They only look flat because we're seeing them edge-on.' Yatima had rehearsed the trick with a lower-dimensional analogue: taking the band between a pair of concentric circles and twisting it 90 degrees out of the plane, standing it up on its edge; the extra dimension created room for the entire band to have a uniform radius. With a torus it was much the same; every circle of latitude could have the same radius, so long as they were given different 'heights' in a fourth dimension to keep them apart. Yatima re-colored the whole torus in smoothly varying shades of green to reveal the hidden fourth coordinate. The inner and outer surfaces of the 'cylinder' only matched colors at the top and bottom rims, '—here they met up in the fourth dimension; elsewhere, different hues on either side showed that they remained separated.
Radiya said, 'Very nice. Now can you do the same for a sphere?'
Yatima grimaced with frustration. 'I've tried! Intuitively, it just looks impossible… but I would have said the same thing about the torus, before I found the right trick.' Ve created a sphere as ve spoke, then deformed it into a cube. No good, though—that was just sweeping all the curvature into the singularities of the corners, it didn't make it go away.
'Okay. Here's a hint.' Radiya turned the cube back into a sphere, and drew three great circles on it in black: an equator, and two complete meridians 90 degrees apart.
'What have I divided the surface into?'
'Triangles. Right triangles.' Four in the northern hemisphere, four in the south.
'And whatever you do to the surface—bend it, stretch it, twist it into a thousand other dimensions—you'll always be able to divide it up the same way, won't you? Eight triangles, drawn between six points?'
Yatima experimented, deforming the sphere into a succession of different shapes. 'I think you're right. But how does that help?'
Radiya remained silent. Yatima made the object transparent, so ve could see all the triangles at once. They formed a kind of coarse mesh, a six-pointed net, a closed bag of string. Ve straightened all twelve lines, which certainly flattened the triangles-but it transformed the sphere into an octahedral diamond, which was just as bad as a cube. Each face of the diamond was perfectly Euclidean, but the six sharp points were like infinitely concentrated repositories of curvature.
Ve tried smoothing and flattening the six points. That was easy—but it made the eight triangles as bowed and non-Euclidean as they'd been on the original sphere. It seemed 'obvious' that the points and the triangles could never be made flat simultaneously… but Yatima still couldn't pin down the reason why the two goals were irreconcilable. Ve measured the angles where four triangles met, around what had once been a point of the diamond: 90, 90, 90, 90. That much made perfect sense: to lie flat, and meet nicely without any gaps, they had to add up to 360 degrees. Ve reverted to the unblunted diamond, and measured the same angles again: 60, 60, 60, 60. A total of 240 was too small to lie flat; anything less than a full circle forced the surface to roll up like the point of a cone…
That was it! That was the heart of the contradiction! Every vertex needed angles totaling 360 degrees around it, in order to lie flat… while every flat, Euclidean triangle supplied just 180 degrees. Half as much. So if there'd been exactly twice as many triangles as vertices, everything would have added up perfectly-but with six vertices and only eight triangles, there wasn't enough flatness to go round.
Yatima grinned triumphantly, and recounted vis chain of reasoning. Radiya said calmly, 'Good. You've just discovered the Gauss-Bonnet Theorem, linking the Euler number and total curvature.'
'Really?' Yatima felt a surge of pride; Euler and Gauss were legendary miners—long-dead fleshers, but their skills had rarely been equaled.
'Not quite.' Radiya smiled slightly. 'You should look up the precise statement of it, though; I think you're ready for a formal treatment of Riemannian spaces. But if it all starts to seem too abstract, don't be afraid to back off and play around with some more examples.'
'Okay.' Yatima didn't need to be told that the lesson was over. Ve raised a hand in a gesture of thanks, then withdrew vis icon and viewpoint from the clearing.
For a moment Yatima was scapeless, input channels isolated, alone with vis thoughts. Ve knew ve still didn't understand curvature fully—there were dozens of other ways to think about it—but at least ve'd grasped one more fragment of the whole picture.
Then ve jumped to the Truth Mines.
Ve arrived in a cavernous space with walls of dark rock, aggregates of gray igneous minerals, drab brown clays, streaks of rust red. Embedded in the floor of the cavern was a strange, luminous object: dozens of floating sparks of light, enclosed in an elaborate set of ethereal membranes. The membranes formed nested, concentric families, Daliesque onion layers—each series culminating in a bubble around a single spark, or occasionally a group of two or three. As the sparks drifted, the membranes flowed to accommodate them, in such a way that no spark ever escaped a single level of enclosure.
In one sense, the Truth Mines were just another indexscape. Hundreds of thousands of specialized selections of the library's contents were accessible in similar ways—and Yatima had climbed the Evolutionary Tree, hopscotched the Periodic Table, walked the avenue-like Timelines for the histories of fleshers, gleisners, and citizens. Half a megatau before, ve'd swum through the Eukaryotic Cell; every protein, every nucleotide, even carbohydrate drifting through the cytoplasm had broadcast gestalt tags with references to everything the library had to say about the molecule in question.
In the Truth Mines, though, the tags weren't just references; they included complete statements of the particular definitions, axioms, or theorems the objects represented. The Mines were self-contained: every mathematical result that fleshers and their descendants had ever proven was on display in its entirety. The library's exegesis was helpful—but the truths themselves were all here.
The luminous object buried in the cavern floor broadcast the definition of a topological space: a set of points (the sparks), grouped into 'open subsets' (the contents of one or more of the membranes) which specified how the points were connected to each other—without appealing to notions like 'distance' or 'dimension.' Short of a raw set with no structure at all, this was about as basic as you could get: the common ancestor of virtually every entity worthy of the name 'space,' however exotic. A single tunnel led into the cavern, providing a link to the necessary prior concepts, and half a dozen tunnels led out, slanting gently 'down' into the bedrock, pursuing various implications of the definition. Suppose T is a topological space… then what follows? These routes were paved with small gemstones, each one broadcasting an intermediate result on the way to a theorem. Every tunnel in the Mines was built from the steps of a watertight proof; every theorem, however deeply buried, could be traced back to every one of its assumptions. And to pin down exactly what was meant by a 'proof,' every field of mathematics used its own collection of formal systems: sets of axioms, definitions, and rules of deduction, along with the specialized vocabulary needed to state theorems and conjectures precisely.
When ve'd first met Radiya in the Mines, Yatima had asked ver why some non-sentient program couldn't just take each formal system used by the miners and crank out all its theorems automatically sparing citizens the effort.
Radiya had replied, 'Two is prime. Three is prime. Five is prime. Seven is prime. Eleven is prime. Thirteen is prime. Seventeen is-'
'Stop!'
'If I didn't get bored, I could go on like that until the Big Crunch, and discover nothing else.'
'But we could run a few billion programs at once, all mining in different directions. It wouldn't matter if some of them never found anything interesting.'
'Which 'different directions' would you choose?'
'I don't know. All of them?'
'A few billion blind moles won't let you do that. Suppose you have just one axiom, taken as given, and ten valid logical steps you can use to generate new statements. After one step, you have ten truths to explore.' Radiya had demonstrated, building a miniature, branching mine in the space in front of Yatima. 'After ten steps, you have ten billion, ten to the tenth power.' The fan of tunnels in the toy mine was already an unresolvable smear-but Radiya filled them with ten billion luminous moles, making the coal face glow strongly. 'After twenty steps, you have ten to the twentieth. Too many to explore at once, by a factor of ten billion. How are you going to choose the right ones? Or would you time-share the moles between all of these paths—slowing them down to the point of uselessness?' The moles spread their light out proportionately-and the glow of activity became invisibly feeble.
