Tiles as being like the data tape of a Turing Machine.' Paolo had the library grant him knowledge of the term; it was the original conceptual form of a generalized computing device, an imaginary machine which moved back and forth along a limitless one-dimensional data tape, reading and writing symbols according to a given set of rules.
'With the right set of tiles, to force the right pattern, the next row of the tiling will look like the data tape after the Turing Machine has performed one step of its computation. And the row after that will be the data tape after two steps, and so on. For any given Turing Machine, there's a set of Wang Tiles that can imitate it.'
Paolo nodded amiably. He hadn't heard of this particular quaint result, but it was hardly surprising. 'The carpets must be carrying out billions of acts of computation every second… but then, so are the water molecules around them. There are no physical processes that don't perform arithmetic of some kind.'
'True. But with the carpets, it's not quite the same as random molecular motion.'
'Maybe not.'
Karpal smiled, but said nothing.
'What? You've found a pattern? Don't tell me: our set of twenty thousand polysaccharide Wang Tiles just happens to form the Turing Machine for calculating pi.'
'No. What they form is a universal Turing Machine. They can calculate anything at all—depending on the data they start with. Every daughter fragment is like a program being fed to a chemical computer. Growth executes the program.'
'Ah.' Paolo's curiosity was roused—but he was having some trouble picturing where the hypothetical Turing Machine put its read/write head. 'Are you telling me only one tile changes between any two rows, where the 'machine' leaves its mark on the 'data tape'… The mosaics he'd seen were a riot of complexity, with no two rows remotely the same.
Karpal said, 'No, no. Wang's original example worked exactly like a standard Turing Machine, to simplify the argument… but the carpets are more like an arbitrary number of different computers with overlapping data, all working in parallel. This is biology, not a designed machine—it's as messy and wild as, say, a mammalian genome. In fact, there are mathematical similarities with gene regulation: I've identified Kauffman networks at every level, from the tiling rules up; the whole system's poised on the hyperadaptive edge between frozen and chaotic behavior.'
Paolo absorbed that, with the library's help. Like Earth life, the carpets seemed to have evolved a combination of robustness and flexibility which would have maximized their power to take advantage of natural selection. Thousands of different autocatalytic chemical networks must have arisen soon after the formation of Orpheus, but as the ocean chemistry and the climate changed in the Vegan system's early traumatic millennia, the ability to respond to selection pressure had itself been selected for, and the carpets were the result. Their complexity seemed redundant, now, after a hundred million years of relative stability—and no predators or competition in sight—but the legacy remained.
'So if the carpets have ended up as universal computers… with no real need anymore to respond to their surroundings… what are they doing with all that computing power?'
Karpal said solemnly, 'I'll show you.'
Paolo followed him into a scape where they drifted above a schematic of a carpet, an abstract landscape stretching far into the distance, elaborately wrinkled like the real thing, but otherwise heavily stylized, with each of the polysaccharide building blocks portrayed as a square tile with four different colored edges. The adjoining edges of neighboring tiles bore complementary colors—to represent the complementary, interlocking shapes of the borders of the building blocks.
'One group of microprobes finally managed to sequence an entire daughter fragment,' Karpal explained, 'although the exact edges it started life with are largely guesswork, since the thing was growing while they were trying to map it.' He gestured impatiently, and all the wrinkles and folds were smoothed away, an irrelevant distraction. They moved to one border of the ragged-edged carpet, and Karpal started the simulation running.
Paolo watched the mosaic extending itself, following the tiling rules perfectly—an orderly mathematical process, here: no chance collisions of radicals with catalylytic sites, no mismatched borders between two newgrown neighboring 'tiles' triggering the disintegration of both. Just the distillation of the higher-level consequences of all that random motion.
Karpal led Paolo up to a height where he could see subtle patterns being woven, overlapping multiplexed periodicities drifting across the growing edge, meeting and sometimes interacting, sometimes passing right through each other. Mobile pseudo-attractors, quasi-stable waveforms in a one-dimensional universe. The carpet's second dimension was more like time than space, a permanent record of the history of the edge.
Karpal seemed to read his mind. 'One dimensional. Worse than flatland. No connectivity, no complexity What can possibly happen in a system like that? Nothing of interest, right?' He clapped his hands and the scape exploded around Paolo. Trails of color streaked across his sensorium, entwining, then disintegrating into luminous smoke.
'Wrong. Everything goes on in a multidimensional frequency space. I've Fourier-transformed the edge into over a thousand components, and there's independent information in all of them. We're only in a narrow cross section here, a sixteen-dimensional slice—but it's oriented to show the principal components, the maximum detail.'
Paolo spun in a blur of meaningless color, utterly lost, his surroundings beyond comprehension. 'You're a gleisner robot, Karpal! Only sixteen dimensions! How can you have done this?'
Karpal sounded hurt, wherever he was. 'Why do you think I came to C-Z? I thought you people were flexible!'
'What you're doing is…' What? Heresy? There was no such thing. Officially. 'Have you shown this to anyone else?'
'Of course not. Who did you have in mind? Liesl? Hermann?'
'Good. I know how to keep my mouth shut.' Paolo jumped back to the dodecahedron; Karpal followed. 'How can I put this? The physical universe has three spatial dimensions, plus time. Citizens of Carter-Zimmerman inhabit the physical universe. The false promises of Kozuch Theory kept us from the stars for a thousand years. Higher- dimensional mind games are strictly for the solipsists.' Even as he said it, he realized how pompous he sounded.
Karpal replied, more bemused than offended, 'It's the only way to see what's going on. The only sensible, way to apprehend it. Don't you want to know what the carpets are actually like?'
Paolo felt himself being tempted. Inhabit a sixteen dimensional slice of a thousand-dimensional frequency space? But it was in the service of understanding a real physical system—not a novel experience for its own sake.
And nobody had to find out.
He ran a quick self-predictive model. There was a ninety-three percent chance that he'd give in, after a kilotau spent agonizing over the decision. It hardly seemed fair to keep Karpal waiting that long.
He said, 'You'll have to loan me your mind-shaping algorithm. My exoself wouldn't know where to begin.
When it was done, he steeled himself, and jumped back into Karpal's scape. For a moment, there was nothing but the same meaningless blur as before.
Then everything suddenly crystallized.
Creatures swain around them, elaborately branched tubes like mobile coral, vividly colored in all the hues of Paolo's mental palette—Karpal's attempt to cram in some of the information that a mere sixteen dimensions couldn't show. Paolo glanced down at his own body; nothing was missing, but he could see around it in all the thirteen dimensions in which it was nothing but a pinprick. He quickly looked away. The 'coral' seemed far more natural to his altered sensory map, occupying space in all directions, and shaded with hints that it occupied much more. Paolo had no doubt that it was 'alive'; it looked more organic than the carpets themselves, by far.
Karpal said, 'Every point in this space encodes some kind of quasi-periodic pattern in the tiles. Each dimension represents a different characteristic size—like a wavelength, although the analogy's not precise. The position of each dimension represents other attributes of the pattern, relating to the particular tiles it employs. So the localized systems you see around you are clusters of a few billion patterns with broadly similar attributes at similar wavelengths.'
