The Natural History of Impossible Fish

The first fish was known as the Great Shadow, although no one had ever seen its body.

It inhabited a shallow river running through a valley of white stones. At midday, when the sun stood directly overhead, its shadow passed over the riverbed, darkening the stones and the water between them. The shadow was shaped like a fish, with a long dorsal fin and a tail divided into three lobes. It was so large that the river disappeared beneath it, and the white stones became black.

At first, the villagers believed it to be the shadow of a cloud. But clouds passed overhead without changing it, and on moonless nights the shadow remained.

One morning, a boy saw the shadow move against the current. He followed it upstream until it entered a narrow gorge, where it narrowed to a dark thread and vanished between two stones.

The next day, the gorge was filled with water.

The river had risen, though no rain had fallen, and beneath its surface swam a fish no larger than a finger. It was entirely transparent, except for a single black fin. Wherever the fin moved, the water deepened.

The boy watched it disappear into the distance. Far downstream, the villagers found that the river had become deep enough to float a ship.

They called the small fish the Shadow’s Child, and entered it in the village register as a separate species.

No one thought to record the ship.

---

The second fish was called the Cartographer.

Its scales contained landscapes. Each scale held a different country, complete with mountains, rivers, cities and forests. The landscapes were not representations of places that existed elsewhere. They possessed their own weather, their own seasons, their own migrations of birds.

When the fish swam, the countries shifted beneath its skin. A mountain range passed from one scale to the next; a river divided into several rivers, each entering a different sea. Cities grew at the borders of forests, and forests advanced across cities, covering their towers with leaves.

The fish was captured in a net of fine silver thread and placed in a glass tank. It did not struggle. Its landscapes continued to change.

A scholar was summoned to describe it. He examined the scales through a magnifying lens and drew a series of maps, one for each visible country. He gave the rivers names and numbered the mountains. He classified the cities according to their shapes and the forests according to their foliage.

When he returned the following morning, the maps were wrong.

The fish had moved a single fin during the night. This had caused the countries to rearrange themselves. A river now passed through a mountain, a city stood beneath a lake, and a forest had grown around the fish’s heart.

The scholar made new maps.

By evening, the maps had become obsolete again.

On the third day, he removed the fish from the tank and released it into the river. It swam away, its scales glittering in the afternoon sun.

The scholar kept his maps. He bound them in a leather volume and placed the volume in the library.

Years later, when the library was demolished, the book was found to contain a small quantity of seawater. Between its pages lay a single silver scale. Within the scale was a country whose mountains had not yet appeared on any of the maps.

---

The third fish was beautiful from a distance.

It moved through the ocean with the slow grace of a floating flower. Its body was a delicate rose colour, and its long fins trailed behind it like translucent veils. Tiny points of light shimmered beneath its skin, forming patterns that resembled constellations. When it turned, the ocean brightened around it.

Sailors who saw it from their ships described it as the loveliest creature in the sea.

Those who came closer gave a different account.

Its scales were not scales but thin, overlapping mouths. Each mouth contained a row of small white teeth. Its fins were composed of countless pale fingers, each bearing a transparent nail. Beneath its rose-coloured skin, a multitude of tiny faces moved slowly towards the surface, pressed together as if trying to escape.

The fish did not attack. It drifted among the sailors, its many mouths opening and closing without sound. Wherever its fins brushed the water, the faces beneath its skin became briefly visible.

One sailor fell overboard. His companions saw him floating beside the fish, apparently unharmed. Then the fish turned its head, and the sailor’s face appeared among the faces beneath its skin.

The others hauled him aboard.

He was unchanged in every outward respect. He spoke, ate and slept as before. But when he looked into a mirror, his reflection seemed to be standing a little farther away than it should have been.

After some weeks, he left the ship at a foreign port and was never seen again.

The captain entered the fish in his log as Rosa mirabilis, the marvellous rose fish. He added a note concerning the unusual structure of its scales, but made no mention of the sailor.

The log was preserved in a maritime museum. Visitors admired the captain’s handwriting and the elegance of the drawing he had made of the fish.

In the museum’s great glass case, beneath the logbook, lay a single rose-coloured scale.

Sometimes, when the galleries were empty, a face appeared within it.

---

The fourth fish was difficult to describe because it seemed to be made from the absence of everything around it.

It was first observed in a lake surrounded by black reeds. Wherever it passed, the water appeared to have lost its colour. The reeds became invisible against the banks, and the mountains reflected in the lake disappeared from the surface.

The fish itself had no visible outline. Its presence could be inferred only from what was missing.

A flock of white birds flew over the lake. As they passed above the fish, one bird vanished from the flock. The others continued on their way, but their formation had changed, and the space left by the missing bird seemed larger than the bird itself had been.

A fisherman cast his net into the lake. When he drew it up, the net was empty. Yet one of its meshes had disappeared, leaving a hole through which the fisherman could see the lake’s bed, although the water between them remained.

He repaired the net and went home.

That night, the lake disappeared.

In its place lay a hollow in the ground, perfectly dry, with the black reeds standing around its rim. The mountains were gone from the horizon. The moon shone above the hollow, but its reflection was nowhere to be seen.

By morning, the lake had returned. The reeds were green, the mountains stood in their accustomed places, and the fisherman’s net hung intact in his shed.

He never returned to the lake.

Some naturalists maintained that the fish had been an optical phenomenon, others that it was a species adapted to conditions in which visibility itself had become a form of nourishment. A third school proposed that it was not a fish at all, but a temporary failure of the world to provide an appearance.

None of these explanations was accepted universally.

In the official catalogue, the species was entered as Piscis absens.

Beside the name, the space reserved for its illustration remained blank.

---

The fifth fish was discovered by a woman who had been gathering shells on a beach after a storm.

It lay in a shallow pool among the rocks, its body made of clear water, although the pool itself was filled with seawater. Within it moved a smaller fish, and within that fish another, and within that one another still, each swimming in a different direction.

The woman bent over the pool.

The outer fish opened its mouth. The innermost fish turned and opened its mouth in precisely the same way. Between them, the smaller fish continued swimming, their movements neither identical nor wholly separate.

The woman noticed that one of the fish was missing a fin.

She looked again. The fin was present, but it belonged to a fish farther inside. Another fish bore the shadow of the missing fin, and another contained a small pale shape that might have been the fin’s reflection.

The woman stayed until the tide began to rise.

When the sea entered the pool, the fish separated. Some swam towards the open water, others towards the rocks, and one slipped into a crack so narrow that it could not possibly have contained it.

The woman returned the following day. The pool was empty, but a small circular depression had appeared in the rock. At its centre lay a drop of water that did not evaporate, even in the hottest weather.

She visited the pool for many years. The drop remained.

Eventually, she brought a naturalist to see it. He examined the depression, tested the water, and made careful measurements. He could find nothing remarkable about it.

When he left, the woman looked down and saw that the drop had become a tiny transparent fish.

It was swimming in a sea no larger than a fingernail.

---

The current connected all these fish, although no river or ocean could be traced between their habitats.

It passed through the valley of white stones, beneath the Cartographer’s ocean, around the black reeds and through the narrow cracks in the rocks. Sometimes it moved visibly, carrying silver threads of light. Sometimes it was detectable only by the direction in which leaves turned, or by the slow bending of grass in a wind that did not touch anything else.

The naturalists attempted to map it.

They marked the places where the fish had been seen, drew lines between them, and proposed a system of underground channels. Their maps became increasingly elaborate. Some included the rivers of the Cartographer’s scales; others showed the dry hollow where the lake had disappeared. One map contained a blank space so large that it extended beyond the edges of the paper.

The maps disagreed.

The naturalists met to discuss the discrepancies. They argued about habitats, migration, reproduction and the possibility of a common ancestor. They divided the fish into families and subfamilies, then subdivided the families again. They devised new terms for transparent flesh, landscapes contained within scales, and the apparent absence of a creature from the space it occupied.

At last, they produced a great volume entitled A Complete Natural History of Impossible Fish.

It contained hundreds of descriptions, dozens of maps, and an index that referred to several species not mentioned anywhere in the text. Its final chapter discussed the current, concluding that although its existence could not be demonstrated by conventional means, the evidence suggested a common medium through which the fish might be related.

The volume was placed in a locked cabinet in the museum.

For many years, nothing unusual occurred.

Then, one morning, the curator discovered a small wet patch on the floor beneath the cabinet. He wiped it away. The following day, another appeared. On the third day, a thin line of water ran from the cabinet to the door.

He unlocked the cabinet and removed the book.

The volume had swollen. Its pages were damp, and the ink had begun to spread. When he opened it, the maps appeared to have altered. Rivers had moved. Mountains had shifted. In the blank space at the edge of the last map, a small silver fish was swimming.

The curator turned the page.

The fish was gone.

He turned back. It had reappeared, this time with a dark fin and a long shadow that extended beyond the map’s border.

The curator shut the book and called for assistance.

By the time the other museum staff arrived, the water had disappeared. The cabinet stood dry and empty. The book was found on a nearby table, its pages clean, its maps unchanged.

Only the index had been altered.

At the very end, beneath the final entry, a new name had appeared.

There was no description, no reference number, no indication of where the fish had been seen.

The name was written in a hand none of the staff recognised.

It read:

The Fish That Has Not Yet Appeared.

---

Far beneath the oceans, where the current passed through the dark between one world and another, a fish opened its eyes.

Its body was covered in scales that reflected no light. Within each scale lay a small, still landscape: a valley, a mountain, a river, a city, a plain of white stones.

The landscapes were empty.

Then, in one of them, a small shadow moved across the ground.

The fish turned.

All the landscapes turned with it, and in each one something began to swim.

The Fish of the Imagination

The first fish appeared in a field of cabbages.

It lay between two broad leaves, its body transparent as blown glass, its fins moving with the slow deliberation of silk underwater. Inside it, a small red wheel revolved. Each time the wheel completed a turn, a drop of rain fell from a cloudless sky.

The farmer stood looking down at it. He had never seen a fish like this before, but he knew better than to touch things that appeared without explanation. He fetched a bucket. By the time he returned, the fish had vanished, and every cabbage in the field had opened to reveal a tiny blue eye.

For three days the eyes watched the sky. On the fourth, they closed, and the cabbages grew white feathers.

The second fish appeared in a church bell.

It had been living inside the bronze for so long that its scales had acquired the colour of old bells, green in the hollows, gold where the light struck them. When the bell rang, the fish swam through its sound. The note emerged as a flock of pale birds, each with a human hand in place of a tail.

They flew over the village and settled in the trees. Wherever one folded its wings, a flower opened. Some flowers were exquisite, with petals as delicate as the ears of sleeping children. Others resembled wounds, and from their centres came the faint sound of distant laughter.

The bell was never rung again. Nevertheless, on still nights the villagers heard something moving inside it, turning slowly in the dark.

The third fish was found in a mirror.

It swam behind the glass in a room that had been empty for many years. Its body was long and black, and it carried a crown of branching antlers from which hung tiny silver bells. It had no eyes, but wherever its head turned, the room changed.

First, the wallpaper became a forest of white trunks. Then the floor filled with dark water in which the ceiling floated upside down. A staircase grew from the mantelpiece, winding through the room and disappearing into a sky crowded with unfamiliar stars.

At last the fish opened its mouth.

A second room appeared inside it, and in that room stood another mirror. A smaller black fish swam behind the smaller glass, carrying a crown of still smaller antlers. Within the antlers hung bells, and within each bell something moved.

The first mirror cracked from corner to corner. The fish continued swimming.

By morning the room had returned to its former emptiness, except for a small silver bell lying on the floor. No one knew where it had come from. When the bell was lifted, it contained a single white feather.

The fourth fish appeared in the shadow of a mountain.

It was enormous. Its shadow spread across the plain, covering villages, orchards and rivers, though the mountain itself was scarcely taller than a house. The fish could not be seen, but its shadow moved through the grass, and wherever its fins passed, the earth rose in long blue ridges.

The ridges became waves. The waves became forests. The forests opened like dark green mouths, releasing flocks of birds whose wings were covered in eyes.

For seven days the shadow travelled across the plain. On the eighth, it turned towards the mountain and slowly climbed its side. At the summit it narrowed to a black thread, then disappeared into a crack in the stone.

That evening, every shadow in the valley pointed upwards.

The fifth fish appeared in a bowl of milk.

It was small enough to rest on a child’s palm, although no one could say how it had entered the bowl. Its body was made of a substance that seemed to be neither flesh nor light, but the absence of everything surrounding it. Where it swam, the milk became transparent, and through the white porcelain the child could see a vast dark ocean filled with slowly turning stars.

The fish had no fins. Instead, thin openings appeared along its sides, and through them emerged little creatures shaped like folded leaves. They unfolded as they rose, becoming moths, flowers, hands, tiny white faces. Each drifted upwards until it reached the ceiling, where it opened into a small round window.

Through the windows came a wind that smelled of rain on distant hills.

The child watched until the bowl was empty. Then she carried it outside and placed it beneath the night sky. By morning, the windows had closed, but the bowl was full of seawater, and a small black stone lay at the bottom.

She kept the stone for the rest of her life. Sometimes, when she held it in her hand, she heard waves breaking on a shore that could not be found.

After these things, the fish appeared more frequently.

One swam through a shaft of sunlight in a baker’s shop, trailing a long ribbon of darkness that smelled of roses. Another emerged from a tear in a painted portrait and left behind a scattering of tiny green teeth. A third drifted above a field of poppies, its transparent body containing a storm in which pale horses ran along the undersides of clouds.

There were fish with faces, and fish with forests growing beneath their scales. There were fish that seemed to be made entirely of eyes, each reflecting a different landscape. There were fish so beautiful that the sight of them caused flowers to open in the mouths of sleeping people, and others whose passing left the taste of iron on the tongue.

They did not remain. They moved through the world as if following currents that passed through stone, air, flesh and glass without regard for their differences. Sometimes they disappeared into the earth. Sometimes they rose into the sky. Sometimes they entered things that had no apparent opening and were never seen again.

Yet something always remained.

A blue scale appeared beneath a pillow. A white flower grew in a chimney. A child woke with the memory of a staircase descending through the sea, and for the rest of his life he could never look at a staircase without seeing, for an instant, a small silver fish swimming down its steps.

The remains were seldom beautiful. Some were ugly, some inconvenient, some impossible to distinguish from ordinary rubbish. A broken tooth. A wet feather. A seed that produced glass instead of leaves. But they were never quite inert. Given time, they began to change.

In a disused greenhouse, a scale became a pane of blue glass. Behind it grew a garden of tiny white trees. In an abandoned well, a tooth sprouted black roots that reached down through the stone and emerged in a cellar three villages away. The glass seed lay for years in a drawer before splitting open to release a thin green shoot, which grew into a transparent vine bearing little bells.

And the bells, when they rang, sounded like water moving through a field of cabbages.

By then, the village had grown accustomed to the fish. People no longer ran into the streets when one appeared, nor gathered in crowds to stare. They went on with their work, although occasionally someone would pause in the middle of a sentence, or look twice at a patch of sunlight, or leave a door open for no particular reason.

The fish themselves seemed to have become less frequent. Weeks passed without a single appearance. The cabbages grew ordinary leaves. The church bell remained silent. The mirrors reflected only what stood before them.

Then, one winter morning, a fish appeared in the frost on a window.

It was pale and delicate, with long translucent fins and a body that seemed to contain a distant landscape. Snow-covered mountains stood inside it beneath a green sky. Between the mountains ran a narrow river, and beside the river grew a tree bearing black flowers.

The fish moved slowly through the frost. As it passed, the mountains folded into the river, the river climbed the tree, and the black flowers opened to reveal tiny suns.

For a moment the whole landscape shone with a light so clear and strange that the room seemed to have no walls.

Then the fish turned.

Its tail brushed the edge of the window, and the frost flowered into a thousand delicate branches. The mountains vanished. The river vanished. The tree became a single black line, and the suns became small white points of light.

The fish swam upwards and disappeared beyond the frame.

For a long time the window remained covered in frost. Then the branches began to melt, leaving a clear patch in the centre of the glass.

Beyond it lay the ordinary garden, the bare trees, the path to the gate.

And in the garden, where the last of the snow had melted, something small and silver was moving beneath the soil.

A Golden Opportunity

Mr Blottisham entered the Senior Common Room carrying a sunflower.

He held it aloft with the satisfaction of a man who had brought decisive evidence to a long-running dispute.

“I believe,” he announced, “that I can now settle our recent discussion of geometry.”

Professor Quillibrace lowered his newspaper.

“Which discussion?”

“The one about whether geometry governs the natural world.”

Miss Stray looked at the sunflower. “And you believe this will settle it?”

“Look at it!” said Blottisham. “The spirals. The proportions. The mathematical regularity. The Fibonacci sequence itself, growing quite happily in a garden without the assistance of a mathematician.”

“An admirable achievement,” said Quillibrace.

Blottisham frowned. “You are making fun of me.”

“Not at all. I was congratulating the sunflower.”

Miss Stray accepted the flower and turned it gently in her hands.

“The remarkable thing,” she said, “is that the pattern need not have been imposed on the plant. It may emerge from the way the plant grows.”

“Precisely!” cried Blottisham. “Nature follows mathematics.”

“Does it?” asked Quillibrace. “Or do mathematical relationships help us describe what happens when nature grows?”

Blottisham considered this distinction.

“I fail to see that it matters.”

“Then let us suppose,” said Quillibrace, “that the plant is not consulting a mathematical textbook.”

“Obviously not. Plants don't read.”

“Yet its developing structures interact in ways that can produce recognisable patterns. In many sunflower heads, the spirals running in opposite directions have counts corresponding to neighbouring numbers in the Fibonacci sequence.”

“Exactly!”

“But that does not mean every plant follows the sequence, nor that the plant is attempting to reproduce a mathematical ideal.”

Blottisham looked at the sunflower with renewed suspicion.

“Are you suggesting that it has done this accidentally?”

“I am suggesting,” said Miss Stray, “that accidentally may not be the right word. A pattern can arise from the relations between things as they develop, without having been designed in advance.”

Blottisham sat down.

“I had hoped for a simpler conclusion.”

“You have a sunflower,” said Quillibrace. “Surely that is sufficient for one afternoon.”

“It is not merely a sunflower. It is an example of the golden ratio.”

“Ah,” said Quillibrace. “Now we approach the second difficulty.”

Blottisham brightened. “The golden spiral!”

“Not quite the same thing.”

“Surely a golden spiral is a spiral governed by the golden ratio?”

“In a particular mathematical sense, yes. Its dimensions increase according to a precise relationship. But not every spiral found in nature is a golden spiral.”

“Then what of the nautilus?”

Miss Stray smiled. “I wondered how long it would take you to reach the nautilus.”

“I have read extensively on the subject.”

“Have you measured any shells?”

“Certainly not. I am not a zoologist.”

“Nor, I think,” said Quillibrace, “are you yet a mathematician.”

Blottisham ignored this.

“But the shell grows in a spiral!”

“Many shells grow in logarithmic spirals,” said Miss Stray. “That does not establish that their spirals have precisely the proportions of a golden spiral. The resemblance is often more persuasive in diagrams than in measurements.”

Blottisham looked from one to the other.

“Are you telling me that the golden ratio is not responsible for the beauty of nature?”

“I am telling you,” said Quillibrace, “that beauty does not constitute a measurement.”

For a moment, nobody spoke.

Outside, a gardener was pruning a rosebush.

Miss Stray placed the sunflower on the table.

“Perhaps we have been asking the wrong question. We keep asking whether nature follows geometry, as though geometry were a set of instructions delivered from elsewhere. But what if certain geometrical relationships arise from the ways things grow, occupy space, and affect one another?”

“Then geometry would be descriptive rather than governing,” said Blottisham.

“Not necessarily,” said Quillibrace. “A description can reveal constraints. A relationship can help explain why certain arrangements are possible and others difficult to sustain. But we must distinguish the pattern we identify from the processes that produce it.”

“And the processes,” Miss Stray added, “may be more interesting than the pattern alone.”

Blottisham regarded the sunflower again.

“I begin to regret bringing this.”

“You have brought us something quite valuable,” said Quillibrace.

Blottisham looked pleased.

“What, precisely?”

“A reason to be cautious.”

The pleasure disappeared.

Miss Stray laughed. “I think the flower has done enough for one day.”

She took it home with her.

After she had gone, Blottisham turned to Quillibrace.

“Still, you must admit that the Fibonacci sequence is not imaginary.”

“Certainly not.”

“And the golden ratio is a genuine mathematical relationship.”

“Entirely genuine.”

“And geometry is present in the natural world.”

“Mathematical relationships are certainly found in the patterns we observe.”

Blottisham stood up.

“Then I shall write an essay demonstrating that the golden ratio is the key to understanding the universe.”

“Which universe?” asked Quillibrace.

Blottisham paused at the door.

“I shall have to think about that.”

It was, Professor Quillibrace reflected, the most promising thing Mr Blottisham had said all afternoon.

On the Proper Uses of Geometry

Professor Quillibrace was examining a small drawing when Mr Blottisham entered the Senior Common Room.

"Ah," said Blottisham, removing his coat. "Sacred geometry. I've been meaning to have a word about that."

Miss Elowen Stray looked up from her tea.

"Have you?"

"Naturally. Four stories, all involving circles, proportions, shapes, and other such things. I presume we are expected to discover the underlying message."

"Are we?" asked Quillibrace.

Blottisham regarded him with mild impatience.

"One doesn't write four stories about geometry without intending to communicate something about geometry."

"One might," said Quillibrace, "write four stories about geometry without intending to communicate the same thing about geometry."

Blottisham sat down.

"That," he said, "is precisely the sort of distinction that makes literary criticism unnecessarily complicated."

Miss Stray smiled. "Which story did you prefer?"

"The house."

"Why?"

"Because it was the only one that seemed to have a proper architectural foundation. The garden was fanciful, the stone was positively irresponsible, and the final story appeared to suggest that people dream things into existence. One must draw the line somewhere."

"Where would you draw it?" asked Quillibrace.

"At the point where the author begins to confuse imagination with reality."

"An interesting location," said Quillibrace. "Did you find it marked on any of the plans?"

Blottisham ignored him.

"The house, at least, had a perfectly straightforward moral. We ought to be grateful for the stability of the world we inhabit."

"Was that the moral?" Miss Stray asked.

"What else could it have been?"

"Perhaps," she said, "the difficulty was that the house was stable in ways its inhabitants couldn't adequately describe."

Blottisham frowned.

"If a house cannot be adequately described, I fail to see how anyone can be expected to live in it."

"The narrator managed."

"By abandoning the attempt to understand it."

"Yes," said Quillibrace. "A method occasionally recommended for the understanding of difficult things."

Blottisham looked from one to the other.

"I see. We are to admire ignorance."

"No," said Miss Stray. "We are to distinguish between not understanding something and being mistaken about what understanding requires."

Quillibrace put down his drawing.

"That is a useful distinction. The architects' plans were accurate. Their difficulty was not in measuring the house incorrectly, but in assuming that accurate measurements must produce a complete account of it."

"Then the architects were incompetent."

"On the contrary. They were competent enough to discover that their competence had limits."

Blottisham considered this.

"An alarming precedent."

"Only for those who believe competence ought to guarantee omniscience," said Quillibrace.

Miss Stray poured herself more tea.

"I thought the garden was more disturbing."

"Because of the woman?" asked Blottisham.

"Because of the second circle."

"Surely the woman was the important part. A mysterious woman in a garden is a familiar literary device."

"She may be," said Miss Stray. "But the second circle changes the garden itself. The path moves, the stream divides, and the woman goes elsewhere. The new circle doesn't simply give the narrator more of the same world. It changes the relations within it."

Blottisham looked relieved.

"Then the lesson is obvious. One should be careful before making alterations to one's property."

"That is certainly one possible lesson," said Quillibrace.

"Thank you."

"It is not necessarily the most interesting one."

Blottisham's relief vanished.

Miss Stray continued. "I wondered whether the narrator's decision not to draw a third circle was a form of wisdom."

"Or cowardice," said Blottisham.

"Possibly. We aren't told enough to decide."

"Surely one must decide something."

"Why?"

"Otherwise the story remains unfinished."

Quillibrace glanced at the drawing before him.

"Does it?"

"Of course. The narrator has an unexplained garden, an unexplained woman, and a compass. He could have continued."

"That is true."

"Yet he doesn't. Very unsatisfactory."

"Perhaps," said Miss Stray, "the story finishes precisely where the narrator decides not to continue."

Blottisham shook his head.

"I cannot approve of a protagonist who refuses to investigate the central mystery of his own experience."

"Would you have drawn the third circle?" Quillibrace asked.

"Certainly."

"Why?"

"To discover what happened."

"And if drawing it changed everything?"

"Then I should have discovered that."

"Even if there were no longer anyone capable of making the discovery?"

Blottisham opened his mouth, then closed it.

"I don't see why the author should introduce such a complication."

"Neither," said Quillibrace, "did the surveyor."

There was a brief silence.

Miss Stray turned towards him.

"The stone?"

"Precisely. The surveyor's difficulty was not that he couldn't calculate the missing lines. He could. His difficulty was deciding whether completing the figure would restore the world or destroy something that the incompleteness was preserving."

"An absurd dilemma," said Blottisham. "If the figure was intended to be complete, it should have been completed."

"Intended by whom?" asked Miss Stray.

"The person who made it."

"Do we know that anyone made it?"

Blottisham paused.

"It is a geometrical figure. Geometrical figures do not grow on trees."

"Neither do the trees grow according to geometrical figures in any simple sense," said Quillibrace. "Yet we are continually tempted to treat a pattern we can recognise as an explanation of what we have recognised."

Blottisham looked suspicious.

"Are you suggesting that geometry is unreliable?"

"Not at all. The surveyor's calculations were correct. The question was whether correctness in calculation entitled him to alter the stone."

"Then the lesson is that one should never interfere with things one doesn't understand."

"Would you include medicine?" asked Miss Stray.

"Certainly not."

"Engineering?"

"Obviously not."

"Education?"

Blottisham drew himself up.

"Education is an entirely different matter."

"Why?"

"Because one must know what one is doing."

Quillibrace returned to his drawing.

"An admirable principle. We need only establish how much knowledge is sufficient."

Blottisham decided that the discussion was becoming unhelpfully abstract.

"At any rate, the final story was the most objectionable. A pattern that dreams people into existence is an inversion of the natural order. People dream. Patterns do not."

"Do people dream patterns?" asked Miss Stray.

"Metaphorically."

"Sometimes literally."

"That proves nothing."

"Perhaps not. But the story doesn't say that the geometry creates the people. It leaves open the possibility that the people create the geometry, or that each changes through their relation to the other."

Blottisham looked appalled.

"That is not an explanation. It is three explanations."

"Or three possibilities," said Quillibrace.

"A possibility is not an explanation."

"Quite so."

"Then why introduce it?"

"Because the story may be concerned with what can happen before an explanation has been settled."

Blottisham leaned back in his chair.

"I begin to understand the difficulty. These stories are designed to prevent the reader from reaching a conclusion."

"No," said Miss Stray. "They prevent the reader from reaching certain conclusions too quickly."

She took a sip of tea.

"I thought it significant that the collector eventually stopped trying to find the original pattern. He began arranging the drawings according to their differences rather than their similarities. That was when the collection became interesting to him."

"An untidy method," said Blottisham.

"Perhaps. But the drawings seemed to form a relationship that none of them contained individually."

"Then the relationship was the real pattern."

"Perhaps."

"Well, there you are. We have arrived at the answer."

Quillibrace looked up.

"Which answer?"

"The pattern is the relationship."

"That is a proposition."

"Indeed."

"It may be worth considering. But it doesn't tell us whether the relationship existed before the drawings, emerged through them, or depends upon both."

Blottisham stared at him.

"Must everything have three possibilities?"

"Only when the evidence permits them."

"I find this very tiresome."

"I daresay," said Quillibrace.

For a while they sat quietly. Outside, the afternoon light moved across the lawn, dividing it into patches of sun and shadow.

Miss Stray watched the branches of an old tree shift in the breeze.

"I wonder," she said, "whether the four stories are really about geometry at all."

Blottisham seized upon this.

"At last. A sensible observation."

"I don't mean that geometry is incidental. I mean that each story begins with an assumption about how the world ought to fit together. The garden should remain inside its circle. The house should fit its plans. The stone should contain a complete figure. The pattern should have an original form."

"And each assumption fails," said Quillibrace.

"Not quite. Each works, until someone assumes that it must work without remainder."

Blottisham considered this.

"So the message is that rules are useful until they become inconvenient."

"No," said Miss Stray. "That would be too easy."

"Then what is the message?"

She looked at him with some sympathy.

"I don't think the stories have one message."

Blottisham sighed.

"Four stories, no single message, and a geometrical pattern that may or may not exist independently of the people who dream it. I cannot imagine what a reader is supposed to take away."

"Perhaps," said Quillibrace, "the reader is supposed to take away something that the characters have not managed to settle."

Blottisham stood and retrieved his coat.

"I shall have to think about that."

"An excellent beginning," said Miss Stray.

He paused at the door.

"One final question. If the shape that held the world had been completed, would the world have ended?"

Quillibrace and Miss Stray looked at one another.

"We don't know," said Quillibrace.

Blottisham nodded gravely.

"Then I shall assume that it would."

He left the room, satisfied that he had reached a conclusion.

Miss Stray watched the door close.

"Do you think he'll write to the author?"

"Almost certainly," said Quillibrace.

"To explain the moral?"

"To request one."

Miss Stray laughed.

Quillibrace returned to his drawing. It consisted of several circles, none of which quite touched the others.

After a moment, he added another.

Then he put down his pencil, leaving the figure unfinished.

IV. The Geometry That Dreamed Us

The first person to dream the pattern was a child.

She drew it in the dust beside her mother's door: seven circles, each touching its neighbours, with a smaller circle at the centre. She had no name for it and did not remember having seen it before. When her mother asked what it was, she said it was a place she had visited in her sleep.

"What sort of place?"

"A place where everything knows where it belongs."

Her mother smiled and swept the drawing away.

The following night the child dreamed the pattern again. This time she walked between the circles. Each opened into a landscape: a field of tall grass, a shore without waves, a forest in which every tree had the same number of branches. At the centre was a small, empty room.

She returned to the room several times in her dreams, but never found anything inside it.

When she was older, she forgot the pattern.

Years later, a potter in a distant town began dreaming of a design he could not reproduce. It consisted of circles and lines arranged with such precision that he felt certain he had seen it somewhere before. He tried to draw it upon his pots, but each attempt was slightly different. The pattern seemed to change whenever he looked directly at it.

He eventually produced a bowl whose decoration was almost correct.

When he fired it, the bowl cracked along seven perfectly curved lines. The pieces fell apart in his hands, and for a moment he saw, beneath the glaze, the outline of a room.

He threw the fragments away.

A mathematician dreamed the pattern next. He woke with the conviction that it contained a relationship he had been seeking for years. He spent months working through its possible symmetries, testing proportions and constructing diagrams. Every solution led to another problem. The pattern appeared to possess an inexhaustible order, but no single arrangement could account for all its features.

He published a paper describing his results. It attracted little attention.

In the final paragraph, he mentioned that he had first encountered the pattern in a dream. His colleagues advised him to remove the reference from any future publication.

He did so.

By then, the pattern had begun appearing in other places.

A child drew it on a school desk. A woman traced it in the condensation on a window. An architect discovered it hidden in the proportions of a building he had designed years earlier. None of them recognised it as the same figure, because each had drawn it differently.

Yet when their drawings were placed side by side, the differences seemed to belong to a larger design.

A collector began travelling between towns to acquire them. He filled a room with sketches, bowls, scraps of paper, and photographs of markings on walls. He arranged the objects according to their apparent similarities, then rearranged them according to their differences.

Neither arrangement satisfied him.

At night he dreamed of a room containing all the drawings. The walls were covered with patterns, each one incomplete. In the centre stood a table on which lay a blank sheet of paper.

He began to dread the dream, because he knew what would happen next.

Someone would enter the room and begin to draw.

One morning he woke to find that he had covered the wall beside his bed with circles. He could not remember drawing them. The lines were faint and uneven, but they formed the pattern more accurately than anything he had collected.

He sat looking at the wall until the light changed.

Then he took a cloth and wiped the drawings away.

For several weeks nothing happened. The dreams ceased, and the collector returned to his work. He told himself that he had been tired, that the mind was capable of producing elaborate designs without any particular meaning, and that coincidence could become persuasive when one was inclined to look for it.

Then he received a letter from the mathematician.

The letter contained no explanation, only a drawing.

The collector recognised the pattern at once. It was not identical to the figure on his bedroom wall, but it seemed to complete something that the wall had left unresolved.

He placed the letter among his collection.

That night he dreamed of the empty room again.

The blank sheet remained on the table. This time, however, the person who entered the room was the collector himself. He approached the table and looked down at the paper. He knew that if he drew the pattern, the dream would end.

He picked up a pencil.

Before he could begin, he noticed that the paper was not blank. There were faint lines upon it, almost invisible, extending beyond its edges. They seemed to continue through the table and into the floor.

He put down the pencil.

The room grew brighter.

He could see that the walls were covered not with drawings, but with openings. Beyond each opening lay another room, and beyond each room another. The pattern was not repeated exactly. Each occurrence differed from the others, sometimes by a single line, sometimes by an entire circle.

Yet all the rooms belonged to one another.

At the far end of the last room, he saw a small figure standing beside a door.

It was a child.

She was drawing circles in the dust.

The collector woke before he could see what she drew.

In the morning he returned to his collection. He took down every drawing, every photograph, every fragment of pottery. He spread them across the floor and began to arrange them, not according to resemblance, but according to the differences between them.

By evening, he had found a sequence.

Each pattern differed from the next in a way that made the next one possible. No drawing was complete, yet none could be removed without altering the arrangement of the others.

He stood in the middle of the room and looked at the collection.

For the first time, he felt no desire to discover the original pattern. The drawings seemed to be doing something that none of them could do alone.

He left them where they were and went to bed.

That night he dreamed of the child again.

She was standing in the empty room, holding a piece of chalk. The circles were already drawn upon the floor, but they did not form the pattern he had come to know. There were more of them now, and some overlapped in unfamiliar ways.

The child looked up at him.

"Is this the right one?" she asked.

He wanted to answer, but could not decide what the question meant.

The child waited.

Then she began to draw another circle.

The collector woke with the feeling that something had changed. He lay still, listening to the sounds of the house. Beyond the window, the first birds were calling, and the sky was beginning to lighten.

He went to the room where he kept his collection.

The drawings were still there, scattered across the floor. But the arrangement had altered. He could not tell whether something had moved during the night or whether he had simply failed to notice the pattern that was already present.

He sat down among the papers.

After a while, he found a blank sheet and began to draw.

He did not attempt to reproduce the pattern he remembered. Instead, he drew a circle, then another, allowing the second to overlap the first. He continued until he had made a figure that seemed unfamiliar and yet, in some way, necessary.

When he finished, he placed the drawing among the others.

Nothing happened.

He returned to his ordinary life. He wrote letters, visited friends, and occasionally travelled to acquire new objects for his collection. He no longer tried to arrange the drawings into a final sequence. Sometimes he added one; sometimes he removed another. The collection grew, although he could not have said what it was becoming.

Many years later, when the collector was an old man, a young woman came to see him.

She had heard of his collection and wished to study it. He showed her the drawings and explained how they had come into his possession, though he did not tell her about the dreams.

She examined the papers for several hours.

At last she pointed to a drawing near the bottom of the pile.

"I dreamed this when I was a child," she said.

The collector looked at the figure. He recognised it, but not as one of the drawings he had collected. It was a pattern he had seen in his own dreams, though it had never appeared quite like this.

"Do you remember where you saw it?" he asked.

"In a room," she said. "There was a table, and a blank sheet of paper. Someone was standing beside it."

The collector said nothing.

"Do you know what it means?" she asked.

He looked at the drawing for a long time.

"No," he said. "But I think it has changed."

After she had gone, he returned to the collection. He found the drawing he had made on the morning after his last dream and placed it beside the young woman's.

The two patterns were not the same.

Yet between them, something seemed to have appeared that neither drawing contained.

He sat watching the light move across the papers until the room grew dark.

That night, he dreamed of no pattern at all.

He dreamed of a wide, quiet landscape beneath an unfamiliar sky. The hills rose and fell without symmetry. A river wandered through the valley, dividing and joining again. Trees grew where they could, and the paths between them were made by those who walked.

In the distance stood a small house.

Its windows were lit.

The collector approached it, but before he reached the door, he woke.

He lay still for a while, trying to remember the shape of the house.

Then he understood that he had not seen it from the outside.

He had been looking through one of its windows.

Someone inside was asleep, dreaming of a room filled with drawings.

And on the table, beside the sleeping figure, lay a blank sheet of paper.

III. The Shape That Held the World

In the oldest village, there was a stone that nobody could move.

It stood in a field above the river, half buried in the earth. It was neither large nor beautiful, and the villagers had long ago ceased to notice it. Children climbed upon it, sheep scratched themselves against it, and generations of farmers had ploughed around it without giving it a second thought.

Then, one spring, the river changed its course.

It did not flood or dry up. It simply began to bend towards the village, although the ground sloped in the opposite direction. The miller was the first to notice. He watched the water for three days, then went to the village elders.

They examined the river and agreed that it was behaving strangely.

A week later, the road to the neighbouring village became longer. Travellers who had walked it all their lives found themselves arriving after nightfall, though they had set out at the usual hour. One man claimed that he had passed the same oak tree three times without turning back. Another insisted that the road had narrowed behind him.

The elders sent for a surveyor.

He arrived with measuring rods, a chain, and a book in which he recorded distances. He measured the road, the river, and the field above the village. His figures were accurate, but they contradicted one another. The distance between the village and the river was shorter when measured from the east than when measured from the west. The road was longer when approached from the neighbouring village than when approached from home.

The surveyor blamed his instruments and returned with new ones.

The results were the same.

At last he noticed the stone.

He had measured the field twice, and on each occasion the stone occupied a different position in his calculations. It had not moved, yet the distances around it had altered. He asked the elders whether it had always been there.

"As long as anyone remembers," they said.

The surveyor knelt beside it and brushed away the earth. Beneath the soil, he found a line cut into the stone. He cleared more earth and discovered another line crossing the first, then a third, then a fourth.

By sunset he had uncovered a figure.

It was a pattern of straight lines joined at their ends, forming a shape that seemed at first to be a square. But one corner was missing, and the lines did not quite meet. The surveyor could not determine whether the figure had been left unfinished or whether the stone had broken along an existing fault.

He made a drawing and took it to the elders.

None had seen the figure before.

An old woman who lived near the mill studied the drawing for a long time.

"My grandmother told me a story," she said. "She said that the world was once held together by a shape. Nobody knew who had made it, or what it was made of. But while it remained whole, the rivers flowed, the roads led where they should, and people could find their way home."

"What happened to it?" asked the surveyor.

"Someone tried to finish it."

The elders laughed at this. It was an old woman's tale, and they had more pressing concerns. The river was approaching the lower fields, and the road to the neighbouring village had become almost impassable.

But the surveyor could think of nothing else.

He returned to the stone with a hammer and chisel. If the figure had once been whole, perhaps the missing corner could be restored. He spent several days studying the lines, measuring their angles, and calculating the proportions. Eventually he became convinced that he understood the design.

He drew the missing lines on paper.

The drawing was beautiful. Every angle corresponded to another, every length to a proportion elsewhere in the figure. The pattern possessed a balance that seemed to make the surrounding landscape inevitable. He could not explain why, but he felt that he had discovered something fundamental.

He showed the drawing to the elders.

They were impressed by its symmetry and agreed that he should complete the figure on the stone.

The surveyor returned to the field and began cutting.

As he worked, the river gradually resumed its former course. The road shortened. The villagers found that familiar landmarks had returned to their proper distances. By the time the final line was ready to be cut, the world appeared to have recovered its ordinary order.

The surveyor paused.

The missing corner lay before him, marked in chalk. He raised his chisel, then hesitated.

The old woman had come to watch him.

"Why did the person in your grandmother's story try to finish the shape?" he asked.

"Because it was unfinished."

"And what happened when the shape was completed?"

She looked towards the river.

"The world had nowhere left to go."

The surveyor lowered his chisel.

For several days he did nothing. The villagers grew impatient. The road was passable again, but they wanted the matter settled. They had seen the drawing, and they wanted the stone to match it. An unfinished figure seemed to them an untidy thing, especially one that appeared to govern the behaviour of the world.

The surveyor began to doubt himself. Perhaps the old woman had misunderstood the story. Perhaps the shape had been damaged, and its incompleteness was the source of the disturbance. Perhaps finishing it would restore the world permanently to its proper order.

He returned to the stone at dawn.

The field was empty, and the air was still. He placed the chisel against the chalk line and struck.

A small piece of stone fell away.

Nothing happened.

He struck again.

The missing line began to emerge, pale against the weathered surface. He worked carefully, following his calculations. The line approached the corner, and the figure began to look complete.

Then the river stopped.

Not gradually, as water might stop behind a dam, but all at once. Its surface became still, and the reflections of the trees stood motionless upon it. A bird hung above the field with its wings extended. In the village, a door remained half open, and the person who had been closing it could not move.

The surveyor dropped the chisel.

He looked at the stone. The final line had not yet reached the corner, but the figure was almost complete.

He understood then that the shape was not holding the world in place.

It was holding the world together.

The distinction seemed small, but he could feel its importance. A thing held in place could be moved. A thing held together could not be altered without changing everything that depended upon it.

He took up the chisel and struck the stone again, this time at the end of the unfinished line.

A fragment broke away.

The figure was damaged.

The bird moved. The river resumed its course, and somewhere in the village the door closed with a bang. The surveyor sat down in the grass, shaking.

He did not attempt to repair the stone.

In the years that followed, the village changed. The river occasionally wandered into unfamiliar channels, and the road sometimes took travellers longer than expected. New paths appeared across the fields, while old ones disappeared beneath the grass. The villagers grew accustomed to these things. They learned to leave earlier when travelling, to ask directions even on familiar roads, and not to assume that a place would always be where they remembered it.

The surveyor grew old.

He never told anyone exactly what had happened on the morning he struck the stone. When children asked about the lines, he told them that the figure was a map, though he could not say of what. When adults asked whether it should be completed, he replied that he had not yet found the correct proportions.

Near the end of his life, he returned to the field alone.

The stone was almost covered by earth again. Grass grew between its lines, and rain had softened the edges of the cuts. The missing corner was difficult to distinguish from the rest of the weathered surface.

He knelt and brushed the earth away.

For a moment, he thought he saw another line beneath the stone, extending down into the ground. He cleared the soil, but the line disappeared. He tried again from another angle, with the same result.

At last he sat back on his heels.

He had spent much of his life believing that the world could be understood by finding the pattern that made it whole. Now he wondered whether the pattern had ever been whole at all.

The stone lay quietly in the field.

The river moved below it, taking a course that no map could permanently record. The road wound towards the neighbouring village, passing between trees that had grown in places where no trees had stood before. Beyond them, the hills rose against the sky, and the sky extended beyond the hills.

The surveyor stood and began the walk home.

He did not know whether the world was held together by the shape, or whether the shape was only the place where the world had briefly become visible to itself.

He no longer thought the distinction needed to be settled.

Behind him, the stone remained unfinished.

Ahead of him, the road continued.

II. The House of Perfect Proportions

The house was left to me by an aunt I had never met.

I knew very little about her beyond the fact that she had lived alone, disliked visitors, and had once sent my mother a letter complaining that the family had become careless about symmetry. I had assumed this was a reference to the arrangement of furniture.

The house stood at the end of a long drive, surrounded by neglected gardens. It was a square building of pale stone, with a central doorway and identical windows on either side. Even in its dilapidated state, it possessed a curious air of completeness, as though the weather had been unable to find a suitable place to begin its work.

I engaged an architect to inspect it.

He arrived on a Tuesday with a measuring tape, a camera, and a young assistant who carried a clipboard. They spent the morning examining the exterior, then proceeded through the rooms, measuring walls, ceilings, windows, and doorways.

At lunchtime the architect emerged looking pleased.

"An extraordinary building," he said. "Every proportion is exact."

"Is that unusual?"

"Extremely. The windows are correctly proportioned to the walls, the rooms to the corridors, the floors to the staircase. Even the smallest details appear to follow a consistent scheme."

He showed me his measurements. They were arranged in columns, accompanied by calculations that I did not understand.

"Whoever designed this house," he said, "knew precisely what he was doing."

I asked whether he had found anything wrong.

"Nothing at all."

That afternoon I discovered the room.

It was at the end of the first-floor corridor, between the library and a small bedroom overlooking the garden. I had passed the door several times without noticing it. It was narrow, painted white, and fitted with a brass handle.

Inside was a square room containing a table, a chair, and a window.

The room was perfectly ordinary, except that the window looked out upon a view of the front garden, although the wall in which it stood faced the rear of the house. Through it I could see the drive, the gate, and a large elm tree that had been felled many years before.

I went outside to check. The elm was gone, and the drive lay on the opposite side of the building.

When I returned, the room was unchanged.

I called the architect.

He examined the room without comment, then measured it twice. He measured the corridor outside, the adjoining bedroom, and the library. He took further measurements of the staircase and returned to the room.

Finally, he asked me for the original plans.

There were no original plans among my aunt's papers, but he agreed to draw a new set. He worked for several days, producing a beautifully detailed representation of the house. Every wall, doorway, and window was accounted for.

The mysterious room did not appear.

"Perhaps you have overlooked it," I suggested.

He laid the plans on the table and showed me the relevant section.

"Here is the corridor. Here is the library. Here is the bedroom. There is no room between them."

"But I have just come from it."

"I don't doubt your experience," he said, with the patient expression of a man who has decided to be kind. "But the dimensions leave no space for it."

We went upstairs together.

The door was still there.

He opened it, entered the room, and stood looking through the window.

For the first time, he seemed uncertain.

He returned to the plans and began measuring again.

Over the next week he produced three further versions. Each was more exact than the last. In the second, the corridor was slightly longer. In the third, the library was slightly smaller. In the fourth, the proportions of the entire first floor had been adjusted by fractions so small that I could barely distinguish the drawings.

All four plans were internally consistent.

None contained the room.

The architect eventually withdrew from the commission. He left his plans behind, explaining that he would be unable to proceed until the discrepancy had been resolved.

I tried another architect, then a surveyor. The surveyor brought a laser measuring device and spent two days recording the dimensions of the house. His results agreed with those of the first architect, except for the room, which he measured as having no measurable extent.

He did not mean that it was too small to measure. He meant that every measurement he took was contradicted by another.

The room was four metres wide, according to the wall facing the garden. It was three metres wide, according to the wall facing the corridor. When he measured diagonally, he obtained a third figure. When he repeated the measurements, he obtained the original figures again.

He was sufficiently disturbed by this to return with a colleague.

They spent the afternoon arguing over the instrument and eventually concluded that the room was an artefact of faulty measurement.

I suggested that they might leave it out of the report.

They were grateful.

For several weeks I avoided the room. I had other things to attend to, and I found that I could live quite comfortably in the house so long as I did not attempt to account for every part of it.

Then, one evening, I noticed that the door had moved.

It was no longer between the library and the bedroom. It stood opposite the staircase, at a point where the wall had previously been unbroken.

I opened it.

The room was unchanged. The table, the chair, the window, and the view of the vanished elm were all exactly as I remembered them.

I closed the door and went downstairs.

The following morning I found the architects' plans spread across the dining table. I had put them away in a drawer, but now they lay side by side, their edges precisely aligned.

The four drawings were identical.

I carried them upstairs and compared them with the house. The proportions were exact. The corridor, the staircase, the library, and the bedroom all corresponded perfectly to the plans.

I stood before the door opposite the staircase.

It was not represented in any of the drawings.

I began to understand that the problem was not that the plans were inaccurate. They were accurate in every detail. Nor was the problem that the room was impossible. It was there whenever I opened the door.

The difficulty was that the house could not accommodate both facts at once.

I decided to leave the plans as they were.

Over the following months, the house settled into a new routine. I stopped asking where the room was, and it stopped troubling me with its movements. Sometimes it appeared at the end of the corridor; sometimes beside the staircase; once, memorably, behind the pantry door.

I never found it in the same place twice.

The room itself remained unchanged. I began to suspect that it was not the room that moved, but the rest of the house.

One afternoon, while sitting in the library, I noticed a small door in the wall behind my chair. It was so narrow that I had to turn sideways to enter.

Beyond it lay a room I had never seen before.

There was no window, no furniture, and no visible door except the one through which I had entered. The walls were white, and the floor was covered with pale stone tiles arranged in a pattern of interlocking squares.

I stood in the middle of the room, trying to decide whether to call someone.

Then I noticed that the tiles were not squares.

Each was a slightly different shape, and yet the pattern fitted together perfectly. I could follow the lines across the floor indefinitely without finding a gap or an overlap. The pattern seemed to continue beyond the walls, though I could not see how.

I returned to the library and fetched a ruler.

When I came back, the door had disappeared.

I searched the wall for a handle, a seam, any indication that an opening had ever been there. I found nothing.

I have since had the house redecorated, but I have made no further alterations to its structure. The architects' plans remain in a drawer in my study, and I occasionally take them out to admire their precision.

I have come to appreciate my aunt's concern about symmetry.

The house is a very comfortable place to live, provided one does not insist on knowing exactly where one is.

I have also learned not to be alarmed when I find a door where there was none before.

After all, the house has never been wrong.

It is only my understanding of it that keeps changing.

I. The Garden Inside the Circle

I drew the circle by accident.

I was clearing the old study when I found a compass in the bottom of a drawer. It belonged to my father, who had been a surveyor, and I remembered him using it to draw circles on the backs of envelopes while he spoke on the telephone. I had never understood why he did this. Perhaps he needed something to occupy his hands.

The compass still worked. I found a sheet of paper, fixed the point near its centre, and turned the pencil around it. The result was almost perfect.

When I lifted the compass, I noticed that the circle enclosed a small patch of green. I thought at first that the paper had been stained, but the green had texture. There were blades of grass, a narrow path, and the top of a tree whose branches extended beyond the boundary of the circle and disappeared.

I fetched a magnifying glass.

The garden was much larger than the circle could contain. I could see a stone bench beside the path, a little gate, and what appeared to be a greenhouse. Beyond these lay a wooded slope, descending towards a strip of water. The further I looked, the more the garden seemed to recede, although the circle remained no larger than a teacup.

I turned the paper over. The garden disappeared.

When I turned it back, it was still there.

I am not an imaginative man, and I dislike being made to feel that I ought to be. I put the paper aside and went to make tea.

For the next few days, I tried not to think about it. The paper lay on my desk beneath a book, and I occasionally caught myself wondering whether the garden required watering. This was absurd, of course. I had never watered a drawing in my life.

On Thursday, I lifted the book.

The garden had changed. A small wooden table now stood beside the bench, and a cup rested upon it. There was a wisp of steam above the cup.

I took the paper outside and placed it on the garden table. The afternoon was windy, but the paper remained perfectly still. I leaned closer.

A woman was sitting on the bench. She wore a pale dress and had her back to me. Her hair was pinned up, and one hand rested on the table beside her.

I called out, though I knew she could not hear me.

She turned her head.

I dropped the paper.

When I picked it up, the bench was empty.

That evening I put the paper in a drawer and locked it. I slept badly, dreaming of a path that narrowed as I walked along it, though the trees on either side grew further apart.

The following morning, I took the paper out again. I had decided that I would destroy it, but before doing so I wanted to examine it once more.

The woman had returned. She was standing at the little gate, looking towards the circle's edge.

I noticed something else. The gate was open.

I had the strange impression that she was waiting for me.

I placed the paper on the floor and crouched beside it. The circle was no longer a circle in the ordinary sense. Its boundary had become a thin, dark line, and beyond it I could see the faint suggestion of a path continuing into the room.

I moved my hand towards the paper.

The path extended a little further.

I withdrew my hand. The path stopped.

For some time I sat on the floor, trying to decide what to do. I had the distinct feeling that whatever happened next would be my responsibility, although I could not have said why.

At last I took the compass from the drawer.

I set its point on the paper, just outside the original circle, and drew a second circle that overlapped the first.

Nothing happened immediately.

Then the garden changed.

The path that had led to the gate now curved away from it. The woman turned and walked towards the wooded slope. The little stream, which had previously run towards the edge of the garden, divided into two branches, one flowing uphill.

I watched until she disappeared among the trees.

The new circle enclosed a small part of the study floor. I could see the grain of the wood, a dark knot, and a thread of dust beneath the skirting board. But where the two circles overlapped, the floor had become grass.

I put my hand down.

The grass was cool and damp.

I did not step through. I am still not sure why. Perhaps I was afraid of what I might find on the other side. Perhaps I was afraid that there would be no other side.

Instead, I went to the kitchen and made another cup of tea.

When I returned, the garden had grown quiet. The woman was sitting on the bench again. Beside her stood a second chair, facing the first.

I looked at the compass in my hand.

Then I looked at the empty chair.

I have not drawn a third circle.

Not yet.

A Dog Is a Dog, Surely?

Mr Blottisham entered the Senior Common Room carrying a printed manuscript in one hand and a dog biscuit in the other.

Professor Quillibrace looked up from his newspaper.

“Have you taken to feeding the literature, Blottisham?”

“It's for the dog.”

“What dog?”

“The dog in the story.”

Professor Quillibrace folded his newspaper.

“I'm relieved to hear that it has remained a dog.”

“Don't be facetious. I've read the story.”

Miss Elowen Stray, who had been arranging a small vase of flowers on the mantelpiece, turned towards him.

“And what did you think?”

Blottisham placed the manuscript on the table with the air of a man depositing a body.

“I think the author has been at it again.”

“At what?” asked Quillibrace.

“Whatever it was that produced the previous four stories.”

“Have you discovered any evidence of chemical interference?”

“No. And I consider that extremely suspicious.”

Miss Stray sat down.

“Perhaps you could tell us what troubles you about this one.”

Blottisham took a deep breath.

“I specifically requested a perfectly ordinary story. A house, a man, a woman, and a dog. I stipulated that the house should remain a house, the man should remain a man, the woman should remain a woman, and the dog should remain a dog.”

“An admirably comprehensive set of conditions,” said Quillibrace.

“And the author has complied!”

Miss Stray raised her eyebrows.

“Then what is the problem?”

“Everything else!”

Quillibrace reached for his tea.

“Ah. The familiar difficulty of the entirely satisfactory disaster.”

“Don't interrupt. The house has windows that reflect landscapes that aren't there. Shadows wander about the premises without their owners. A pantry contains a passage that disappears when you try to enter it. The woman sees things that the man cannot see. The dog behaves as though it knows something that the author refuses to tell us.”

“Does the dog know something?” Miss Stray asked.

“Of course it knows something! It growls at the window!”

“Dogs do that.”

“Not at windows containing other dogs that aren't there!”

“Dogs sometimes bark at their reflections.”

“This was not a reflection. It was another dog in another world!”

Quillibrace looked thoughtful.

“Did the author establish that?”

“No! That's the point!”

“Then perhaps you should distinguish between what the story establishes and what you infer from it.”

Blottisham sat down abruptly.

“I refuse to be lectured on the difference between a dog and an imaginary dog.”

“An admirable refusal,” said Quillibrace. “Though it may prove difficult to sustain in the present circumstances.”

Miss Stray picked up the manuscript.

“I wonder whether the author has done something rather clever here.”

Blottisham groaned.

“Please don't.”

“The earlier stories allowed things to transform openly. Fish emerged from the current; flowers contained oceans; the palace became a flower. In this story, the house remains ordinary in every observable respect. Yet the reflections, shadows, and spaces make its ordinariness uncertain.”

“Exactly!” Blottisham exclaimed. “The author has concealed the nonsense instead of removing it!”

“Perhaps,” said Miss Stray, “the author has discovered that the ordinary world can be stranger than it appears without ceasing to be ordinary.”

Blottisham stared at her.

“That is a very dangerous proposition.”

“Why?”

“Because it makes nonsense impossible to detect.”

“Does it?”

“Certainly. If a house can contain anything without ceasing to be a house, then what is to prevent every house in the country from containing an ocean, a forest, or an entire population of invisible people?”

“Nothing, perhaps,” said Quillibrace.

Blottisham looked at him in horror.

“Professor!”

“I merely observe that architecture has long contained rooms whose contents are not visible from the street.”

“You know perfectly well what I mean!”

“I'm afraid I don't. But I am beginning to enjoy the uncertainty.”

Miss Stray turned to the passage about Margaret at the window.

“There's another curious thing. The woman doesn't seem frightened by what she sees. She becomes attentive to it.”

“Which is entirely unreasonable,” said Blottisham. “Any sensible person would call a glazier.”

“Why?”

“To repair the window!”

“What would you ask him to repair?”

“The glass!”

“Suppose the glass is perfectly sound?”

“Then the reflection!”

“Can a glazier repair a reflection?”

Blottisham opened his mouth.

Quillibrace intervened.

“We mustn't be too hard on the trades. A competent glazier may be able to repair a great many things, but metaphysical discrepancies are not generally covered by the standard guarantee.”

“Then the story is impossible to resolve.”

“Is it required to resolve?”

“Of course! Otherwise, the reader is left in a state of uncertainty.”

Miss Stray looked at him with gentle interest.

“And is uncertainty always a defect?”

“Certainly not. There is uncertainty about the weather, for example, but one can still carry an umbrella.”

“And what umbrella would you carry against a world that might be larger than it appears?”

Blottisham considered this with grave attention.

“A sufficiently large umbrella.”

Quillibrace put down his cup.

“I believe we have reached the limits of practical metaphysics.”

But Blottisham was not finished.

“The dog is the most objectionable feature.”

“I thought the dog was the most satisfactory feature,” said Quillibrace. “You insisted upon it.”

“I insisted upon a dog that remained a dog. I did not insist upon a dog that had access to information denied to the human characters.”

“Perhaps the dog has no such access.”

“Then why does it growl at the window?”

“Perhaps it sees something.”

“Something that isn't there?”

“Something that isn't there for Edward,” said Miss Stray.

Blottisham looked at her.

“You mean that the dog sees what the man cannot?”

“Perhaps.”

“Then the dog is more perceptive than the man!”

“Dogs often are.”

“That is not the point!”

“No,” she said. “I suspect the point is that the dog doesn't need to decide what the thing is before responding to it.”

Quillibrace smiled.

“An enviable freedom. The dog need not classify the phenomenon, establish its ontological status, or determine whether its existence is consistent with the prevailing account of reality. It can simply bark.”

Blottisham clutched the dog biscuit.

“I will not have it implied that a dog possesses superior philosophical judgement.”

“Then we must be careful,” said Quillibrace. “It might hear you.”

At this, Blottisham put the biscuit into his pocket.

Miss Stray returned to the ending.

“I rather like the fact that Thomas remains a dog even in his dreams. The house has no walls, the garden has no boundaries, and the black tree grows from a seed that was never planted. Yet the dog doesn't become a tree, or a flower, or a staircase.”

“Precisely!” said Blottisham, seizing upon this unexpected support. “At last, a sensible arrangement!”

“Although,” she continued, “the dog runs through a world in which the usual boundaries have disappeared.”

Blottisham's expression fell.

“You see? You see? Even the dog's dreams have been contaminated!”

“Must dreams be ordinary?” asked Quillibrace.

“Dreams are perfectly ordinary. Everyone has them.”

“Then perhaps we should ask everyone to submit their dreams for inspection.”

“Don't be absurd.”

“I'm relieved to hear that you still recognise the category.”

Blottisham rose and walked to the window.

Outside, the college garden lay beneath a pale morning sky. A gardener was pruning the roses, cutting away dead wood with small, precise movements.

Blottisham watched him for a moment.

“Now there,” he said, pointing. “There is a man doing something entirely comprehensible. He is pruning a rose. The rose is a rose. The shears are shears. The gardener is a gardener. Nobody is turning into anything.”

Miss Stray joined him at the window.

“Look at the rosebush beside the wall.”

Blottisham looked.

One pale rose had opened among the thorns. Its petals were almost white, with a faint green shadow at their edges.

“It's a rose,” he said warily.

“Yes.”

“And it will remain a rose.”

“Almost certainly.”

Blottisham turned to her.

“What do you mean, almost certainly?”

“I mean that it will open, fade, and lose its petals. Perhaps it will produce a rose hip. Perhaps a bird will eat it. Perhaps the gardener will cut the branch back.”

“None of which involves metamorphosis!”

“Not the sort you object to.”

Blottisham looked back at the flower.

For a moment, he seemed to be examining it more carefully than he intended.

Then he stepped away from the window.

“I shall not be reading any more stories by this author.”

“An understandable precaution,” said Quillibrace.

“I mean it. If I encounter another house, another garden, another animal behaving as though it possesses forbidden knowledge, I shall stop reading immediately.”

“Then I recommend,” said Miss Stray, “that you avoid looking too closely at the world outside.”

Blottisham put on his hat.

“I have spent my entire life looking at the world outside, Miss Stray, and I have never encountered a single thing that required me to revise my understanding of it.”

“Perhaps,” she said, “that is what you have been looking for.”

He departed without replying.

Quillibrace watched him go.

“Do you suppose he will inspect his own house when he gets home?”

“I imagine he will check every window.”

“And the pantry?”

“Certainly.”

“And the shadows?”

Miss Stray smiled.

“I think he will be especially careful about those.”

Quillibrace returned to his newspaper.

“Then the story has accomplished something. It has made an ordinary man look at an ordinary house with extraordinary suspicion.”

Outside, the gardener finished pruning the roses.

He gathered the cut branches and carried them away.

The pale rose remained on the bush.

For the moment, it was simply a rose.

And in the Senior Common Room, Professor Quillibrace began to wonder whether Blottisham's house had a pantry.