"I've been thinking about the numbers all night. Sums, quotients, products. I don't see anything. Mathematically, they're arranged at random. Cryptographic gibberish."
"And yet they're all part of the Fibonacci sequence. That can't be coincidence."
"It's not. Using Fibonacci numbers was my grandfather's way of waving another flag at me - like writing the message in English, or arranging himself like my favorite piece of art, or drawing a pentacle on himself. All of it was to catch my attention."
"The pentacle has meaning to you?"
"Yes. I didn't get a chance to tell you, but the pentacle was a special symbol between my grandfather and me when I was growing up. We used to play Tarot cards for fun, and my indicator card always turned out to be from the suit of pentacles. I'm sure he stacked the deck, but pentacles got to be our little joke."
Langdon felt a chill. They played Tarot? The medieval Italian card game was so replete with hidden heretical symbolism that Langdon had dedicated an entire chapter in his new manuscript to the Tarot. The game's twenty-two cards bore names like The Female Pope, The Empress, and The Star.Originally, Tarot had been devised as a secret means to pass along ideologies banned by the Church. Now, Tarot's mystical qualities were passed on by modern fortune-tellers.
The Tarot indicator suit for feminine divinity is pentacles, Langdon thought, realizing that if Sauniere had been stacking his granddaughter's deck for fun, pentacles was an apropos inside joke.
They arrived at the emergency stairwell, and Sophie carefully pulled open the door. No alarm sounded. Only the doors to the outside were wired. Sophie led Langdon down a tight set of switchback stairs toward the ground level, picking up speed as they went.
"Your grandfather," Langdon said, hurrying behind her," when he told you about the pentacle, did he mention goddess worship or any resentment of the Catholic Church?"
Sophie shook her head. "I was more interested in the mathematics of it - the Divine Proportion, PHI, Fibonacci sequences, that sort of thing."
Langdon was surprised. "Your grandfather taught you about the number PHI?"
"Of course. The Divine Proportion." Her expression turned sheepish. "In fact, he used to joke that I was half divine... you know, because of the letters in my name." Langdon considered it a moment and then groaned.
s-o-PHI-e.
Still descending, Langdon refocused on PHI.He was starting to realize that Sauniere's clues were even more consistent than he had first imagined.
Da Vinci... Fibonacci numbers... the pentacle.
Incredibly, all of these things were connected by a single concept so fundamental to art history that Langdon often spent several class periods on the topic.
PHI.
He felt himself suddenly reeling back to Harvard, standing in front of his" Symbolism in Art" class, writing his favorite number on the chalkboard.
1. 618
Langdon turned to face his sea of eager students. "Who can tell me what this number is?"
A long-legged math major in back raised his hand. "That's the number PHI." He pronounced it fee.
"Nice job, Stettner," Langdon said. "Everyone, meet PHI."
"Not to be confused with PI," Stettner added, grinning. "As we mathematicians like to say: PHI is one H of a lot cooler than PI!"
Langdon laughed, but nobody else seemed to get the joke. Stettner slumped." This number PHI," Langdon continued," one-point-six-one-eight, is a very important number in art. Who can tell me why?"
Stettner tried to redeem himself. "Because it's so pretty?" Everyone laughed." Actually," Langdon said," Stettner's right again. PHI is generally considered the most beautiful number in the universe."
The laughter abruptly stopped, and Stettner gloated.
As Langdon loaded his slide projector, he explained that the number PHI was derived from the Fibonacci sequence - a progression famous not only because the sum of adjacent terms equaled the next term, but because the quotients of adjacent terms possessed the astonishing property of approaching the number 1. 618 - PHI!
Despite PHI's seemingly mystical mathematical origins, Langdon explained, the truly mind-boggling aspect of PHI was its role as a fundamental building block in nature. Plants, animals, and even human beings all possessed dimensional properties that adhered with eerie exactitude to the ratio of PHI to 1.
"PHI's ubiquity in nature," Langdon said, killing the lights," clearly exceeds coincidence, and so the ancients assumed the number PHI must have been preordained by the Creator of the universe. Early scientists heralded one-point-six-one-eight as the Divine Proportion."
"Hold on," said a young woman in the front row. "I'm a bio major and I've never seen this Divine Proportion in nature."
"No?" Langdon grinned. "Ever study the relationship between females and males in a honeybee community?"
"Sure. The female bees always outnumber the male bees."
"Correct. And did you know that if you divide the number of female bees by the number of male bees in any beehive in the world, you always get the same number?"
"You do?" "Yup. PHI." The girl gaped. "NO WAY!"
"Way!" Langdon fired back, smiling as he projected a slide of a spiral seashell. "Recognize this?"
"It's a nautilus," the bio major said. "A cephalopod mollusk that pumps gas into its chambered shell to adjust its buoyancy."
"Correct. And can you guess what the ratio is of each spiral's diameter to the next?" The girl looked uncertain as she eyed the concentric arcs of the nautilus spiral. Langdon nodded. "PHI. The Divine Proportion. One-point-six-one-eight to one." The girl looked amazed.
Langdon advanced to the next slide - a close-up of a sunflower's seed head. "Sunflower seeds grow in opposing spirals. Can you guess the ratio of each rotation's diameter to the next?" "PHI?" everyone said." Bingo." Langdon began racing through slides now - spiraled pinecone petals, leaf arrangement on plant stalks, insect segmentation - all displaying astonishing obedience to the Divine Proportion.
"This is amazing!" someone cried out.
"Yeah," someone else said," but what does it have to do with art?"
"Aha!" Langdon said. "Glad you asked." He pulled up another slide - a pale yellow parchment displaying Leonardo Da Vinci's famous male nude - The Vitruvian Man - named for Marcus Vitruvius, the brilliant Roman architect who praised the Divine Proportion in his text De Architectura.
"Nobody understood better than Da Vinci the divine structure of the human body. Da Vinci actually exhumed corpses to measure the exact proportions of human bone structure. He was the first to show that the human body is literally made of building blocks whose proportional ratios always equal PHI."
Everyone in class gave him a dubious look.
"Don't believe me?" Langdon challenged. "Next time you're in the shower, take a tape measure."
A couple of football players snickered.
"Not just you insecure jocks," Langdon prompted. "All of you. Guys and girls. Try it. Measure the distance from the tip of your head to the floor. Then divide that by the distance from your bellybutton to the floor. Guess what number you get."
"Not PHI!" one of the jocks blurted out in disbelief.
"Yes, PHI," Langdon replied. "One-point-six-one-eight. Want another example? Measure the distance from your shoulder to your fingertips, and then divide it by the distance from your elbow to your fingertips. PHI again. Another? Hip to floor divided by knee to floor. PHI again. Finger joints. Toes. Spinal divisions. PHI. PHI. PHI. My friends, each of you is a walking tribute to the Divine Proportion."