As the line on the sphere crossed the equator, the path crossed between the two cones. "That would have been the singularity, but in a moment I'll show you what's become of it." The avatar extended the meridian toward the south pole, and the path through the worm hole continued on through the lower cone, and emerged in the bottom region of ordinary space.

"Okay, that's one geodesic. And in the classical version, all geodesics from one wormhole mouth to the other would converge on the singularity. But now…" It drew a second meridian on the sphere, starting again from the north pole, but heading for a point on the equator 180 degrees away. This time, the colored path that appeared on the wormhole diagram approached the top mouth from the opposite side.

As before, when the meridian crossed the equator of the sphere, the path through the wormhole crossed between the two cones. Since the tips of the cones only touched at a single point, the second path had to pass through the same point as the first—but the avatar produced a magnifying glass and held it up to that point's standard fiber for Blanca to see. The tiny sphere had two colored dots on opposite sides of its equator. The two paths never actually collided; the extra dimensions gave them room to avoid each other, even though they converged on the same point of ordinary space.

The avatar gestured at the diagram, and suddenly the whole surface was color-coded for the extra dimensions. Far from the wormhole mouths the space was uniformly white—indicating that the extra dimensions were unconstrained, and there was no way of knowing any point's position on the standard fiber. Within each cone, though, the space gradually took on a definite hue—red in the top cone, violet in the bottom—and then, close to the meeting point, the color began to vary strikingly with the angle of approach: vivid green on one side of the top cone, sweeping round to magenta 180 degrees away—a pattern that emerged inverted on the cone below, before melding smoothly into the surrounding violet, which in turn faded to white. It was as if every radial path through the wormhole had been lifted "up" out of the plane of this two-dimensional space to a slightly different "height" as it approached, allowing them all to "cross over" at the center without fear of colliding. The only real difference was that the extra-dimensional equivalent of "height above the plane" had to occur in a space that looped back on itself, so that a line rotated through 360 degrees could change "height" smoothly all the way, and still end up exactly where it began.

Blanca gazed at the diagram, trying to see it from a fresh perspective despite the numbing familiarity of the concepts. "And a 6-sphere generates a whole family of particles, because there's room to avoid the singularity in different ways. But you said you started with a 2-sphere. Do you mean later, when you were working with three-dimensional space?"

"No." The avatar seemed somewhat bemused by the question. "I started exactly as you see here: with two-dimensional space, and a 2-sphere for the standard fiber."

"But why a 2-sphere?" Blanca duplicated the diagram, but used a circle as the standard fiber instead of a sphere. Again, no two paths through the wormhole were the same color at the cross-over point; the main difference was that they took on different colors straight from the whiteness of the, surrounding space, because there were no "north and south poles" now from which they could spread out. "In two-dimensional space, you only need one extra dimension to avoid the singularity."

"That's true," the avatar conceded. "But I used a two-dimensional standard fiber because this wormhole possesses two degrees of freedom. One keeps the geodesics from colliding at the center. The other keeps the two mouths of the wormhole itself apart. If I'd used a circle as the standard fiber, then the distance between the mouths would have been fixed at precisely zero—which would have been an absurd constraint, when the whole point of the model was to mimic quantum uncertainty."

Blanca felt vis infotrope firing up, frustrated but ever hopeful. They'd reached the heart of the Distance Problem. The exaggerated size of the cones in the diagram was misleading; the gravitational curvature of ordinary space around an elementary particle was negligible, and contributed virtually nothing to the length of the wormhole. It was the way paths through the wormhole coiled around the extra dimensions of the standard fiber that allowed them to be slightly longer than they would have been if the two mouths had simply been glued together, rim to rim.

Or in reality, much more than slightly.

"Two degrees of freedom," Blanca mused. "The width of the wormhole, and its length. But in your model, each dimension shares those two roles from the start—and if they don't share them equally it gives nonsensical results." Blanca had tried distorting the standard fiber to allow for longer wormholes, but that had been a disaster. Stretching the 6-sphere into a 6-ellipsoid of astronomical proportions allowed for hundred-billion-kilometer wormholes like the Forge had produced, but it also implied the existence of "electrons" shaped like pieces of string of astronomical length. And changing the topology of the standard fiber, rather than just its shape, would have destroyed the correspondence between wormhole mouths and particles. The avatar responded, somewhat defensively, "Maybe I could have done it your way, starting with a circle to keep the geodesics apart. But then I would have had to introduce a second circle to keep the mouth apart-making the standard fiber a 2-torus. If I'd taken that approach, by the time I worked my way up to matching the particle symmetries I would have found myself lumbered with twelve dimensions: six for each purpose. Which would have worked just as well, but it would have been twice as extravagant. And after the debacle of string theory, it was hard enough selling anyone on six."

"I can imagine." Blanca responded automatically, before ve'd fully absorbed what the avatar had said. A moment later, it hit ver.

Twelve dimensions? Ve'd felt so besieged by the realist backlash that ve'd never even considered doing more than defending Kozuch's six against the charge of "abstractionism." Twice as extravagant? It certainly would have been in the twenty-first century, when no one knew how long wormholes really were.

But now?

Blanca shut down the avatar and began a fresh set of calculations. Kozuch herself had never said anything so explicit about higher-dimensional alternatives, but the avatar's educated guess turned out to be perfectly correct. Just as a 2-torus was the result of expanding every point in a circle into another circle perpendicular to the first, turning every point in a 6-sphere into a 6-sphere in its own right created a 12-torus—and a 12-torus as the standard fiber solved everything. The symmetries of the particles, and the Planck-Wheeler size of their wormhole mouths, could arise from one set of six dimensions; the freedom of the wormholes to take on astronomical lengths could then arise from the remaining six.

If the 12-torus was much larger in the six "length" dimensions than the six "width" ones, the two scales became completely independent, the two roles entirely separate. In fact, the easiest way to picture the new model was to split up the whole four-plus-twelve-dimensional universe in much the same way as the ten-dimensional universe of the original Kozuch Theory—but with three levels, instead of two. The smallest six dimensions played the same role as ever: every point in four-dimensional space-time gained six sub-microscopic degrees of freedom. But the six larger dimensions made more sense if the roles were reversed: instead of a separate six-dimensional "macrosphere" for every point in the four-dimensional universe… there was a separate four-dimensional universe for every point in a single, vast, six-dimensional macrosphere.