Diaspora

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 gri

"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 Riema

"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 cha

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 co

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 tu

Yatima had persisted. "You could make the programs more sophisticated. More discriminating. Let them generalize from examples, form conjectures… aim for proofs."

Radiya had conceded, "Perhaps it could be done. Some fleshers tried that approach before the Introdus—and if you're short-lived, slow, and easily distracted, it almost makes sense to let unthinking software find the lodes you'd never hit before you died. For us, though… Why should we sacrifice the opportunity for pleasure?"

Now that ve'd experienced Truth Mining for verself, Yatima could only agree. There was nothing in any scape or library file, any satellite feed or drone image, more beautiful than mathematics. Ve sent the scape a query tag, and it lit the way to the Gauss-Bo