A tale of measurements, misdirections, and one extremely unreasonable staircase
The Royal Society of Cartographers received its commission on a Tuesday, which was generally considered an unsuitable day for anything unprecedented.
The commission was brief.
His Majesty requires an accurate map of the House at the End of Lantern Street, including all rooms, staircases, passages, entrances, exits, and their relations to one another. The map must be suitable for use by a person unfamiliar with the building.
The last sentence was underlined twice.
Master Orwin Bell, Senior Cartographer to the Crown, read it with satisfaction.
“An entirely reasonable requirement,” he announced to his assistant, Pell. “A building is a collection of rooms connected by passages. We record the rooms, measure the passages, and establish their relations. There is no mystery in it.”
“None at all, sir,” said Pell, who had never mapped a building in which he had not become lost.
They found the house behind a row of respectable shops. It was three storeys high, built of pale stone, and possessed a green front door with a brass knocker shaped like a hand holding a smaller brass knocker. A narrow plaque beside the door read:
THE HOUSE
Below it, in smaller letters, someone had added:
Please knock on whichever side you find yourself.
Bell knocked.
The door opened immediately, revealing a long corridor. At its far end stood a staircase ascending towards a window through which sunlight shone upwards.
Bell entered.
“North-facing,” he said, consulting his compass.
“Which north, sir?” asked Pell.
Bell looked at him.
“The north indicated by the compass.”
“Of course, sir.”
They proceeded down the corridor, measured its length, recorded its doors, and reached the staircase. Bell placed his instrument on the first step.
“Rise, seven inches. Tread, eleven. Entirely conventional.”
They climbed to the first landing, turned left, and entered a room containing a desk, a clock, and a portrait of a man whose face had been carefully erased.
Bell measured the room. Pell sketched its windows.
Then they returned to the landing.
The staircase was descending.
Bell frowned.
“Pell, did we turn around?”
“No, sir.”
“Did we descend?”
“No, sir.”
“Then why is the staircase descending?”
Pell examined it.
“Perhaps, sir, it is an unusually accommodating staircase.”
Bell made a note: Staircase changes orientation during survey. Investigate mechanical explanation.
They descended to the ground floor and found the front door exactly where they had left it. Outside, the shops stood in a row beneath the afternoon sun.
Bell took out his watch.
“Eleven minutes,” he said. “We have been inside for eleven minutes.”
“An efficient visit, sir.”
“We shall begin again.”
On the second attempt, they climbed the staircase, turned right instead of left, and entered a library. The shelves contained maps of the house, all drawn by cartographers whose names Bell recognised. Some were beautifully engraved. Others were covered in annotations, corrections and increasingly agitated punctuation.
One map showed a staircase ascending through the roof. Another depicted a corridor that returned to its starting point without turning. A third was simply a drawing of a man sitting on the floor with his head in his hands.
Bell examined the last.
“An unfortunate lapse in professional standards.”
“His name is Wetherby, sir,” said Pell. “He was First Cartographer to the Crown.”
“Then he should have known better.”
Bell spread a fresh sheet of paper across the library table.
“We shall proceed systematically. We will map every room independently, then establish the connections between them. If each room is accurately represented, the whole building must follow.”
Pell looked at the shelves.
“Have the others tried that, sir?”
“Undoubtedly. And have they succeeded?”
Pell pointed to the maps.
“I believe their results are before us.”
Bell rolled the old maps into a drawer.
“Then they have not been sufficiently systematic.”
For the next three days, the cartographers worked methodically.
They numbered the rooms. They measured every wall. They recorded each door, window and passage. They marked the direction of every staircase at the moment of observation. They even measured the distance between the house and the nearest public fountain, in case the difficulty lay in some unnoticed distortion of scale.
The results were excellent.
Every room, considered separately, was mapped with perfect accuracy.
The trouble began when Bell attempted to join the rooms together.
Room Twelve opened into Room Thirteen. Room Thirteen opened into Room Fourteen. Room Fourteen opened into Room Twelve.
This was not, in itself, unusual. A circular arrangement of rooms could easily be represented on paper.
But when Bell followed the route from Twelve to Thirteen to Fourteen, he returned to Twelve from the opposite side. When he retraced the route, he arrived at Thirteen. When Pell attempted to demonstrate the discrepancy, he emerged in a pantry and was unable to explain how he had reached it.
“Perhaps the pantry is connected to all three rooms,” Bell suggested.
“Through what passage, sir?”
“An unrecorded one.”
“But we have measured every wall.”
“Then the passage is concealed.”
“Within the walls?”
“Possibly.”
Pell looked at the stonework.
“That would make for a very crowded wall, sir.”
Bell ignored him and drew a new passage on the map.
The pantry became accessible from all three rooms. The map was now consistent.
Unfortunately, the next morning the pantry had acquired a window overlooking the library, although the two rooms occupied opposite sides of the house. Bell added another passage. The window disappeared. In its place appeared a door opening onto the staircase.
The staircase was now ascending.
Bell sat down.
“I begin to suspect,” he said, “that this house has not been properly constructed.”
A voice came from the doorway.
“On the contrary. It was constructed by the finest builders in the kingdom.”
The speaker was an elderly woman carrying a tray of tea. She wore a grey dress, a ring of keys, and an expression of mild disappointment.
“I am Mrs Vale, the housekeeper. You must be the cartographers.”
Bell stood.
“Master Bell, Senior Cartographer to the Crown. We are attempting to establish the building’s complete arrangement.”
“Are you?” Mrs Vale placed the tray on the table. “How ambitious.”
“Madam, we have encountered certain irregularities.”
“Have you tried the blue staircase?”
Bell stared at her.
“The blue staircase?”
“Near the winter room.”
“There is no blue staircase on our plan.”
“Then you have not found it.”
She left the room.
Pell poured the tea.
“Perhaps we should find the winter room, sir.”
“We shall not be diverted by domestic folklore.”
They found the winter room shortly before noon. It contained a fireplace, two armchairs, a bowl of pears, and a blue staircase rising from the centre of the carpet.
Bell approached it cautiously.
“Where does this lead?”
“Upstairs,” said Pell.
“Thank you. I had not noticed.”
They climbed.
At the top, they found themselves in the winter room.
Bell descended.
At the bottom, they were in the winter room.
He climbed again, this time counting the steps aloud. There were seventeen. At the top, the winter room. At the bottom, the winter room. He tried the staircase backwards. He tried it sideways, though Pell advised against this. The result remained unchanged.
Mrs Vale returned with fresh tea.
“Ah,” she said. “You have found it.”
“It leads nowhere!”
“On the contrary. It leads to the winter room.”
“We are already in the winter room.”
“Then it has brought you home.”
Bell opened his mouth, then closed it again.
“Madam, I must insist that a staircase connect two distinct levels of a building.”
“Must you?”
“It is the purpose of a staircase.”
“Is it? I had thought its purpose was to help people get where they wanted to go.”
“And how, precisely, does one get anywhere in this house?”
Mrs Vale considered the question.
“By going.”
Bell looked at Pell, who was studying the blue staircase with the expression of a man contemplating a promising career in another profession.
“Madam,” Bell said carefully, “have you ever attempted to map this house?”
“Once.”
“And?”
“I found it much easier to live in.”
She took the tray and departed.
That evening, Bell worked late. He spread his measurements across the library floor and began to draw a larger plan. The rooms fitted together. The passages fitted together. The staircases fitted together, provided he recorded their orientations separately at each landing.
But when he joined the floors, the house became impossible.
A room on the first floor occupied the same space as a room on the second. A staircase connected two landings that lay on opposite sides of the plan. A passage that was perfectly straight on one floor crossed itself on another.
He tried changing the scale. He tried rotating the floors. He tried drawing the house from below, from above, and from a point somewhere outside it.
At last he produced a map that was internally impeccable and depicted a building no one could enter.
Pell examined it.
“It is a beautiful map, sir.”
“Thank you.”
“Where is the front door?”
Bell pointed.
“And where is the street?”
Bell pointed again.
“And how does one get from the street to the front door?”
Bell traced a route with his finger. It led through the library, the winter room, a corridor, and a staircase.
“Can one actually do that, sir?”
Bell looked at the map.
“Not at present.”
“Then perhaps it is a map of the house as it ought to be.”
Bell regarded his assistant with cold displeasure.
“A map is not a wish, Pell.”
“No, sir. But this one appears to be a building that wishes to exist.”
Bell sent him to bed.
For the next week, the cartographers tried increasingly sophisticated methods. They drew separate maps for every floor, every staircase, every direction of travel. They used transparent sheets, coloured threads, hinged diagrams and a revolving wooden model. They even commissioned a mirror-maker to construct a device that would show the whole house from several perspectives at once.
The mirror-maker delivered a magnificent instrument consisting of twelve angled mirrors arranged around a central chamber.
When Bell looked into it, he saw the house repeated into infinity. Staircases rose through staircases. Corridors opened onto corridors. In one reflection, Mrs Vale was serving tea to a reflection of herself, who was serving tea to another reflection, who had apparently been waiting for some time.
“Does it help?” asked Pell.
“It reveals every possible perspective.”
“Does it reveal which perspective is correct?”
Bell adjusted one of the mirrors.
“It reveals the need for a more comprehensive perspective.”
“Ah,” said Pell. “Then it is very much like the house.”
Bell ordered the device removed.
On the eighth day, a royal inspector arrived to examine their progress. He was a short, broad man named Sir Dorrick, who carried a leather case containing the official standards for architectural measurement.
He inspected the maps, tested the instruments, and walked through several rooms.
“Each measurement is correct,” he announced.
Bell brightened.
“Precisely.”
“Each room is accurately represented.”
“Precisely.”
“Each connection you have recorded has been observed.”
“Precisely.”
Sir Dorrick placed the maps on the table.
“And yet the complete building cannot be reconstructed from them.”
Bell's brightness diminished.
“That is the difficulty.”
Sir Dorrick frowned.
“Then the problem must lie in the reconstruction.”
“But the reconstruction follows from the measurements.”
“Then the measurements must be incomplete.”
“They are complete.”
“Then your methods must be inadequate.”
“They are the finest methods in the kingdom.”
Sir Dorrick paused.
“I see.”
He wrote a report declaring that the house required further investigation.
This was considered an excellent outcome. It demonstrated that the Crown had identified the problem and established the need for further work.
The inspector departed.
That afternoon, Bell found Mrs Vale in the hall, polishing the brass knocker.
“Madam, I have a question.”
“Only one?”
“Do you believe this house has a correct arrangement?”
Mrs Vale polished the smaller knocker.
“Of course.”
“Then why can we not map it?”
“Perhaps you can.”
“But every map that works in one place fails in another.”
“Does it?”
Bell fetched his latest plan and showed her the impossible junction where three corridors met.
“This passage cannot exist.”
Mrs Vale studied it.
“Why not?”
“Because it would require a person to enter from two directions at once.”
“Does it?”
Bell hesitated.
“I have not observed anyone doing so.”
“Then perhaps you should not draw it that way.”
She handed back the map.
“Your difficulty, Master Bell, is that you keep asking the house to explain itself in the language of your drawings.”
“And what language should I use?”
Mrs Vale smiled.
“I have no idea. I am the housekeeper.”
She returned to her polishing.
Bell went back to the library.
For the first time, he wondered whether the house might not possess a single arrangement that could be represented in the manner he expected. The thought seemed absurd. A building must have walls, floors, passages and connections. It could not be both one thing and another.
Yet every measurement was sound. Every room was where it had been found. The staircases did not become imaginary merely because their relations resisted his plan.
He took a blank sheet of paper and drew a single room.
Then he drew a door.
Then a passage beyond the door.
He stopped.
He could continue indefinitely, recording each new room as he encountered it. He could describe how one place led to another, how a staircase behaved when approached from different directions, how the house changed the expectations of those who moved through it.
But he could not yet see how all these descriptions belonged to one complete drawing.
He put down his pen.
Outside, evening fell. The front door stood open, revealing the street beyond. The corridor stretched away towards the staircase. From somewhere above came the sound of a piano playing a tune Bell almost recognised.
He rose and walked towards it.
At the first landing, he turned left and found the library. At the second, he turned right and found a small room containing a single chair. He sat down and listened.
The piano stopped.
A door opened behind him.
Mrs Vale entered carrying a tray.
“Tea, Master Bell?”
He accepted a cup.
“Mrs Vale, does the house ever surprise you?”
“Every day.”
“And does that trouble you?”
“Not particularly. It is a very old house.”
She placed a plate of biscuits beside him.
“Though I did once spend three weeks looking for the linen cupboard.”
“And did you find it?”
“Oh, yes. It was in the linen cupboard.”
Bell looked at her.
Mrs Vale looked back.
Then, for the first time since arriving, Bell laughed.
The sound travelled down the corridor, passed through the winter room, climbed the blue staircase, and emerged from the front door into Lantern Street, where a passer-by heard it and wondered what could possibly be so amusing.
The following morning, Bell began a new map.
He did not abandon his measurements, nor did he discard the plans that had served him so well. But he left certain connections open, marked some routes as dependent on the direction of approach, and added a note wherever the relation between two accurately described places remained unresolved.
At the bottom of the page he wrote:
The rooms have been measured. The passages have been observed. The relations between them remain under investigation.
Pell read the note.
“Will the king accept it, sir?”
“I have no idea.”
“Will it help someone find their way through the house?”
“I hope so.”
“And will it be the final map?”
Bell looked towards the staircase, which was presently ascending.
“No, Pell. I rather doubt it.”
He signed the map and carried it to the front door.
On the way out, he passed Mrs Vale, who was bringing a new vase of flowers into the hall.
“Master Bell,” she called, “you have forgotten something.”
He turned.
“What is it?”
“Your map.”
He looked down at his empty hands. The map was under his arm.
“No, madam. I have it here.”
Mrs Vale glanced at the paper.
“Ah. Then you have forgotten the house.”
Bell looked at the map, then at the staircase, then at the front door standing open upon the street.
For a moment, he thought of returning to the library to begin again.
Instead, he stepped outside.
Behind him, the door closed gently. Somewhere in the house, a staircase changed direction.
And for the first time, Master Orwin Bell was not entirely certain that he had left it.
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