And the reply came back, not just in words, but in the answering hum of the one symbol among the thousands that reached out to claim all the rest. Not to mirror every thought, but to bind them. To hold them together, like skin.
Who is thinking this?
I am.
2
TRUTH MINING
Konishi polis, Earth
23 387 281 042 016 CST
18 May 2975, 10:10:39.170 UT
"What is it you're having trouble with?"
Radiya's icon was a fleshless skeleton made of twigs and branches, the skull carved from a knotted stump. Vis homescape was a forest of oak; they always met in the same clearing. Yatima wasn't sure if Radiya spent much time here, or whether ve immersed verself completely in abstract mathematical spaces whenever ve was working, but the forest's complex, arbitrary messiness made a curiously harmonious backdrop for the spartan objects they conjured up to explore.
"Spatial curvature. I still don't understand where it comes from." Yatima created a translucent blob, floating between ver and Radiya at chest height, with half a dozen black triangles embedded in it. "If you start out with a manifold, shouldn't you he able to impose any geometry you like on it?" A manifold was a space with nothing but dimension and topology; no angles, no distances, no parallel lines. As ve spoke, the blob stretched and bent, and the sides of the triangles swayed and undulated. "I thought curvature existed on a whole new level, a new set of rules you could write any way you liked. So you could choose zero curvature everywhere, if that' what you wanted." Ve straightened all the triangles into rigid, planar figures. "Now I'm not so sure. There are some simple two-dimensional manifolds, like a sphere, where I can't see how to flatten the geometry. But I can't prove that it's impossible, either."
Radiya said, "What about a torus? Can you give a torus Euclidean geometry?"
"I couldn't at first. But then I found a way."
"Show me."
Yatima banished the blob and created a torus, one delta wide and a quarter of a delta high, its white surface gridded with red meridians and blue circles of latitude. Ve'd found a standard tool in the library for treating the surface of any object as a scape; it re-scaled everything appropriately, forced notional light rays to follow the surface's geodesics, and added a slight thickness so there was no need to become two-dimensional yourself. Politely offering the address so Radiya could follow, Yatima jumped into the torus's scape.
They arrived standing on the outer rim—the torus's "equator"—facing "south." With light rays clinging to the surface, the scape appeared boundless, though Yatima could clearly see the backs of both Radiya's icon and vis own, one short revolution ahead, and ve could just make out a twice-distant Radiya through the gap between the two of them. The forest clearing was nowhere to be seen; above them was nothing but blackness.
Looking due south the perspective was very nearly linear, with the red meridians wrapping the torus appearing to converge toward a distant vanishing point. But to the east and west the blue lines of latitude—which seemed almost straight and parallel nearby—appeared to veer apart wildly as they approached a critical distance. Light rays circumnavigating the torus around the outer rim reconverged, as if focused by a magnifying lens, at the point directly opposite the place where they started out—so the vastly distended image of one tiny spot on the equator, exactly halfway around the torus, was hogging the view and pushing aside the image of everything north or south of it. Beyond the halfway mark the blue lines came together again and exhibited something like normal perspective for a while, before they came full circle and the effect was repeated. But this time the view beyond was blocked by a wide band of purple with a thin rim of black on top, stretching across the horizon: Yatima's own icon, distorted by the curvature. A green and brown streak was also visible, partly obscuring the purple and black one, if Yatima looked directly away from Radiya.
"The geometry of this embedding is non-Euclidean, obviously." Yatima sketched a few triangles on the surface at their feet. "The sum of the angles of a triangle depends on where you put it: more than 180 degrees here, near the outer rim, but less than 180 near the inner rim. In between, it almost balances out."
Radiya nodded. "All right. So how do you balance it out everywhere without changing the topology?"
Yatima sent a stream of tags to the scape object, and the view around them began to he transformed. Their smeared icons on the horizon to the east and west began to shrink, and the blue lines of latitude began to straighten out. To the south, the narrow region of linear perspective was expanding rapidly. "If you bend a cylinder into a torus, the lines parallel to the cylinder's axis get stretched into different-sized circles; that's where the curvature really comes from. And if you tried to keep all those circles the same size, there'd be no way to keep them apart; you'd crush the cylinder flat in the process. But that's only true in three dimensions."
The grid lines were all straight now, the perspective perfectly linear everywhere. They appeared to he standing on a boundless plane, with only the repeated images of their icons to reveal otherwise. The triangles had straightened out, too; Yatima made two identical copies of one of them, then maneuvered the three together into a fan that showed the angles summing to 180 degrees. "Topologically, nothing's changed; I haven't made any cuts or joins in the surface. The only difference is…"
Ve jumped back to the forest clearing. The torus appeared to have been transformed into a short cylindrical band; the large blue circles of latitude were all of equal size now-but the smaller red circles, the meridians, looked like they'd been flattened into straight lines. "I rotated each meridian 90 degrees, into a fourth spatial dimension. 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.