‘Don’t fuck with me, Doc. You know what I mean.’
Trintignant touched a finger to the chin of his helmet. ‘I confess I do not. Unless it is your contention that the Spire has at some point agreed to bind by a set of strictures, which I would ardently suggest is far from the case.’
‘No,’ I said. ‘Hirz is right, in one way. There have been rules. It’s clear that it won’t tolerate us inflicting physical harm against it. And it won’t allow us to enter a room until we’ve all stepped into the preceding one. I think those are pretty fundamental rules.’
‘Then what about the drone, and the door?’ asked Childe.
‘It’s like Trintignant said. It tolerated us playing outside the rules until now, but we shouldn’t have assumed that was always going to be the case.’
Hirz nodded. ‘Great. What else is it tolerating now?’
‘I don’t know.’ I managed a thin smile. ‘I suppose the only way to find out is to keep going.’
We passed through another eight rooms, taking between one and two hours to solve each.
There had been a couple of occasions when we had debated whether to continue, with Hirz usually the least keen of us, but so far the problems had not been insurmountably difficult. And we were making a kind of progress. Mostly the rooms were blank, but every now and then there was a narrow, trellised window, panelled in stained sheets of what was obviously a substance very much more resilient than glass or even diamond. Sometimes these windows opened only into gloomy interior spaces, but on one occasion we were able to look outside, able to sense some of the height we had attained. Forqueray, who had had been monitoring our journey with an inertial compass and gravitometer, confirmed that we had ascended at least fifteen vertical metres since the first chamber. That almost sounded impressive, until one considered the several hundred metres of Spire that undoubtedly lay above us. Another few hundred rooms, each posing a challenge more testing than the last?
And the doors were definitely getting smaller.
It was an effort to squeeze through now, and while the suits were able to reshape themselves to some extent, there was a limit to how compact they could become.
It had taken us sixteen hours to reach this point. At this rate it would take many days to get anywhere near the summit.
But none of us had imagined that this would be over quickly.
‘Tricky,’ Celestine said, after studying the latest puzzle for many minutes. ‘I think I see what’s going on here, but…’
Childe looked at her. ‘You think, or you know?’
‘I mean what I said. It’s not easy, you know. Would you rather I let someone else take first crack at it?’
I put a hand on Celestine’s arm and spoke to her privately. ‘Easy. He’s just anxious, that’s all.’
She brushed my hand away. ‘I didn’t ask you to defend me, Richard.’
‘I’m sorry. I didn’t mean—’
‘Never mind.’ Celestine switched off private mode and addressed the group. ‘I think these markings are shadows. Look.’
By now we had all become reasonably adept at drawing figures using our suits’ visualisation systems. These sketchy hallucinations could be painted on any surface, apparently visible to all.
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 walclass="underline" 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.