A photographic tale

 Artef Fierel and the Garden of Proportions

A JOURNEY THROUGH PHILOSOPHY, SCIENCE AND THE ART OF PHOTOGRAPHY

A proportion can guide me. The responsibility for what I show remains mine.

The morning after his return from Athens, Artef Fierel woke with the feeling that someone was still talking in the courtyard. He listened. From the street came only the rumble of an old van’s engine and the creak of a garage shutter struggling upwards.

On the table, beside the camera, lay the magnifying glass, the notebook and Sophie’s World. The sheet of paper that had invited him to bring along “the room that remembers” had disappeared. Artef opened the notebook. Between two pages he found a slender blade of grass. It might have been there since his days in the meadow, or slipped from his jacket as he put away his notes. He brushed it with a finger, as though its fragility might help him distinguish a memory from a fantasy. He laid it on the windowsill and returned to the camera.

The portrait of the three philosophers appeared on the screen: Plato beside the column, Aristotle gazing towards the passing swallow, Democritus smiling in the light. The photograph was still there. Artef, however, could not have pointed out a route back to that courtyard, which was already beginning to recede in his memory. He lingered over the details. The column held his gaze close to Plato; Aristotle’s eyes led it towards the edge of the image; Democritus’s smile drew it back to the other side. Between the vine and the column, the spider’s thread could still be made out. Artef had chosen the camera’s position, the aperture and the shutter speed. He had not placed that thread there, nor commanded the swallow to pass.

The evening before, he had wondered what continued beyond the edges of a photograph. Now he was curious about what happened within them. Why did those very different presences seem to hold together? Did the order belong to the courtyard, to the image or to his way of looking at it? In the notebook he wrote: “If every framing is a choice, what guides me when I seek harmony?”

He left Sophie’s World open on the table, slipped the notebook and magnifying glass into his jacket pocket, took the camera and went out. This time he had found no invitation between the pages. Those questions had been enough to draw him outside.

The city was already awake, as always. Buses stopped at junctions, passers-by crossed the strips of sunlight between the buildings, shutters rose with a metallic clatter. In a shop window, Artef once again saw his own face superimposed on the windows of the buildings opposite, as he had the evening before. He took a step to one side: the reflection of a cornice crossed his forehead and immediately vanished. It took so little to change the order of things.

He walked without a particular destination until he reached the botanical garden, tucked between a library and a red-brick building. He was certain he had been down that street before, but he did not remember the gate. He went in. On the path he noticed a pine cone that had fallen at the foot of a tree. He picked it up and slowly turned it in his hands. The scales were not arranged in straight rows, but formed spirals winding in opposite directions. He tried to count them. Five one way, eight the other. He started again, afraid he had made a mistake, but the result did not change. A little further on he found another. He turned it in the light and followed the scales with his fingernail: eight spirals in one direction, thirteen in the other. He placed them side by side at the edge of the path and photographed them, so that he could count them again at leisure.

— Five, eight, thirteen — said a voice behind him. — I see you have started counting them too.

Artef turned. An elderly man in a light-coloured hat was sitting on a bench with a book open on his knees. A walking stick rested against the armrest. He kept a finger between the pages.

— Have you counted them too? — asked Artef.

— A long time ago. Then I began looking for those numbers everywhere.

— And did you find them?

— Often enough to become excited. Sometimes even where they weren’t.

The man turned the book towards him. A sequence was written on the page:

1, 1, 2, 3, 5, 8, 13, 21…

— Look closely — he said. — After the first two, each number comes from the sum of the two before it.

Artef followed the sequence with a finger.

— So after twenty-one comes thirty-four.

The old man smiled.

— Thirty-four — he confirmed. — And then fifty-five.

Artef looked back at the pine cones.

— Fibonacci?

— That is the name by which we know this sequence. Leonardo of Pisa, known as Fibonacci, presented it in the Liber Abaci in the early thirteenth century, through a problem about rabbits. He had encountered Hindu-Arabic numerals in the Mediterranean and helped spread their use in Europe.

The man stroked the edge of the page.

— But these numbers already had a history. In India they had appeared in the study of poetic rhythms. Before reaching your pine cones, they had passed through the syllables of a poem.

Artef sighed. From Alice’s White Rabbit to Fibonacci’s rabbits, something had found a way to lead him elsewhere.

— Do you work here?

The old man picked up his walking stick.

— I spend a great deal of time here. Come, I want to show you something.

· · ·

They reached a bed of sunflowers. Some had bowed their heads, their dark discs already full of seeds; others still wore a crown of yellow petals. The man pointed out the interwoven spirals of a flower head and helped Artef follow them without counting the same one twice. They reached thirty-four in one direction and fifty-five in the other. On a nearby flower the pattern was harder to read: an empty patch interrupted the rows.

— Here, too, new elements form as the plant grows — the old man explained. — In many arrangements of this kind, the angle between one element and the next is close to a hundred and thirty-seven and a half degrees. We call it the golden angle.

He traced the path with his finger without touching the flower. That angle helped describe a distribution in which the elements did not line up along just a few directions. The visible spirals emerged as many small elements fitted together. The sunflower had not begun with a complete design: it had developed one element at a time.

Artef brought the camera close to a sunflower. On the screen, the flower head became a landscape of seeds, curves and shadows. He placed its centre exactly in the middle of the image and took a photograph. The result was steady, almost hypnotic. Then he shifted the framing slightly. He let the sunflower’s centre move towards one side and the spirals seem to continue beyond the edge. The second photograph was less regular, but suggested movement. The eye entered the flower and seemed to carry on beyond the frame.

— Which of the two is correct? — asked Artef.

He immediately thought back to the courtyard: he had already learnt that two different images could show different aspects of the same thing.

— Or perhaps I should ask what changes between them?

— Look at them — the man replied. — In the first, you keep returning to the centre. In the second, you look for what continues beyond the edge. Which of these movements interested you?

· · ·

Near the sunflowers grew a small plant whose leaves were arranged around the stem without completely overlapping.

Artef looked at it from above. The leaves faced in different directions, partly limiting their overlap. He moved his head and saw the light reaching even the lower ones.

— Aristotle would have wanted to watch it grow — he murmured.

— That would be a good way to begin — said the old man.

Looking around, however, Artef also saw damaged leaves, flowers with an irregular number of petals, branches broken by the wind and plants that had grown in unpredictable directions.

— Not everything follows Fibonacci — he observed.

— No. If we do not find the numbers we expected in a plant, we must check our reasoning. It is not the plant that has to obey our rules.

The man pointed to the flower beside the path.

— Those numbers recur in certain forms of growth. They do not give flowers marks.

Artef thought of the vine he had photographed in Athens. The image allowed him to examine its leaves, but was not enough to explain its life. A sequence of numbers, too, could aid observation and, at the same time, lead him to believe he had already understood everything.

He bent down before a corolla. One petal was folded, another bore the mark of a bite. Artef moved the camera until the light passed through the petal’s thin edge around the missing part, then took the photograph.

Looking at the image afterwards, he realised that his eye returned to that very spot.

· · ·

The path ended in front of the library.

The man opened the door and invited Artef to follow him. Inside, light fell from the tall windows and divided the room into large rectangles of light and shade. Stairs rose towards a gallery, while the tables formed lines of perspective leading to a distant door.

A volume on geometry lay open on a lectern.

— Long before Fibonacci — said the old man — Greek mathematicians knew of a particular way of dividing a line segment. Euclid called it division in extreme and mean ratio.

He took a sheet of paper and drew a line, dividing it into a longer part and a shorter one.

— In the golden section, the ratio of the whole line to the larger part is equal to the ratio of the larger part to the smaller. This relationship gives us a number whose value is approximately one point six one eight.

Artef looked at the line, then at the sequence in the old man’s book.

— And those numbers we counted? How are they related to this?

The man wrote on the sheet: 8 ÷ 5 = 1.6; 13 ÷ 8 = 1.625; 21 ÷ 13 ≈ 1.615.

— If you divide a term by the one before it, as you move along the sequence, you come closer and closer to the golden ratio. The pine cones brought us to a sequence; the sequence leads us to a ratio. They are different things, but here they meet.

Artef copied the divisions. That step seemed more surprising to him than a simple resemblance: he could repeat it and check what happened.

— And where does the gold come from? — he asked, pointing to the name of the proportion.

— The name came much later. For Euclid, it was chiefly a geometrical problem. In the Renaissance, however, that proportion also acquired symbolic meanings. Luca Pacioli devoted his treatise De divina proportione to it, published in 1509 and illustrated by Leonardo da Vinci. He saw in it a relationship between mathematics, art and the order of creation.

The old man turned a few pages. Pentagons, pentagrams and geometrical solids appeared. In the drawing of a regular pentagon, the diagonals formed a star and enclosed a smaller pentagon. The man showed Artef where the golden ratio reappeared. This time the relationship did not depend on how a flower had grown: it belonged to the geometrical construction.

— The Pythagoreans regarded number as a key to harmony — the man continued. — They had discovered that certain musical intervals, too, could be described through simple numerical ratios. From this arose one of the most enduring ideas in Western philosophy: that beneath the apparent confusion of the world there is an intelligible order.

— Plato told me that an image can make us forget what it does not show — said Artef. — Can a perfect form make us forget something too?

The old man looked at him without asking when he had spoken to Plato.

— For him, geometry helped the mind turn towards what does not change. The line drawn on the paper is not perfect, yet it allows us to conceive of a line that is.

He paused to look at the pencil mark.

— But when you raise your camera, what you encounter is precisely this paper, this light, this slightly crooked stroke. If you seek only the ideal figure, you risk missing everything else.

Artef raised the viewfinder to his eye and framed the room. The tables and the gallery traced a rigorous geometry. A librarian was putting some volumes back in place. Artef showed her the camera and asked whether she would be happy to appear in the photograph he was about to take of the room. She agreed and went back to her work. As she passed between the tables, she opened a volume to consult a page. Artef took the photograph. In the image, her figure interrupted the symmetry and the tilted pages caught the light.

On another table lay a large reproduction of the Parthenon. Artef recognised the columns he had seen overlooking Athens from the hill of olive trees, and stopped. Nearby were images of the Great Pyramid of Khufu, the Mona Lisa and several Renaissance buildings. Someone had drawn golden rectangles and spirals over every image.

— Did they all use the golden section? — asked Artef.

The old man slowly shook his head.

— This drawing is not enough to tell us. In some works, we find proportions close to the golden ratio; in others, all it takes is a careful choice of the endpoints to measure. Look.

He placed a transparent sheet over the reproduction and shifted the rectangle a few millimetres. One coincidence disappeared; another took its place.

— So is it all false?

— It means we must distinguish what we find in the work from what we know about its design. To attribute a choice to the artist, I would look for documents and preparatory drawings; I would check the measurements and the reason they were taken at those particular points. The overlay alone does not tell us how the work came into being.

Artef thought of the philosophers’ portrait. He could have superimposed a spiral on it, enlarged it, rotated it and moved it until it connected the three faces. But that drawing would not have proved that he had composed the photograph according to the golden ratio.

He switched on the camera and showed the portrait to the old man. The man studied it for a long time.

— They look as though they had quite a lot to discuss.

— They had asked not to appear to be in agreement.

— Then the arrangement of these three men already has a reason. Before looking for another, remember that one.

Artef put the camera away. On the transparent sheet the rectangle had remained still; on the page beneath, it continued to prove nothing on its own.

They left the library and reached a small greenhouse.

A snail was slowly making its way across a damp stone. Its shell wound around a centre, growing larger with each turn.

— The golden spiral — said Artef.

— A spiral, certainly. But recognising the way it coils is not enough to call it golden.

Artef looked at him in surprise. The man explained that the growth of many shells could be described, at least in part, by a logarithmic spiral: a curve that widens in a regular way, with the distance from the centre multiplied by the same factor at every quarter-turn. The real shell, however, was a solid body, with thickness and irregularities. Even a good geometrical description showed only certain aspects of it.

— The golden spiral is a particular case — he added. — Not every logarithmic spiral is golden. Not even the nautilus, which appears in so many illustrations, is evidence to be accepted without taking measurements.

Meanwhile, the snail had reached the edge of the stone. The name Artef had tried to give it had not held it back.

Artef crouched down without moving it. In the first photograph he left space in front of its body: the damp streak on the stone led to the shell and helped reveal its short journey. Then he took a second photograph from very close up, cropping out part of the shell. The first showed the relationship between the animal and its surroundings. The second turned the shell into an abstract form in which the animal almost disappeared. Artef returned to the first: that morning he was also interested in knowing where it was going.

When they went back outside, the old man gathered four thin branches and laid them on the ground to form a rectangle.

— Imagine this is your frame.

With more branches he traced two vertical and two horizontal lines. They did not divide the image into exactly three equal parts: they were a little closer to the centre.

— These lines roughly mark the golden divisions. Their intersections can suggest where to place an important subject.

Artef placed a small leaf on one of the intersections.

The composition looked balanced without being perfectly symmetrical. On one side was the subject; on the other, a larger space remained in which the eye could continue to move.

— It looks like the rule of thirds — he observed.

— The lines of the golden grid are a little closer to the centre. The thirds, on the other hand, divide each side into three equal parts. You can try both. But this leaf also carries weight in your gaze because of its colour, the light and what is near it. Coordinates do not decide everything.

Artef remembered some landscapes in which he had placed the horizon too close to the centre without a clear reason. Dividing the image could help him choose how much space to give to the sky and how much to the land.

He looked at the rectangle of branches. A long shadow crossed one of its sides.

A very low horizon could make the sky seem immense. A high horizon could give weight to the land. A central horizon could create stillness, silence or a perfect relationship between a mountain and its reflection.

A gust blew the leaf away from the intersection. Artef started to put it back, then let his hand drop. Even in that position, the leaf created a balance with the shadow.

After stopping at the fountain, they reached the spiral staircase leading to a terrace. The old man climbed slowly, pausing at every turn. Artef followed, listening to the walking stick tapping against the steps.

From above, the railing wound around the central opening. A visitor was about to climb the stairs. Artef asked whether he could photograph him on the staircase; the man raised a hand in agreement.

Artef looked for a position from which the curve of the railing would lead the eye towards the lower steps. He thought of the spiral that cropping software superimposes on images to help with composition. He could use it as a guide without the shape of the staircase having to match the spiral perfectly. He waited.

The figure entered the frame, climbed a few steps and reached the point towards which the lines converged. Artef took the photograph.

On the screen, the illuminated railing led to the figure. Artef could follow that path even without imagining a spiral over it.

A few steps higher, the visitor stopped to catch his breath and looked towards him. Artef shifted the framing and brought him to the centre.

He took another photograph.

The curve now seemed to gather around the figure. The first image accompanied him as he climbed; this one held him in a pause.

— And your grid? — asked the old man.

— I did not need it for this photograph.

— Has the photograph become worse?

Artef compared the two images.

— No. It tells a different story.

The old man nodded. Artef showed the two photographs to the visitor when he reached them. The man preferred the second: he said he did not remember being so hunched over in the first.

From the terrace they could see rooftops, streets, trees and people.

The city no longer seemed to be composed only of straight lines. Cars followed curves, flocks of birds changed direction, passers-by spread across the squares and shadows slowly lengthened.

Artef looked for the golden ratio in everything.

For a few minutes he seemed to find it everywhere: in the shape of a window, in the distance between two trees, in the outline of a building and even in the chance position of a man sitting on a bench.

To make the numbers work, however, he had to keep changing the rectangle he imagined around things.

He realised that he was repeating the gesture he had seen in the library: moving the edges until the world proved him right. He lowered the camera. In the square, the man sitting on the bench stood up and walked out of the proportion Artef had assigned to him.

— Why do we want beauty to depend on a number? — he asked.

The old man was silent for a while before replying.

— Because a formula reassures us. If beauty depended on a precise proportion, we could measure it, repeat it and perhaps even possess it. But aesthetic experience is more unsettled. It depends on form, certainly, but also on memory, culture, the moment and the person looking.

Artef showed him the philosophers’ portrait again. Plato continued to look beyond the scene, Aristotle followed the flight, Democritus smiled. He could have arranged their faces according to a different proportion. That would not have helped him understand their disagreement any better.

— I have read that Aristotle regarded order and proportion as characteristics of beauty — he said. — And yet that flower, even with part of a petal missing, still seems beautiful to me.

— Order can contribute to beauty, but it is not enough to explain it. Think of the question Kant would address many centuries after Aristotle: can we prove that something is beautiful as we prove a theorem? You can measure the parts of an object and verify that they conform to a certain proportion. But that verification alone is not enough to prove its beauty.

The sun was beginning to sink.

Light entered sideways between the buildings. They returned towards the sunflowers, whose shadows had lengthened across the ground. The old man sat on the bench and reopened his book. Artef walked a few steps further and stopped before the same sunflower he had photographed that morning.

He tried placing the flower head close to a golden division, then in the centre. He already knew those two images. A bee was moving along the edge that was still in bloom.

He lowered the camera to follow it, then moved just enough to avoid covering it with his own shadow.

The bee passed from one floret to another and reached a point far from the intersections Artef had imagined. He tilted the framing slightly. A leaf entered the foreground; a patch of light remained in the background. He moved a little more to prevent that brightness from washing out the insect’s outline.

The composition had changed. Artef could still see things he might correct, but the bee was already about to take flight.

He chose a fast shutter speed and focused. He remembered the spider’s thread in the Athens portrait: the world had gone on working even as he decided how to show it.

Artef took the photograph.

When he turned to show the old man the photograph, the bench was empty.

On the wooden seat he found only the book. Its pages fluttered in the wind and came to rest on the sequence he had seen that morning: 1, 1, 2, 3, 5, 8, 13…

Beside the numbers, someone had written in pencil:

“Count what you see. Then look at what has been left out of the count.”

He read the sentence again. The handwriting reminded him of the invitation he had found between the pages of Sophie’s World, but the sheet had disappeared and he could not compare them.

He waited a few more minutes. Then he took the book to the library and entrusted it to the woman he had photographed.

— A gentleman in a light-coloured hat left it. Perhaps he will come back for it.

The librarian placed it behind the desk. Artef left before asking whether she knew the man.

That evening he returned home and printed the photographs of the garden. Beside them he placed a small print of the three philosophers’ portrait.

He arranged the images of the pine cones, the sunflowers, the snail and the staircase on the table. He put the damaged corolla last. He moved them several times to look at them side by side, then separated them: each asked for a different kind of attention.

The photographs he most liked returning to did not all follow the same rule. The grid had suggested possibilities he would not have considered that morning. Symmetry had allowed him to pause; a curve had invited him to continue.

None of those choices, on its own, explained why he returned to one photograph and set another aside. In the philosophers’ portrait he rediscovered his reason for taking it: he had wanted each of them to remain recognisable without appearing to agree with the others. It still seemed a good reason. He did not need to derive a formula from it.

He looked once more at the photograph of the bee. The bright patch in the background drew the eye a little too much. He could tone it down. But first he followed the outline of the wings, the legs resting on the flower, the tiny shadow beneath the body. The image had limitations and something he wanted to preserve. Recognising the former did not prevent him from seeing the rest.

He looked at the corolla with part of a petal missing and remembered Plato’s voice: an image could make us forget what it excluded. In composing an orderly photograph, how often had he left out the very detail that should have prompted him to ask questions? He returned to the question he had written that morning.

Seeking harmony no longer seemed to him either an innocent act or a mistake to avoid. It was a choice to acknowledge, like the focus and the edges of the frame. He put the camera beside the magnifying glass and Sophie’s World. Beneath the question he added: “A proportion can guide me. The responsibility for what I show remains mine.”

Outside, the city’s windows were lighting up. Artef watched them without counting. Behind each rectangle, a life continued that he could not know from where he stood.

On the windowsill, the blade of grass he had found that morning bent beneath a breath of air from the window. For a moment it traced a small curve.

It might have resembled a spiral.

Or it was simply a blade of grass stirred by the wind.

Artef smiled.

He left the notebook open. The blade of grass straightened a little, then changed shape again.