My morning with Fibonacci

The sunlit steps spiraled under my feet as I climbed up the stone tower of Porte Saint Jacques one crisp January morning in Parthenay.  The staircase is narrow, and I kept my left palm resting against the central pillar, more for reassurance than for safety. I was reminded of something I had read about once but had not really understood, about mathematical sequences that create patterns in art and nature.

P1120124

I was deep in these reflections when I arrived at the top of the fortified gate, the northern entrance to the town that once welcome pilgrims from far and wide on their way to Santiago de Compestela in Spain. The walls around me were dotted with tiny square windows, where once, no doubt, lookouts kept watch for enemy intruders. Beyond the tower walls, the town spread out its rosy-roofs to receive whatever the winter sun had to offer. I had never seen anyone here; one of the many places in Parthenay where you are guaranteed a generous dose of solitude.

But there was someone. A young man of perhaps twenty-four sitting on the ground writing furiously. He was well-dressed with a soft plum coloured velvet cap over wavy chestunut shoulder-length locks and a long teal cape of heavy material that he had wrapped over his thigh-length boots, the leather of which was flecked with lines of tiny polished silver studs. I stood there for a while, expecting him to look up from his notebook, but he did not appear to notice me, so absorbed was he in his work.
“Good morning,” I ventured at last.
At that he looked at me. Without any surprise, as if he had been expecting my presence. He had a handsome noble face. Long dark eyelashes. Large expressive chestnut eyes bright as quicksilver. He had the complexion of a man who had be born quite fair but spent years in warmer climes.”Good morning,” he replied.
His accent suggested Italian.
“I hope I am not disturbing you,” I offered.
“Not at all,” he replied, his full lipped mouth curved into a well-defined smile. “I am not doing anything of import.”
And then seeing my curiosity rising, added, “I am writing a thought experiment involving the reproductive cycles of rabbits.”
“Rabbits?” was all I could of to say.
“Yes. How many rabbits would be born from a single pair in one year. All the rabbits are able to produce offspring at one month of age, so at the end of its second month, a female can produce a second pair of rabbits. Any mating pair always produces one new pair which are always one male and one female, every month from the second month onwards.”
“I see,” I said, although I didn’t really. “And how long can the rabbits keep doing this?”
“Oh, it has nothing to do with reality,” he said, pushing his cap back little off his face with the tip of his pencil. “In this experiment, the rabbits can mate for ever. They’re immortal!”
“That’s a lot of rabbits.”
“Yes, it is. But the point is that it produces a rather elegant-looking sequence. I think I will put it in the book I’m writing. I came across something similar in India, an 8th century mathematician named Virahanka.”

“This mathematical pattern that you get from your rabbit experiment. Do you see it anywhere else?”
He looked at me curiously.
“What do you mean?”
“Does the word Fibonnaci mean anything to you?”
“I can’t say that it does.”
“There’s a sequence in mathematics. It’s called the Fibonacci sequence. It was named after an Italian mathematician.”
“I see.”
“It’s a series of numbers that builds by adding up the two previous numbers.”
“What does it start from?”
“1, 1 I think.”
“Ah, the original pair! So then it would go from 1, 1, to 2 to 3, as the sum of 1 and 2, to 5 as the sum of 2 and 3, then 8, 13, 21, 34 and so on. How marvelous! It’s just like my rabbits!”

P1120135He began scrawling furiously in his notebook, then looked at me again, his face flushed with excitement. He ran to the spiral staircase a gazed down it, then returned to his scribblings.
“Do you know that this sequence you just described appears in equiangular spirals.”
“Like the staircase?”
“No, not this one. You’ll notice this staircase doesn’t scale up, it’s steps remain equal througout. But there is an example of such a spiral in the Vatican library, I believe.”
“What’s so special about these spirals?”
“An equiangular spiral is one where a radial line from the centre always makes the same angle to the curve. Archimedes wrote about. It creates levels of developmental that are always self…self…” he grappled for the term, “self-similar.”
I was feeling in over my head, but I could not stop now.
“What does that mean?”
“It means that growth occurs but the fundamental shape is maintained.”

“I see,” I said again, now more sure than ever that I did not. But my mind was grappling with other problems. This did not seem to be a man in fancy dress and yet his fashion was clearly of a far earlier century, 13th or 14th, she guessed.
Was it possible?
He kept writing for a few more minutes, so lost in contemplation that it was as if their conversation had never happened and she began to wonder if she should leave him to it. Then he leaped suddenly up onto his feet.
“I am so happy to meet you! You have really made my day,” he exclaimed, offering his hand.
It was the perfect opportunity.
“Jude,” she replied. “Jude Windsor.”
“Leonardo.”
He shook her hand vigorously with both of his.
“Look at this,” he said, excitedly pointing to his notebook where he had sketched a series of interconnecting squares.

fibo

“Okay. It is far less complicated than it looks. Think of a square that equals the measurement 1, whatever that measurement is, it doesn’t matter. Remember the rabbits. You need two to create more. So add another square beneath it, with the same measurement of 1. Then next to it you create a square that is double the size of one of the original pair. 1+1 = 2. Yes, it is that simple. Then beneath that you add a square that is the result of adding the first square to the second. 1 + 2 = 3. Easy! Now, next to this we add another square that is five by five. Why? Because it is the result of the third and fourth square, in other words, the result of 2 + 3, which equals 5. Above you create another square which is eight by eight, created from adding 3 to 5. So, then we go 5 + 8 = 13, and so on. Each next term is the sum of the two previous terms. When you draw a quadrant in each square, the result is…”
“…a spiral.”
“Exactly.”
“It looks like a shell.”

nautilus

“Yes. It looks like a shell.”
His eyes then turned inwards again, as if he could only maintain an exterior engagement for a brief period of time before falling back into his natural state of musing. A chill tinged the air around us, and I hugged my coat more closely around me. I could not shrug off the feeling that I was talking to a man out of time. As if sensing my doubts, he quickly made his excuses and descended the spiral staircase. I watched from the tower, waiting for him to appear, either along the promenade of the river or across the bridge, or walking south along the cobbled street towards Eglise Saint Laurent. But he appeared in neither quadrant, and I surmised that he must have taken a quick turn around the corner into a parallel street.

Later that evening, as I poured over Google, I found the story of Fibonacci. I froze as I read that his given name was Leonardo. I wanted to run back to the tower and up those spiral stairs right there and then, but it would be no use, I knew. What I could have learned from him! All the questions I could have asked! It plagues me to this day. He had been born into a wealthy trading family in Pisa. In his travels throughout the Middle East in his twenties, he was explored the mathematical ideas that had originated in India and had been passed on through Arabic countries. His 1202 book, Liber Abaci, revolutionized Western mathematics, by introducing Hindu system of numbers (that we now call Arabic notation) to the West replacing Roman Numerals.

The Fibonacci sequence is found in sunflowers and artichokes, in leaves, in music, in art and architecture, in the branching of trees (trees grow by spiraling), and even in clouds and solar systems. Mathematics becomes much more interesting when we view it as the science of patterns. Remembering that Fibonacci numbers are:

1, 1, 2, 3, 5, 8, 13, 21, 34, 55, 89, 144, etc.

The number of spirals on a sunflower or a pineapple is typically a Fibonacci number, as is the number of petals on a flower. A lily has 3 petals. Buttercups have 5. Chicory has 21. Daisies have 34. The DNA molecule measures 34 angstroms long by 21 angstroms wide for each full cycle of its double helix spiral, both numbers in the Fibonacci series.

In geometry, a golden spiral grows by a factor called the ‘golden ratio’ or φ (phi, after Phidias, a sculptor who is said to have employed the golden ratio in his work). The Golden ratio is a geometric relationship between two parts. It is a special number found by dividing a line into two parts so that the longer part divided by the smaller part is also equal to the whole length divided by the longer part. A golden spiral gets wider or further from its origin every quarter turn it makes. It was used to create great architecture such as the Parthenon and the Pyramids of Giza.

Golden ratio was used to achieve symmetry and beauty in Renaissance art. Da Vinci used the Golden ratio in all the proportions of his Last Supper. And yes, as my friend Leonardo had suspected, there is it, all nautilus-like in the spiral staircase of the Vatican library.

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