Deep Isolation

“Are you conscious again?”

The laughter came from above, back beyond in the upper portion of the horn-shaped room that was stripped in orange bands that became deeper the further in. I was as far in as one could get, at the very bottom of the cell. They descended toward me and rained down water, the signal to wake up!

They stood atop shelves that descended from within the higher levels of the prison, which was one of the gnomic prisons in Rigel, a very efficient packing space. I was in the isolation ward of the 9th Wing, a quite friendly place, as a matter of fact. The horn-shaped room that unfurled in higher-order space gave one the distinct impression of a being inside a fusion ramjet, or at the base of a horn antenna, or some combination thereof, and, again, it had bands of orange that became deeper as it descended toward me, until it was a kind of rust color in the shadows here at the very bottom. In deep isolation, we 9th Wing prisoners entered dream-like states that lasted for who knew how long.

The effects of the curved geometry were subtle here, but I could see how the cavernous metal room curved backwards into a dark oblivion from whence the caretakers descended, and where they hosed me down with the water and nutrient fluid.

“How was your journey?”

“Quite pleasant,” I responded. “Will we be taking tea?”

“There’s a lecture on in half-an hour.”

“One on n-ordered domains in a Lobachevsky space?” I inquired. “I do love that subject.”

Gradually, I scaled the platforms, from broad metal shelf to metal shelf. They were spaced far apart, and I had been in stasis for a long time, so I required assistance from the caretakers. As I ascended, I sensed the bustle and commotion of the prisoners above, all gathering for the first time in eons for the Lecture.

***

The broad lecture hall occupied the center of a vast interior space formed from the hull, a paraboloid of revolution. This was a kind of node which was not separated from the cells and chambers that descended from it (such as the one from which I emerged), like a metallic nucleus at the center of dendrites. The seats were as in any amphitheater style classroom. We gradually filed into our seats in a jolly mood. Of course, the mood was one of jollity because it had been so long since our last convening and because it was time for a stimulating lecture related to Lobachevsky space, or at least I hoped.

Down in the pit of the amphitheater, the lecturer stood before three chalkboards, a petite, mousy woman. Her gray hair was done up in a bun and she wore dark-rimmed spectacles and a pumpkin colored scarf that complemented the orange, brown and vermillion stripes of the cavernous architecture that curved and disappeared into darkness far behind her, the recesses of some metal axon beyond the nucleus of the lecture hall.

“This lecture will focus on n-ordered domains in a Lobachevsky space and is entitled ‘The Negative Dimension,’” she began, “and it is based on the findings of the Geometry Supra-Division, Galactic One. In the following lecture, we will elaborate the complement to knots. Knots abound in the everyday, and researchers have discovered that studying even the simplest of these can lead us into spaces of almost unimaginable quality…”

So the lecture would be regarding n-ordered domains in a Lobachevsky space as I had anticipated. I nodded to the prisoner next to me and we exchanged a knowing glance. “The keys to the city,” I muttered and we chuckled.

“But we should begin by painting a simpler picture.” The Lecturer took up the chalk and began marking out cyphers on the chalkboard with assured strokes. “What is life like with a single point missing, or a single line, or a plane, i.e., with a single dimension removed? We can illustrate this with hyper-cone. As the apex steepens, the four space around the hyper-cone increases but the nappes (which are just the four-dimensional analogs of the three-dimensional surface — these are the future light-cone and past light-cone, respectively, in the temporal interpretation) remain constant. Do you see that for one living within the space tangent to the conical hypervolume that the light rays remain straight, but for the external observer situated in the manifold — they would see the curvature of the light rays?

“Now supposing a spaceship is traveling around the central 3-ball of the hyper-cone that runs through the apex. Sometimes the fundamental domains fit perfectly around the 3-ball and this is called ordered — order and symmetry with n fundamental domains fitting together. When the fundamental domains are ordered, the ship travels around the central 3-ball and the light rays around the apex reach the intrinsic observer from multiple different vantages simultaneously, resulting in what appears to be a multiplicity of ships. Now recall her question of what life is like in a space with a single line missing, or a plane that has the single point removed, etc.? As the order increases the missing dimension disappears in the limit, as the 3-ball vanishes to infinity. The endemic observer witnesses an endless array of spaceships that seem to pass through and invert through the very subject position of the observer herself: but this is all a single ship seen by light traveling on various paths corresponding to the light cones.

“Well, what happens when you take away a knot? What happens to the space? We mathematicians call the union of several loops a link. What is life like in a space where the link has been removed? With some rudimentary exceptions, all knots and links have complements that emit hyperbolic structures, according to a theorem of Bill Thurston.”

I raised my hand. “May I ask a question?”

“Please,” said the lecturer. She adjusted her bun and crossed her arms.

“How do we know that the 3-ball vanishes to infinity in the limit?”

“Differentiation is defined for functions that are differentiable — ”

“This is analogous to understanding the basic function of the integral,” I continued.

“It works by definition,” she concluded.

“Why does differentiation work? Is there any intuitive proof?”

“What is an ‘intuitive proof’?” she scoffed. “Something either works, or it doesn’t.”

“What I’m saying is, suppose we consider the fringes in the interface where the hyper-cone is adjacent with 3-space — trust me, I’ve had eons in the cryo-sleep to iterate the problem.”

The audience laughed at this.

Standing now, egged on by the crowd’s reaction, I continued: “I think we should take a closer look at the limit in this context.”

“But it works.”

“But there’s no evidence for it!”

“This is purely conjecture!”

There was some chatter at this, among the observers. I felt validated, satisfied by having my viewpoint expressed and considered. By the assembly, and by the lector.

“Analogizing from the hyper-cone to 3-space, from inside the rhombic dodecahedron, the link has become infinitely far, so far that the rays can never touch us. In terms of the hyperbolic structure for the link complement this is the picture of the complement of the Borromean rings.”

It occurred to me at this point that our worlds were not so different, she and I, but the connection did not exist. Standing before the audience, in the interval between waiting for acknowledgement and her getting on with the lecture, I felt the scaffolding inside me quake: I can’t go on like this. The feeling would have been recognizable to those familiar with a state of deep isolation, such as being alone for many winters, that of losing sense of the boundaries between the body and Nature; like an aura of nested, diminishing spaces, the sense of receding into afterimages of oneself, reminiscent of the brown and ochre bands of the curved metal space that grew dimmer as one withdrew.

Then, and only then, the alarms began.

I saw before me the spindly, smooth orange bridge that descended into the structure, like part of a honeycomb structure of smooth orange surfaces and bridges, or what it looks like to rush into the darkness of space with order N axis along the edges of the dodecahedron, or at worst, the hyperbolic manifold in the complement to the Borromean rings, a uniform honeycomb.

And then just as quickly as we fled together the light was gone. In the terror and the darkness of all hell breaking loose in the maze-like confines, the cacophony of billions rushing into these bending corridors, fragile orangish bridges in the dark honeycomb, if we let go of each other we would certainly perish.