the pineapple frequently has 8 clockwise spirals and 13 counterclockwise spirals.
Exhibit C: Take a close look at the way leaves are located on the stems of trees and plants. The leaves are located on a spiral path that winds around the stem. Starting at one leaf, count how many complete turns of the spiral it takes before you find a second leaf directly above the first. Let p be that number. Also count the number of leaves you encounter (excluding the first one itself). That gives you another number, q. The quotient p/q is called the divergence of the plant. Here’s the surprising part. If you calculate the divergence for different species of plants, you find that both the numerator and the denominator are usually Fibonacci numbers. In particular, 1/2, 1/3, 2/5, 3/8, 5/13, and 8/21 are all common divergence ratios. For instance, common grasses have a divergence of 1/2, sedges have 1/3, many fruit trees (including the apple) have a divergence of 2/5, plantains have 3/8, and leeks come in at 5/13. Clearly, there is something going on here.
Now, as in any good mystery novel, it’s time to bring the two threads together, to show how the ancient Greeks’ golden ratio and the 13th-century Fibonacci sequence are connected. Take the Fibonacci sequence and divide each number into the one that follows it. Here is what you get:
1/1 = 1; 2/1 = 2; 3/2 = 1.5; 5/3 = 1.666 ... ; 8/5 = 1.6; 13/8 = 1.625; 21/13 = 1.615 ... ; 34/21 = 1.619 ... ; 55/34 = 1.6176 ... ; 89/55 = 1.6181 ...
Wait a minute. 1, 1.6, 1.61, 1.618. This is starting to look like the golden ratio. This is no accident. Mathematicians have proved conclusively that, like two shy lovers in a Hollywood romance, the Fibonacci ratios get slowly closer to φ and finally kiss at infinity. Now that our two stories have merged, it’s time to unravel the mystery. What is so special about the Fibonacci sequence and the golden ratio?
A Golden Ratio Quiz
There are so many false claims made about the golden ratio, and so many surprising truths, that it can be hard to separate fact from fiction. Here are some of the most common statements you will find in the literature, besides the ones mentioned in the article. See how many you can correctly guess are true or false. (True means known for sure to be true; false means there is insufficient evidence to justify the statement.)
1. The Egyptian pyramids were constructed using the golden ratio.
2. Some Egyptian tombs were constructed using the golden ratio.
3. Some stone tablets show the Babylonians knew about the golden ratio.
4. The cubists based much of their work on the golden ratio.
5. The famous French architect Le Corbusier advocated and used the golden ratio in architecture.
6. The Secretariat building at the United Nations headquarters in New York comprises a stack of three golden rectangles.
7. Some Gregorian chants are based on the golden ratio.
8. Mozart used the golden ratio in some of his music.
9. Béla Bartók used the golden ratio in some of his music.
10. When a falcon attacks its prey, it swoops in along a path that is mathematically related to the golden ratio.
11. The poet Virgil based the meter of his poem The Aeneid on the golden ratio.
12. Some 12th-century Sanskrit poems have a meter based on the Fibonacci sequence (and hence are related to the golden ratio).
13. The golden ratio occurs in certain crystal structures.
14. There is a Fibonacci number with exactly 666 digits.
15. If you square any Fibonacci number, the answer will differ by at most 1 from the product of the two adjacent Fibonacci numbers.
1. F 2. F 3. F 4. F 5. T 6. F 7. F 8. F
9. F 10. T 11. F 12. T 13. T 14. T 15. T
First of all, you need to separate the truly significant facts from the accidental or the spurious. Whether or not the ancient Greeks felt that the golden ratio was the most perfect proportion for a rectangle, many modern humans do not. Numerous tests have failed to show that most observers prefer any one rectangle, and preferences are easily influenced by other factors.
Another spurious appearance of the golden ratio is one that appears in The Da Vinci Code. In his Harvard lecture, Langdon says: “Measure the distance from the tip of your head to the floor. Then divide that by the distance from your belly button to the floor. Guess what number you get.” His answer is φ. Why is this spurious? First of all, you won’t get exactly the number φ. You never can; remember, f is irrational. But in the case of measuring the human body, there is a lot of variation. True, the answers will always be fairly close to 1.6. But there’s nothing special about 1.6. Why not say the answer is 1.603? Or 1.698? More to the point, there’s no reason to divide the human body by the navel. If you spend a half hour or so taking measurements of various parts of the body and tabulating the results, you will find any number of pairs of figures whose ratio is close to 1.6 — or 1.2. Or 1.8.
Some artists have flirted with φ, but as with the Greeks and their architecture, you have to be careful to distinguish fact from fiction. The oft-repeated claims (which appear again in The Da Vinci Code) that Leonardo da Vinci believed the golden ratio is the ratio of the height of a perfect human face to its width and that he used φ in his drawing Vitruvian Man, which has a prominent role in the early part of Brown’s novel, seem to be without foundation. So, too, are the equally common claims that Sandro Botticelli used φ to proportion Venus in his famous painting The Birth of Venus and that Georges Seurat based his painting The Side Show on φ. Painters who definitely did make use of φ include the 20th-century artists Louis-Paul-Henri Sérusier, Juan Gris, Gino Severini, and Salvador Dalí; but all four seem to have been experimenting with φ for its own sake rather than for some intrinsic aesthetic reason.
The data on flowers and plants, however — Exhibits A, B, and C — are another matter. Such frequent appearances of Fibonacci numbers in nature cannot be an accident. Although many of those observations were made a hundred years or more ago, it was only in the 1990s that mathematicians and scientists were finally able to figure out what is going on. It’s a question of nature being efficient.
To achieve maximum efficiency, flower heads and plant leaves grow in a spiral fashion governed by the golden ratio. Since φ is an irrational number and the number of petals, spirals, or stamens in any plant or flower has to be a whole number, nature “rounds off” to the nearest whole number. Because of the kissing property, this will be a Fibonacci number.
That still leaves two basic questions. Why spirals? And why spirals that follow the golden ratio?
In the case of leaves, each new leaf is added so that it least obscures the leaves already below and is least obscured by any future leaves above it. Hence the leaves spiral around the stem. For seeds in the seed head of a plant, nature wants to pack in as many as possible. The way to do this is to add new seeds in a spiral fashion.
As early as the 18th century, mathematicians suspected that a single angle of rotation can make all this happen in the most efficient way: the golden ratio (measured in the number of turns per leaf, and so on). However, it took a long time to put together all the pieces of the puzzle. The final step came just over a decade ago, in 1993, with some experimental work by two French scientists, Stéphane Douady and Yves Couder.
Today, then, we have a good scientific explanation of why φ plays such a crucial role in plant growth: It’s the ratio that gives the optimal solution to growth equations. The mathematical explanation is that of all irrational numbers, φ is, in a very precise, technical sense, the furthest from being representable as a fraction.
And so the story ends. Or does it? In addition to bringing the plot to a satisfactory conclusion, a good novel should leave you wondering about some aspect of life. For The Da Vinci Code, the lingering questions left behind are how much of traditional church teachings are fact, and why people are so willing to accept things that are probably not true. Our φ Code ends with two similar questions. The literature on the golden ratio is full of claims that have little or no basis in fact and in some cases are known to be false. Why is it that these myths live on? And why are we so eager to believe everything we are told?
How the Greeks Found Φ
In his book Elements, Euclid shows how to divide a straight line running between points A and B into two by a point P so that the ratio of the longer segment (AP) to the shorter one (PB) is exactly the same as the ratio of the entire line (AB) to the longer segment (AP).
A P B
AB = AP
It doesn’t matter if the actual length of the line AB is one foot, one meter, or one shoelace length. It’s the ratios that count. So let’s say the length of PB is 1 to simplify our discussion.
With PB = 1, the length (x) of AP is what we now call the golden ratio. To calculate its value, we have to do a bit of algebra. The length of AB will be x + 1. This means we can rewrite the above geometric identity as the equation
x + 1 = x
This can be rearranged by cross multiplying to give 1 (x + 1) = (x) x, which becomes x + 1 = x2. We can then rearrange this to give the quadratic equation x2 – x – 1 = 0.
If you think back to your high school algebra class, quadratic equations have two solutions, and there is a formula to give you those solutions. When you apply this formula to the above equation, you get the two answers:
x = 1 + √5 and x = 1 - √5
Using a calculator to three decimal places, the answers are 1.618 and –0.618, respectively. The golden ratio, φ, is the first of these two solutions — the positive number.
You start to suspect there’s more to φ than meets the eye when you ask what happened to the negative solution to the quadratic equation, –0.618, which also goes on forever as a decimal. Apart from the minus sign, it looks the same as the first solution (φ) but with the initial 1 missing. But that turns out to be a false lead. Calculate a few more decimals and you will see that the two numbers are not the same. But if you dig a bit deeper still, you will find a surprising identity. The negative solution is equal to 1 - 1/φ. Hmmm. That doesn’t usually happen with quadratic equations. — K. D.