228 Rue de Lyon was a five-storey townhouse of maybe a century in age. Next to its siblings it was identical in colour and architecture and distinguished only by the crookedness of the floors. For crooked it was: the third floor sagged alarmingly in the middle, and the very face of the building seemed to lean out into the street all the way up to the bowed eaves of the attic window high above our heads.
M. Bouchard muttered under his breath: “What a death trap.”
Mme Herat met us on the stair. She wore her engineer’s overalls and was fiddling with a pencil in one hand and a cigarette in the other. (Gauloises - I'd run enough errands for her to know she’d smoke no other.) She was, as my mother said, a New Woman, who’d spent the war fixing aircraft first for the Air Force and then for the British, and now lived in an apartment in the Latin Quarter where instead of a husband she kept exotic reptiles and (it was rumoured) female lovers.
“Maurice, thank God, someone with an inch of common sense to rescue this damsel in distress.” She waved the pencil vaguely up the stairs. “Top floor, and be careful as you pass the third, there’s this old hag living there who keeps sticking her neck out to complain about the noise.”
Bouchard blinked at her as she stuck the pencil in her mouth, realised her mistake, scowled at it, then stuck it in an ashtray balanced on the bannister and took a furious puff on the cigarette instead.
“I’ll need help with my measurements.”
She waved at me. “Sure, take the kid, whatever.”
As we climbed the stairs Bouchard looked at me. “What's your name, then?”
“Darius, monsieur,” I said.
He paused to look at me strangely and I felt very exposed. “And did your parents choose that name for you, or did you choose it for yourself?”
I flushed, which is a habit I hate in myself. Almost as much as I hate adults who think they see right through me. I grit my teeth and remembered Georg's advice. “Sir. It's my name.”
“Huh.” He continued up. “Funny the names they pick for young girls nowadays.”
“Sir, I'm not a girl.”
He turned to look at me. His gaze had an intensity I'd never witnessed before, which I would later learn was scientific observation, precise measurement and analysis. I burned under it like an ant under a magnifying glass. There was nothing I wanted to do less than spend an hour or more in the company of this man helping him at his work. But I clenched my fist on the bannister as if to brace myself against a rushing tide and forced myself to meet his eyes.
It gave me great joy that he turned away first.
“My apologies,” he said, keeping on up like nothing had happened. “And how old are you?”
“Fourteen, monsieur.”
We'd just reached the fourth floor and I realised he was slightly out of breath; maybe he was more frail than he looked. “Well, Darius, tell me about this room.”
So I did. I told him about how Mme Herat’s employer, a Dutchman named De Vries, had acquired the property at auction, along with a fake Toulouse-Lautrec and a genuine dodo egg (or was it the other way around?). Before that it had been the possession of a luckless landlord who’d lost all his money in speculative investments in Togo and had to evict all his tenants to keep from bankruptcy. Now nobody could remember who’d last resided in the attic apartment, but it was generally agreed that it must have been a miserable place to live, plagued as it was with a leaky roof, uneven floors, and a draft that nobody could seem to find the source of or put a stopper to.
“Kid, I’m not looking to buy the damn place, I want to know about the construction project.”
Well, De Vries had poked his fine Netherlandish nose into each of the apartments and declared them more or less salvageable. Except, that is, for the attic. He’d been found hurrying down the stairs shaking his head and pressing his fingers to his temples in consternation, as one does with a headache. “No good,” he announced. “We’ll have to rip it all out and start again. Total renovation. It won’t suffice to keep it vacant; I want it unrecognisable.”
Which is where Mme Herat came in. Having quit the Arsenal de l'Aéronautique in a blaze of scandal, she’d come to apply her talents first to agricultural machinery (“I hate cows,” she declared, “they smell almost as bad as Parisian men”) then to architectural structures. She took on odd jobs all across the city and specialised in the ones nobody else wanted: the slums, the drunk holes, the apartments just downwind from the municipal waste works where all residents acquired a certain ripeness of smell which refused to be washed out. And this was just one such job, because De Vries wasn’t a particularly tactful man, and not long after he’d published the advertisement requesting a man to perform the renovation on the attic, it had become a well-understood fact that there was something wrong with the attic room at 228 Rue de Lyon and the task should be avoided at all costs.
Mme Herat, obviously, laughed.
“Well, if no man will perform the renovation, he’ll have to make do with a woman instead, won’t he!” Jobs of this kind didn’t faze Mme Herat, even when she started to realise what precisely was wrong with the room and understand that her work was cut out for her, because she could stick it all on expenses and it provided an adequate intellectual challenge.
We reached the attic.
The stairs had narrowed and become rickety, the sort you were afraid at every turn that you might put a foot through. From the door was hung a “No Entry: Works in progress” sign, but the door itself was ajar. M Bouchard pushed it gently open and we stood in the doorway to watch.
Inside was a plain, unfurnished room, perhaps five or six paces across on each side. The floorboards were bare and thick with sawdust and old age. One of Mme Herat’s boys had started to strip the ugly rose wallpaper from one wall, but hadn’t quite finished the job, his tools left abandoned in a pile on the floor. A tattered electric lamp hung from the ceiling, but it was switched off. A slight draft came from the room, very gentle, but cold. That was about it.
I made to enter the room, but Bouchard held a hand across the frame to stop me. “Whoa, whoa, whoa, little hothead,” he said. “Cool yourself a moment and you can go in when I say you can.”
I’d talked myself out of being irritated at him, but it all came flooding back. I went and sat down on the stairs to fume in peace. But soon I found myself irresistibly drawn to the geometeer’s work, which resembled nothing I’d seen before.
First Bouchard waved a hand in front of him, to and fro, like he was feeling for an invisible curtain. Satisfied, he reached into his satchel and withdrew a measuring tape which he extended into the room, all the way to the far wall, noted the result, nodded, and zipped it back in to its holder. He then took his first step into the room, carefully, as though testing the floorboards for solidity. He stood in the middle of the room and looked all about himself in wonder, like a child in a museum. All the while he was whistling tunelessly to himself.
Then he remembered I was here and gestured impatiently for me to follow.
“Come on, help me with this.”
“This” turned out to be a length of twine and some nails. Under his instruction, I wedged the nails into the cracks between floorboards - incurring his ire when I got one of them in a spot that was a few centimetres from where he’d first indicated, though the spots seemed to be chosen randomly and without regard for their proximity to each other - and then wound the twine around them to form a triangle. Bouchard fished around in his bag for something which turned out to be a familiar plastic protractor, the kind we got in school. Quickly and carefully he measured the angles of our string triangle and noted them on a notepad.
“Make yourself useful, Darius,” he said as he wrote. “How’s your mental arithmetic?”
How was I supposed to answer that? “It’s very good, monsieur,” I said.
“Well, tell me what’s the sum of sixty-seven, fifty-five, and seventy-one.”
I juggled digits in a panic and to my own surprise dropped none. “One hundred and ninety-three.”
Bouchard looked up at me, surprised and - I like to think - impressed. “Just so. And remind me, from your maths classes, how many degrees do the interior angles of a triangle sum to?”
I looked from him, to the triangle, and back again. “One hundred and eighty.”
“Mm. A deviation of thirteen degrees here, then. Very poorly behaved.”
There must be something wrong with his protractor, I thought, or his reading of it. But when I unwound the twine and picked another three pins, and just as before measured out the triangle drawn between them, I got the same result. Or a little less - more like one-hundred and ninety one.
“It’s not possible!” I exclaimed, infuriated with the indignant little numbers before me.
Bouchard chuckled. “You wouldn’t be surprised if you’d looked at the walls more carefully when we entered.”
I spun around but the walls were just where they’d been before. “They’re just walls, aren’t they, sir?”
“And how many walls are those?” said Bouchard.
I looked around, trying to figure out what the trick question was. “Three, sir.”
“Ok, three. Doesn’t seem odd to you? What room has three walls? And what angle would you say separates those walls, eh?”
“A right angle, or close to it.”
Bouchard looked pleased. “Right enough. All three, though? And each wall a straight line?”
He was right. As I spun around I felt myself starting to get more and more dizzy, like the room was changing around me as I looked.
“And of course you’ve seen the most important problem with the room, I assume?”
Worse than these angles? “Er - the ceiling and the floor? Is something wrong with them too?”
“Not good enough. Look again,” said Bouchard.
I strained to see but all I saw were the three walls. There was the draft, which still remained unplaceable - one instant it felt like it was coming from the left, then the next from the right. And the delapidated light fitting, which - I noticed - didn’t even have a bulb inside.
But if there was no bulb, what light were we seeing by?
“The light, monsieur. Where is it coming from?”
Bouchard clapped his hands. “Bravo! We’ll make a novice geometeer out of you yet. Yes, and if you observe the shadows of the nails you drove into the floor, they seem hazy and blurred, but all on the whole leaning in one particular direction. What does that suggest?”
“A window, sir. I saw a window in the attic as we entered, but I can’t see it now.”
“A window, indeed. It will be somewhere in the room, but perhaps not easily reachable from the entrance - designed, I hypothesise, with obfuscation in mind. We must find the window and once that is done we can start to plan how to collapse the room back into regular space.”
“Designed, sir? You’ve seen such a chamber before?”
“Of course, little Darius! You see before you a fine example of deviant architecture, non-Euclidean architecture. There are better built examples of the thing throughout Paris, but often they go unnoticed even by their owners for many years.”
As we searched, Bouchard told me happily about the history of his field.
“It’s thought that many older buildings utilized deviant techniques, but the theory wasn’t well understood at the time so such architecture tended to fall prey to instability and structural issues. For instance, many castles and fortresses of the princes of Savoy contained small amounts of curvature, though whether this was introduced deliberately or is an accident of the construction techniques is still a matter of debate. In the Far East, certain Khmer temples and a number of Hindoo palaces contain deviant chambers specifically designed to be invisible to the uninitiated or unworthy, which can be reached only through a particular ritual of spatial traversal - often one with a spiritual significance. The workmanship of these chambers, though, was kept most secret by those who knew it; so secret that its very existence was ultimately forgotten, along with any practical knowledge of how to reproduce them. It’s speculated that the Pyramids of Giza contain a labyrinth of deviant chambers geometrically isolated from Euclidean space, in which the darkest secrets of the Pharaohs were immured, but if you ask me that’s just nonsense.”
He stopped to cough into a handkerchief before continuing.
"But it’s the great geometeers of the last century, Lobachevsky and Bolyai, Riemann and Gauss, they were the real masters of the art. They built the mathematics upon which modern deviant architecture is founded, the metrics and the manifolds. They imagined of strange shapes and impossible forms, and when others dismissed their discoveries as fancy or even heresy, they found ways to demonstrate such forms in the real world and create spaces hitherto the realm of dreams.
"Soon, developers of property and architects sought to exploit the new mathematics. They didn't care about the aesthetic foundations, they just wanted to make an extra franc or two from fitting a fifth apartment into a block that used only to fit four. And as when any work of pure mathematics is turned to commerce, things start to fall apart within a year or two. The foundations of buildings were unable to support the weight of extra rooms brought in on top, and many collapsed altogether, causing great loss of life. And so deviant architecture fell out of vogue, and was forgotten in the minds of many. But those of us who are skilled in the geometric arts remember. Hey, Darius, are you still listening?"
I turned to face him and found - to my shock - he wasn’t there. I was alone in the room. “Monsieur Bouchard?”
“Darius? Oh - drat.”
I turned to look for the door and it wasn’t there either. I was alone in a featureless attic, the walls bare, the bulb-less electric light its only feature. I started to panic. “Monsieur Bouchard, the door’s disapppeared. Where am I?”
“Stay calm, Darius. You’ve probably entered a spatial corridor.” His voice seemed to come from everywhere. “Where were you just moving? Can you continue moving in the same direction?”
Walls, walls, on all sides. “I don’t know. It all looks the same.”
“Right. Well, Darius, you’re probably near the window, so just move away from your shadow and towards the light. Can you do that?”
I closed my eyes and forced myself to breathe. “Uhhh. I think so.”
“Slowly, Darius. We have all the time in the world.”
I looked at the lamp hanging from the centre of the room. It had a clear shadow stretching away to the left. I started to move right, but through some trick of perspective the lamp seemed to retreat from view as I moved. Eyes fixed, I followed the shadow as it bent around from the left to the back, and then to the right. I took another step and it stretched suddenly away from me. I turned around and there it was: the attic window.
“Monsieur Bouchard, I found it!”
As I put my hands up to the glass I felt the strange draft stronger than ever before. I unlatched the window and the damp spring air came gushing into the room, blowing up the dust and sending the light fitting swinging. Below us came the sounds of the city: car horns, trains, people. I stuck my head out of the window and breathed it happily in.
Soon Bouchard appeared beside me. “Excellent work.” He started to measure the dimensions of the window, jotting even more incomprehensible numbers down in his book. “I think we have about everything we need.”
But I’d just noticed something else. “Monsieur Bouchard, look here. Scratch marks in the frame. What do you think?”
It was shallow and untidily scrawled, as perhaps a child. But the name was clear: Margot. And a shape, which reminded me at first of the strange geometric stone orbs of Monsieur Bouchard’s shop: two triangles atop each other, a Star of David.