— Both are attempts to make sense of things, to shape aesthetic universes that bear directly upon our own.
Musical Qualities of Mathematics—
Hardy—“Beauty is the first test: there is no permanent place in the world for ugly mathematics.”
Potncaré—“The feeling of mathematical beauty, of the harmony of numbers, of forms, of geometric elegance.”
The Golden Ratio was considered by the Pythagoreans to be the most beautiful of proportions.
The golden rectangle with sides in that ratio has been linked with the proportions of the Parthenon.
Luca Pacioli—1509—De Divma Proportione Da Vinci
The Golden Ratio
A line is divided into a golden section when the ratio of its 2 parts is the same as the ratio between 1 part and the whole. The ratio, that is, reproduces itself within itself. The diagonals of a pentagon divide each other in this ratio.
As a fraction it is composed entirely of 1 ‘s layered in an infinite series. The number becomes a sort of arithmetic “image” of the geometric property of the ratio. It is represented endlessly within itself. If a square formed by 1 side of a golden rectangle is cut off, a golden rectangle remains. If squares are continually removed, there is an infinite spiral of golden rectangles contained within each other.
If a curve is drawn based upon the golden rectangle, it is precisely the shape of a chambered-nautilus shell. It is a logarithmic curve of continuous growth. Any two segments of the curve are the same shape; they are just different sizes.
As a snail grows, it produces shell material in the same formation. Similar curves lie in the center of a sunflower, in the shape of a fir cone, and in other natural forms that contain the golden ratio.
In the manipulation of abstract material, which reveals new relations and structures, math and music find their common formal ground.
In Beethoven piano sonatas there is the sense that a concentrated exploration of musical elements is taking place as one listens; when a theme returns in a recapitulation, it is no longer heard as it was in the beginning.
— This aesthetic has been central to the West and is implicit in the golden ratio. This concept of beauty involves proportion between various elements and a relation between parts and whole — a reproduction of macrocosm in microcosm.
I never asked my father about the golden rectangle. I wonder if he still thinks about that divine proportion now as he stares out to sea and the waves crash against his ocean liner and the sky begins to darken for what will seem like forever.
It is too late for me to ask him, and, even if he were here, I probably would not dare to. I violated his privacy that day, the method of his life. I stepped right into one of his unspoken obsessions, though I did not mean to do so. I was only a child that day looking for boots, only a curious child who loved this kind stranger called Father.
Now the dark is coming on. My father bends down and puts his hand in the frigid water. The snow starts to throw the towers and gables of the Baltic into romantic relief. Ships move about the harbor. A waltz plays. My father in a white tuxedo stares mesmerized by the dance of shadow and light.
Daddy, I would label every leaf on every tree for you. I would wedge my fingers into the wind and bring it to your ears so that you might hear what it whispers. I would build fires around your cool body and teach you to sing. I would shape your soft skull into the fleshy bulbs of lilies or tulips that bloom, then rest, then bloom again. I would make the daylight fluid and let you swim in its secrets, if I could.
“She is princess twice,” my father reads from the paper, “a duchess four times, nine times a baroness, eight times a countess. However, since a majority of the prince’s domain now exists in name only, her kingdom, in reality, is indeed a small one, covering three towns and 22,000 people.”
He comes forward. He hesitates. He stops. He must be walking in his sleep again. He barely looks like my father. He seems shorter somehow — older. “Daddy,” I say, hoping he might speak more easily in his sleep, hoping he might tell me what dream makes him this way.
My grandfather lilts his ax. When it is poised above his head, my father, just a boy, freezes the scene. He is afraid to watch the ax drop, for my grandfather is not chopping wood as one might expect. My father pulls himself from the bed and moves closer to the window. He rubs his eyes just to be sure and then he sees it: his father is cutting down the beautiful tomato plants, grown from seed, hacking them down to the ground. Earlier that season they had put up stakes together for those fragile plants to hold on to.
Is this what my father means when he says there are things it is better to forget? Is this what he is forgetting — his own father out in the garden chopping the tomato plants into pieces, insisting that they are Americans now, not Italians? Did his father announce that there will be no more Italian spoken in his house? No more wine drunk with lunch, as he burned the grape wines? Did he tell his wife there would be no more sad songs from the old country? How much she must have wept, hugging her small son to her breast!
My grandfather takes his ax from the toolshed, and when he lifts it above his head the scene freezes — but only for a moment. He hacks down those sweet tomatoes while the small boy looks on from his bedroom window and the eggplant and the peppers cower in terror.
“Vivaldi,” my father says. “Albinoni.”
“Albinoni,” I say.
“Paganini.”
“Paganini.”
“Corelli.”
In November the turkey industry presented a fifty-five-pound turkey to the President, but Kennedy spared its life, my father read.
“It’s local fair time in Ashtabula, Ohio,” the fat man reads from the newspaper, “where you will find the prize bulls, homemade pies, merry-go-rounds, animal freak shows, vegetable contests — prize pumpkins.
“Pumpkins,” the fat man puzzles.
“You know, Father,” the girls shout, “what we carve and put in the windows at Halloween.”
He turns the page to the next article. “Ah, yes,” he says.
“Alabama,” the father reads.
“Alabama,” Christine says. Such a pretty name, she thinks. She says it out loud, “Alabama. Alabama.”
I picture my father being an avid newspaper reader once, opening it over a breakfast of cereal and eggs and folding it expertly so as not to get it in our faces. I seem to remember that: his daily origami ritual as he routinely turned the New York Times into a square, a rectangle, a bird, while we watched.
We had a television in those days, too, I think, and he liked to watch the news in the evenings until the news event occurred that made all other news unnecessary — the news event so great that it allowed his mind to wander around it tirelessly for years.
He sits transfixed, watching the six gray horses draw the caisson that holds the flag-draped coffin Behind the caisson is a riderless, chestnut-brown horse. Empty boots pointing backwards hold themselves somehow in the stirrups. A beautiful mother in a black veil holds the hands of two small children.
My father moves to the piano, a giant dwarfing the keys. Hunched over, he plays the Goldberg Variations with a heavy-handed deliberateness. He moves back to the television.
I grew up regretting in a mild way the death of our handsome president but mourning the realization that my father was not a happy man and that he probably never had been. I have linked in my mind, unfairly, the death of President Kennedy with my father’s great sadness because I never really noticed it before that day. Surely at that moment as my father sat listlessly in front of the TV set, his head in his hands, he must have abandoned the dream of the golden rectangle forever. Still — he did not destroy the notebook. It was there for me to find on that fleeting, snowy afternoon a few years later.