Celestine, who was the best at this, drew a short red hyphen on the wall.
‘See this? A one-dimensional line. Now watch.’ She made the line become a square; splitting into two parallel lines joined at their ends. Then she made the square rotate until it was edge-on again, and all we could see was the line.
‘We see it…’ Childe said.
‘You can think of a line as the one-dimensional shadow of a two-dimensional object, in this case a square. Understand?’
‘I think we get the gist,’ Trintigant said.
Celestine made the square freeze, and then slide diagonally, leaving a copy of itself to which it was joined at the corners. ‘Now. We’re looking at a two-dimensional figure this time; the shadow of a three-dimensional cube. See how it changes if I rotate the cube, how it elongates and contracts?’
‘Yes. Got that,’ Childe said, watching the two joined squares slide across each other with a hypnotically smooth motion, only one square visible as the imagined cube presented itself face-on to the wall.
‘Well, I think these figures…’ Celestine sketched a hand an inch over the intricate designs worked into the frame, ‘I think what these figures represent are two-dimensional shadows of four-dimensional objects.’
‘Fuck off,’ Hirz said.
‘Look, just concentrate, will you? This one’s easy. It’s a hypercube. That’s the four-dimensional analogue of a cube. You just take a cube and extend it outwards; just the same way that you make a cube from a square.’ Celestine paused, and for a moment I thought she was going to throw up her hands in despair. ‘Look. Look at this.’ And then she sketched something on the wall: a cube set inside a slightly larger one, to which it was joined by diagonal lines. ‘That’s what the three-dimensional shadow of a hypercube would look like. Now all you have to do is collapse that shadow by one more dimension, down to two, to get this—’ and she jabbed at the beguiling design marked on the door.
‘I think I see it,’ Childe said, without anything resembling confidence.
Maybe I did, too — though I felt the same lack of certainty. Childe and I had certainly taunted each other with higher-dimensional puzzles in our youth, but never had so much deended on an intuitive grasp of those mind- shattering mathematical realms. ‘All right,’ I said. ‘Supposing that is the shadow of a tesseract… what’s the puzzle?’
‘This,’ Celestine said, pointing to the other side of the door, to what seemed like an utterly different — though no less complex — design. ‘It’s the same object, after a rotation.’
‘The shadow changes that drastically?’
‘Start getting used to it, Richard.’
‘All right.’ I realised she was still annoyed with me for touching her. ‘What about the others?’
‘They’re all four-dimensional objects; relatively simple geometric forms. This one’s a 4-simplex; a hypertetrahedon. It’s a hyper-pyramid with five tetrahedral faces…’ Celestine trailed off, looking at us with an odd expression on her face. ‘Never mind. The point is, all the corresponding forms on the right should be the shadows of the same polytopes after a simple rotation through higher-dimensional space. But one isn’t.’
‘Which is?’
She pointed to one of the forms. ‘This one.’
‘And you’re certain of that?’ Hirz said. ‘Because I’m sure as fuck not.’
Celestine nodded. ‘Yes. I’m completely sure of it now.’
‘But you can’t make any of us see that this is the case?’
She shrugged. ‘I guess you either see it or you don’t.’
‘Yeah? Well maybe we should have all taken a trip to the Pattern Jugglers. Then maybe I wouldn’t be about to shit myself.’
Celestine said nothing, but merely reached out and touched the errant figure.
‘There’s good news and there’s bad news,’ Forqueray said after we had traversed another dozen or so rooms without injury.
‘Give us the bad news first,’ Celestine said.
Forqueray obliged, with what sounded like the tiniest degree of pleasure. ‘We won’t be able to get through more than two or three more doors. Not with these suits on.’
There had been no real need to tell us that. It had become crushingly obvious during the last three or four rooms that we were near the limit; that the Spire’s subtly shifting internal architecture would not permit further movement within the bulky suits. It had been an effort to squeeze through the last door; only Hirz was oblivious to these difficulties.
‘Then we might as well give up,’ I said.
‘Not exactly.’ Forqueray smiled his vampiric smile. ‘I said there was good news as well, didn’t I?’
‘Which is?’ Childe said.
‘You remember when we sent Hirz back to the beginning, to see if the Spire was going to allow us to leave at any point?’
‘Yes,’ Childe said. Hirz had not repeated the complete exercise since, but she had gone back a dozen rooms, and found that the Spire was just as co-operative as it had been before. There was no reason to think she would not have been able to make her way to the exit, had she wished.
‘Something bothered me,’ Forqueray said. ‘When she went back, the Spire opened and closed doors in sequence to allow her to pass. I couldn’t see the sense in that. Why not just open all the doors along her route?’
‘I confess it troubled me as well,’ Trintignant said.
‘So I thought about it, and decided there must be a reason not to have all the doors open at once.’
Childe sighed. ‘Which was?’
‘Air,’ Forqueray said.
‘You’re kidding, aren’t you?’
The Ultra shook his head. ‘When we began, we were moving in vacuum — or at least through air that was as thin as that on Golgotha’s surface. That continued to be the case for the next few rooms. Then it began to change. Very slowly, I’ll grant you — but my suit sensors picked up on it immediately.’
Childe pulled a face. ‘And it didn’t cross your mind to tell any of us about this?’
‘I thought it best to wait until a pattern became apparent.’ Forqueray glanced at Celestine, whose face was impassive.
‘He’s right,’ Trintignant said. ‘I too have become aware of the changing atmospheric conditions. Forqueray has also doubtless noticed that the temperature in each room has been a little warmer than the last. I have extrapolated these trends and arrived at a tentative conclusion. Within two — possibly three — rooms, we will be able to discard our suits and breathe normally.’
‘Discard our suits?’ Hirz looked at him as if he were insane. ‘You have got to be fucking kidding.’
Childe raised a hand. ‘Wait a minute. When you said air, Doctor Trintignant, you didn’t say it was anything we’d be able to breathe.’
The Doctor’s answer was a melodious piped refrain. ‘Except it is. The ratios of the various gases are remarkably close to those we employ in our suits.’
‘Which isn’t possible. I don’t remember providing a sample.’
Trintignant dipped his head in a nod. ‘Nonetheless, it appears that one has been taken. The mix, incidentally, corresponds to precisely the atmospheric preferences of Ultras. Argyle’s expedition would surely have employed a slightly different mix, so it is not simply the case that the Spire has a long memory.’
I shivered.
The thought that the Spire — this vast breathing thing through which we were scurrying like rats — had somehow reached inside the hard armour of our suits to snatch a sample of air, without our knowing, made my guts turn cold. It not only knew of our presence, but it knew — intimately — what we were.
It understood our fragility.
As if wishing to reward Forqueray for his observation, the next room contained a substantially thicker atmosphere than any of its predecessors, and was also much warmer. It was not yet capable of supporting life, but one would not have died instantly without the protection of a suit.
