Diaspora

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.