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The Math of The Da Vinci Code

Perhaps you have already read this page-turner of a murder mystery by Dan Brown [Doubleday, NY, 2003]. It has been a bestseller for well over a year. I usually shy away from popular fiction, but having received two copies for Christmas, I felt obliged this summer to read one of them. Perhaps you are planning to read it. Let me assure you that I will give away nothing of the plot by writing about some of the mathematical references in this compelling story; and I assure you that you don’t need to understand the math to follow the story.

Let’s look at sheer numbers. In the first chapter there is a murder in the Louvre, the famed art museum in Paris. This edifice warehouses over 65,000 works of fine art. The building is horseshoe-shaped, and a walk around it would entail a three-mile jaunt. There are a million square feet of space outside between the two wings of the horseshoe. The main exhibit hall, called the Grand Gallery, is visited by millions each year. Leonardo daVinci’s Mona Lisa is there. This painting and the extraordinary size of the gallery – it’s length is more than a quarter mile – are significant to the plot.

One more number to consider is 666. In the Book of Revelations, we read that this number is associated with a beast, possibly the Devil. Throughout history, kooky numerologists have tried to tie this number to a real person like the emperor Nero, Pope Leo X, or Martin Luther. At the entrance to the Louvre is a great glass pyramid, commissioned by France’s President Francois Mitterrand and designed by I.M.Pei. This precise geometric figure stands in contrast to the rococo palace that houses the museum (art imitates Las Vegas?). According to the author, Mitterrand insisted that 666 panes of glass be used. Louvre officials dispute the actual count, if not the entire story.

Crosses appear in several parts of the story. The police inspector wears a cross gemmata, containing 13 diamonds and strongly symbolic of Christ’s crucifixion. Later, an equal-armed cross appears. Its squarish shape seems to symbolize the fairness expressed in the phrases “fair and square” and “a square deal.”

I have a cross poser for you. Huntley [The Divine Proportion, Dover, 1970] credits Martin Gardner for it. In the figure below is a schematic of the Cross of Lorraine, a symbol made famous by General Charles de Gaulle, a one-time Mitterrand political rival,. It is made of 13 squares and has two transverse beams. The challenge is to draw a straight line through point P that cuts the area of the cross into two equal parts. Line UX in the figure clearly favors the lower portion, while YZ favors the upper right. I’ll reveal the surprising solution near the end of this article.

Of course, the human body with outstretched arms resembles a cross. Leonardo made a drawing called Vitruvian Man, in which the human figure also has his legs splayed apart. For a picture and related information check out Ralph M Larmann's page. (It will open in a new window. Close window to return here.) The figure fits neatly in a circle, and the limbs and the head suggest five points on it. The murder victim is found naked and posed like Vitruvian Man on the floor of the Grand Gallery. A five-pointed star, or pentangle, is drawn on his chest with blood. Written on the floor next to the body is a coded message and the numbers

13, 3, 2, 21, 1, 1, 8, 5.

I instantly recognized this special set which, albeit scrambled, belongs to the Fibonacci sequence. If you did also, my mathematical hat is off to you. The sequence in proper order runs 1, 1, 2, 3, 5, 8, 13, 21, 34, and so on to infinity. The rule for producing it is simple enough. Just add the two greatest members to get the next one (55 would be next, right?) An interesting property of the sequence is that the ratio of two consecutive members gets closer and closer to the “golden number,” discovered by Greek geometers and thought to be an important secret of Nature. To make the ratios, just divide 5 by 8, then 8 by 13, and so on. You will get ratios that converge on 0.61803398 or so. I hope you are using a calculator for this. The higher you go in the sequence, the closer you get to this irrational number – like pi, one that we can never give you the exact decimal for. Here’s how the Greeks defined it. Cut a segment into one part that measures unity, so that unity is the mean proportional of the smaller segment and the original length.

The golden number shows up in so many places in math and nature, that many think the Greeks were onto something. Is it part of a Grand Plan for the universe? Allowing for individual differences, we find the height of a human is divided into golden parts at the navel. We see Fibonacci counts in branch and seed distributions. Since it is evident in living things, has it been built into the genetic code? It is in fact built into every pentangle. Try this investigation: Mark a large perfect circle at five equally spaced points. You can do this by measuring 72-degree angles around the center. Draw a 5-pointed star inside. Find the measures (in millimeters or eighths of an inch) of a set of differing segments, and arrange them in increasing order. Calculate the ratios of consecutive measures, the smaller divided by the next larger. What do you find?

As the book’s title implies, codes play a vital role in the story. Julius Caesar is said to have used a shift cipher to protect military communiqués. All the letters in his messages were coded with letters three in advance alphabetically. Surprisingly, the same exact cipher was used by Confederate generals in our Civil War. Another ancient cipher, named Athbash, employed reversing the alphabet to encode the text. In this system A would be coded as Z, B as Y, and so on. Finally, we have a code that daVinci himself used. If you attended the exhibition of his scientific notes, as I did when it was at the Boston Museum of Science in the 1990s, you saw it in use firsthand.

But I promised not to give away any of the plot of the novel, so I can’t tell you how to crack his code. Wait! I have a cunning plan! If I show you a sample and ask you to crack it, I won’t have given anything away. What follows is a message I wrote, and thanks to some tricks applied with help of our home computer, I have encoded it as Leonardo would have. Can you read it? Click here for the solution.

The solution to the Cross of Lorraine bisection involves the golden ratio once again. If segment RS is divided into this ratio at V, and there is a way to do this, then VP extended to W will do the job. In other words, if RS measures one (1), then RV measures 0.618 approximately.

Finally, I wondered whether author Brown would employ the golden ratio in the making of his book. I took the 454-page novel and divided it into golden parts. Would something interesting happen at pages 279 or 280? I note that some composers (Bartok, for one) structured some musical works to reach a climax at about 62% of the duration. I also note that the book The Da Vinci Code has not been printed in paperback yet. I didn’t jump to page 279, but I read feverishly to reach it. Sure enough, the sympathetic characters in the story, on those two pages, escape from mortal danger, and run straight into... oh, that’s right... I promised not to tell.