Thank you Fibonacci for Playing with Numbers

What do play, The DaVinci Code, multiplying rabbits, double-entry bookkeeping, and Roman numerals have to do with discovering the secrets of the universe or at least have to do with finding an easier way to do arithmetic?

In the whopper of an opening scene in Dan Brown’s alternate history page-turner The DaVinci Code, a creepy robed figure corners a terrified curator in the Louvre’s famed Grande Gallerie, a room that displays the Mona Lisa and other acclaimed works. After a short interrogation the curator comes to a bad end, but before he expires, he inscribes himself in a circular symbol drawn on the floor in his own blood. Yikes! Weird! OK, sure, the victim could have more easily written “le moine a fait”!—the monk did it!—but in that case the story wouldn’t have needed the expert “symbologist” from Harvard (Tom Hanks’s character in the movie) to exclaim, “The Vitruvian Man; it’s one of Leonardo DaVinci’s most famous sketches!” And the drawing gives the tale the opportunity to introduce Officer Sophie Nevue Saint-Claire, a slinky Parisian cryptologist who works for the French National Police. It’s she who explains that, when unscrambled and set in ascending order, the series of numbers that an ultraviolet lamp reveals on the parquetry nearby—1-1-2-3-5-8-13-21—is a numerical joke where the sum of the first two numbers equals the next. (Try it; it works.) Why would a dying man leave investigators with a plaisanterie numerique? What kind of twisted mind would play a prank like that? What do the numbers have to do with Leonardo Da Vinci? And how is Sophie Nevue involved?

We’ll get to Sophie Nevue in a bit. But for the answers to the other questions we have to refer back to another, earlier Leonardo, Leonardo Da Pisa, later known as Fibonacci. We now call that list of numbers—1-1-2-3-5-8-13-21 and so on—the Fibonacci sequence. He explored the list in his 13th-century book titled Liber Abacci (known in English as The Book of Calculations but best translated as “Abacus Book”). It’s one of the most important books ever written and one of the most forgotten. In this book Leonardo played with mathematical problems. If one began with a pair of rabbits, he asked, “How many pairs of rabbits can be produced from that pair in a year if it is supposed that every month each pair begets a new pair which from the second month on becomes productive?” Leonardo found that the rabbits would multiply over the next 12 months in the sequence 1-1-2-3-5-8-13-21-34-59 and so on until the product reached 377 if you include the initial pair.

It may or may not be useful to understand how rabbits multiply, but there is something special about the numbers of this Fibonacci sequence; they occur in other patterns in nature. The spirals of a sunflower expand in a 137º curve—a Fibonacci spiral—that is based pretty nearly in the Fibonacci sequence, and if you also count the number of lines on the sunflower you’ll likely find that 21 rows run clockwise and 34 or 55 run counterclockwise—all Fibonacci numbers. Indeed, if you count the number of petals on any flower, it’s probably a Fibonacci number. The same goes for the twists on a pine cone and the rows on a corn cob. If you slice a nautilus shell in half, lengthwise, you’ll get an almost perfect Fibonacci spiral. Remarkable, isn’t it?

But if you play with the numbers further you find that something else very interesting happens. When you divide the succeeding number pairs, 5 by 3, 8 by 5, 13 by 8, and 21 by 13 you get 1.666, 1.60, 1.625, and 1.161 respectively. The rest of the products are very similar and they tend toward the figure 1.161. From the 15th century, this figure has been called the Divine Proportion. The Greeks used it to help design their temples; they thought it the most pleasing to the human eye and designated 1.161 by the letter Φ, phi, the way π represents another key number. The Acropolis, for example, often hailed as the world’s most perfect building, is 1.161 times as wide as it is high. And if you look back to nature you find that the ratio of each spiral in a chambered nautilus to the next spiral is approximately 1to 1.61: Φ. Something’s up.

But there’s more. It is said that the length of your leg from knee to floor compared to the length from hip to floor is, you guessed it, Φ. Same goes for the measurement of the elbow to the fingertips to the measurement of the shoulder to the fingertips: Φ again. Leonardo DaVinci certainly thought this was so, and his Vitruvian Man is shot through with Φ because Leonardo took his mathematical principles from the Roman architect, Vitruvius, who claimed to have discovered the divine proportion by examining and measuring human figures. But likely, Vitruvius took the look of the proportion from the Greeks. Incidentally this neat ratio is unlikely to occur very precisely in real people in all their diversity; try measuring yourself and you’ll see how Leonardo DaVinci was playing with the truth as well as Vitruvius’s mathematics.

I promised to get back to Sophie Nevue Saint-Claire, the police code-breaker; Φ solves a riddle here, too. She’s also the grand-daughter of the late curator who inscribed himself within the bloody figure to evoke the Vitruvian Man. When she was very young, he called her “Princess.” Sophie, as it transpires in The DaVinci Code, is also apparently descended from a line of Merovingian kings who some claim to have been descendants of Mary Magdalene and (ahem…) Jesus. (Oh, it’s a long story, and I can tell you without giving anything away that, besides the Fibonacci sequence, the Knights Templar figure in it, too.) Her ancestry makes her “half-divine.” And note a little joke that the author of The DaVinci Code, who in real life is the son of a mathematics professor, plays on us: half the letters in her name, soPHIe, represent the Greek letter phi—Φ—that stands for, wait for it, the Divine Ratio! Whew!

Just one last connection to make with Fibonacci and the benefits of playing with numbers that go beyond literary thrillers: let’s talk Roman numerals. In his book Liber Abacci, Fibonacci introduced Europeans to the Hindu-Arabic system of numerical notation, the one that begins with 0 and ends with 9—the system we use today that replaced Roman numerical system. In fact, this system is enormously handy compared to Roman numerals. Leonardo of Pisa also invented double-entry bookkeeping, the method we still use to make out our office budgets, and you can read about his other contributions in Keith Devlin’s fine book, Man of Numbers. But for the people of Italy who were just beginning to enter into the commercial revival of the late Middle Ages and early Renaissance, an easily taught and easily managed number-system was absolutely essential to conducting business. Why?

Because however wonderful Roman numerals look in carvings they’re murder to manage in practice. It you want to see just how useful the numbers 0-9 are, try adding these Roman numerals:

 

      XIV

+ XVII

 

No wonder we should thank old Fibonacci every day of our lives.