The Inverted World

  The pointer was almost exactly above the central mark.

  “We’re at optimum,” he said. “Know what that means?”

  “Not exactly.”

  “You’ve been down past, haven’t you?” I confirmed this. “There’s always centrifugal force to contend with on this world. The further south one travels, the greater that force is. It’s always present anywhere south of optimum, but it makes no practicable difference to normal operations for about twelve miles south of optimum. Anything further than that, and the city would have real problems. You know that anyway, if you’ve felt the centrifugal force.”

  He took further readings from his instrument.

  “Eight and a half miles,” he said. “That’s the distance between here and the city…or how much ground the city has to make up.”

  I said: “How is the optimum measured?”

  “By its null gravitational distortions. It serves as the standard by which we measure the city’s progress. In physical terms, imagine it as a line drawn around the world.”

  “And the optimum is always moving?”

  “No. The optimum is stationary…but the ground moves away from it.”

  “Oh yes.”

  We packed our gear and continued northwards. Just before sunset we made camp for the night.

  4

  The surveying work was undemanding mentally, and as we moved slowly northwards I found that my only external preoccupation was a need to be ever-watchful for signs of hostile inhabitants. Denton told me that an attack on us was unlikely, but nevertheless we were on our guard.

  I was still thinking about the awesome experience of seeing the whole world lying before me. As an event, it was enough; understanding it was something else.

  On our third day out from the city I suddenly started to think about the education I had been given as a child. I’m not sure what started the train of thought; it was possibly a number of things, not least the shock of seeing how utterly the crèche had been destroyed.

  I had thought little of my education since leaving the crèche. At the time I had felt, in common with most of the children in the crèche, that the teaching we were given was not much more than a penance, in which time was served as of necessity. But looking back, much of the education pushed into our unwilling heads took on a new dimension in the context of the city.

  For instance, one of the subjects which had inspired in us the most boredom was what the teachers referred to as “geography.” Most of these lessons had been concerned with the techniques of cartography and surveying; in the enclosed environment of the crèche, such exercises had been almost wholly theoretical. Now, though, those hours of tedium took on their relevance at long last. With a little concentration and a certain amount of digging into my often faulty memory, I grasped quickly the principles of the work Denton was showing me.

  We had had many other subjects taught to us theoretically, and I saw now how those too had practical relevance. Any new guild apprentice would already have a background knowledge of the work his own guild would expect him to do, and in addition would have similar information about many of the other functions of the city.

  Nothing could have prepared me for the sheer physical grind of working on the tracks, but I’d had an almost instinctive understanding of the actual machinery used to haul the city along those tracks.

  I cared not at all for the compulsory training with the Militia, but the puzzling—at the time—emphasis placed on military strategy during our education would clearly help those men who later took arms for the security of the city.

  This process of thought led me to wonder whether there had been anything in my education that could possibly have prepared me for the sight of a world which was shaped the way this one appeared to be.

  The lessons we had been given which specifically referred to astrophysics and astronomy had always spoken of planets as spheres. Earth—the planet, not our city—was described as an oblate spheroid, and we had been shown maps of some of its land surface area. This aspect of physical science was not dwelt upon; I’d grown up to assume that the world on which Earth city existed was a sphere like Earth planet, and nothing I had been taught had contradicted this assumption. Indeed, the nature of the world was never discussed openly at all.

  I knew that Earth planet was part of a system of planets, which were orbiting a spherical sun. Earth planet itself was circled by a spherical satellite. Again, this information seemed always to be theoretical…and this lack of practical application had not concerned me even when I left the city, for it was always clear that a different circumstance obtained. The sun and moon were not spherical, and neither was the world on which we lived.

  The question remained: where were we?

  The solution lay perhaps in the past.

  This too had been covered comprehensively, although the histories we were taught were exclusively about Earth planet. Much of what we learned concerned military manoeuvres, the transference of power and government from one state to another. We knew that time was measured in terms of years and centuries on Earth planet, that recorded history existed for about twenty centuries. Perhaps unfairly, I formed an impression that I should not care to live on Earth planet, as most of its existence seemed to be a series of disputes, wars, territorial claims, economic pressures. The concept of civilization was far advanced, and explained to us as the state in which mankind congregated within cities. By definition, we of Earth city were civilized, but there seemed to be no resemblance between our existence and theirs. Civilization on Earth planet was equated with selfishness and greed; those people who lived in a civilized state exploited those who did not. There were shortages of vital commodities on Earth planet, and the people in the civilized nations were able to monopolize those commodities by reason of their greater economic strength. This imbalance appeared to be at the root of the disputes.

  I suddenly saw parallels between our civilization and theirs. The city was undoubtedly on a war footing as a result of the situation with the tooks, and that in its turn was a product of our barter system. We did not exploit them through wealth, but we had a surplus of the commodities in short supply on Earth planet: food, fuel energy, raw materials. Our shortage was manpower, and we paid for that with our surplus commodities.

  The process was inverted, but the product was the same.

  Following my line of reasoning, I saw that the examination of the history of Earth planet prepared the way for those who would become Barter guildsmen, but it took me no further along my own search for understanding. The histories began and ended on Earth planet, with no mention of how the city came to be on this world, nor how the city had been built, nor about who its founders were and where they came from.

  A deliberate omission? Or forgotten knowledge?

  I imagined that many guildsmen had tried to construct their own patterns of logic, and for all I knew either the answers were available somewhere in the city, or there was a commonly accepted hypothesis which I had not yet encountered. But I had fallen naturally into the ways of the guildsmen. Survival on this world was a matter of initiative: on the grand scale, by hauling the city northwards away from that zone of amazing distortion behind us, and on the personal scale by deriving for oneself a pattern of life that was self-determined. Future Denton was a self-sufficient man, and so had been most of those I had met. I wanted to be one with them, and comprehend things on my own account. I supposed that I could discuss my thoughts with Denton, but I chose not to.

  The journey northwards was slow and meandering. We did not take a route due north, but followed many diversions to east and west. Periodically Denton would measure our position against optimum, and never at any time were we further north than about fifteen miles.

  I asked him if there were any reason why we should not strike even further north of optimum.

  “Normally, we would go as far north as we can,” he said. “But the city’s in a special circumstance. As well as seeking the easiest northwards route, we nee
d terrain that will allow us to defend ourselves best.”

  The map we were compiling was becoming more complete and detailed every day. Denton allowed me to operate the equipment whenever I wished, and soon I was as adept as he. I learned how to triangulate the land with the surveying instrument, how to estimate the elevation of hills, and how to calculate the distance we were north or south of optimum. I was growing to like working the camera, in spite of the fact I was forced to curb my enthusiasm to conserve the energy in the batteries.

  It was peaceful and agreeable away from the tensions of the city, and I discovered that Denton, in spite of his long silences, was an amiable and intelligent man.

  I had lost track of the days we had been away, but it was certainly at least twenty. Denton showed no sign of wanting to return.

  We encountered a small settlement, nestling in a shallow valley. We made no attempt to approach it. Denton merely marked it on the map, with a rough estimate of the population.

  The countryside was greener and fresher than that to which I had grown accustomed, although the sun was no cooler. It rained more often here, usually during the night, and there were many different sizes of streams and rivers.

  All the features of the region, natural or man-made, difficult for the city to pass through or suitable to its peculiar needs, Denton marked without comment on his map. It was not the job of the Future Surveyors to decide which route the city should take; we worked simply to establish the actual nature of future terrain.

  The atmosphere was restful and soporific, the natural beauty of our surroundings seductive. I knew the city would travel through this region in the miles to come, and pass it without registering appreciation. For the city’s aesthetics, this verdant and gentle countryside might equally be a windswept desert.

  During the hours when I was not actually engaged in any of our routine tasks, I was still lost in speculative thought. I could not get out of my mind that spectacle of the manifest appearance of the world on which we existed. There must have been something, somewhere in those long years of tedious education that would, subconsciously, have prepared me for that sight. We live by our assumptions; if one took for granted that the world we travelled across was like any other, could any education ever prepare one for a total reversal of that assumption?

  The preparation for that sight had begun the day Future Denton had taken me outside the city for the first time, to see for myself a sun that revealed itself to be any shape but that of a sphere.

  But I still felt there must have been an earlier clue.

  I waited for a few more days, still worrying at the problem when I found time, then had an idea. Denton and I had camped one evening in open country beside a broad, shallow river, and as sunset approached I took the video camera and recorder and walked alone up the side of a low hill about half a mile away. At the top there was a clear view towards the north-eastern horizon.

  As the sun neared the horizon, the atmospheric haze dimmed its glare and its shape became visible: as ever, a broad disk spiked top and bottom. I switched on the camera, and took a long shot of it. Later I replayed the tape, checking that the picture was clear and steady.

  I never tired of the spectacle. The sky was reddening, and after the main disk had passed beneath the horizon, the upright pinnacle of light slid quickly down. For a few minutes after its passing there was an impression of a bright focus of orange-white at the centre of the red glow…but soon this passed and night came on quickly.

  I played the tape again, watching the image of the sun on the recorder’s tiny monitor. I froze the picture, then adjusted the brightness control, dimming the image until only the white shape remained.

  There in miniature was an image of the world. My world. I had seen that shape before…long before leaving the confines of the crèche. Those weird symmetrical curves made an overall pattern that someone had once shown me.

  I stared at the monitor screen for a long time, then conscience struck me and I switched off to conserve the batteries. I did not return to Denton straight away: I was straining my memory for some key to that faint recollection of the time when someone had drawn four lines on a sheet of card, and held up for all to see the place where Earth city struggled to survive.

  The map that Denton and I were compiling was taking on a definite shape.

  Drawn on the long roll of stiff paper he had brought, the plan took the form of a long, narrow funnel, with its narrowest point at the patch of woodland a mile or so to the north of where the city had been when we left it. Our travels had all been within the funnel, enabling us to make measurements of large features from all sides, to ensure that we compiled information as nearly accurate as possible.

  Soon the work was finished, and Denton said that we would return at once to the city.

  I had, in the video recorder, a complete and cross-referenced visual record of all the terrain we had covered. In the city, the Council of Navigators would examine as much as they felt necessary to plan the city’s next route. Denton told me that other Futures would go north soon, draw another funnel map of the terrain. Perhaps it too would start at the patch of woodland, and take a course five or ten degrees to east or west, or, if the Navigators felt that a safe route could be found in the terrain we had surveyed, the new map would start further up the known territory, and push forward again the frontier of the future we had surveyed.

  We headed back towards the city. I had expected, in some melodramatic way, that now we had the information we had been despatched to obtain, we would ride through day and night with no regard to safety or comfort; instead, the leisurely ride through the countryside continued.

  “Shouldn’t we hurry?” I said in the end, thinking that perhaps Denton was idling for some reason connected with me; I wished to show that I was willing to move with speed.

  “There’s never any hurry up future,” he said.

  I didn’t argue with him, but it had occurred to me that we had been away from the city for at least thirty days. In that time, the movement of the ground would have taken the city another three miles away from the optimum, and consequently the city would have had to travel at least that distance to stay within safety limits.

  I knew that the unsurveyed territory began only a mile or so beyond the city’s last position.

  In short, the city would need the information we had.

  The return journey took three days. On the third day, as we loaded the horses and continued on south, the memory I had been seeking came to me. It came unbidden, as is often the case when trying for something buried in the subconscious.

  I felt I had exhausted all my conscious memories of the lessons I’d had, and sorting through the memory of the long academic courses had been as fruitless as the sessions had been tedious at the time.

  Then, from a subject I had not even considered, the answer came.

  I remembered a period in my last few miles inside the crèche, when our teacher had taken us into the realms of calculus. All aspects of mathematics had induced the same response in me—I showed neither interest nor success—and this further development of abstract concepts had seemed no different.

  The teaching had covered a kind of calculus known as functions, and we were taught how to draw graphs representing these functions. It was the graphs that had provided the memory key: I had always had a moderate talent for drawing, and for a few days my interest had flickered into life. It died almost immediately, for I discovered that the graphs were not an end in themselves but were drawn to provide a means of finding out more about the function…and I didn’t know what a function was.

  One graph in particular had been discussed in great and onerous detail.

  It showed the curve of an equation where one value was represented as a reciprocal—or an inverse—of the other. The graph for this was a hyperbola. One part of the graph was drawn in the positive quadrant, one in the negative. Each end of the curve had an infinite value, both positive and negative.

  The teacher had dis
cussed what would happen if that graph were to be rotated about one of its axes. I had neither understood why graphs should be drawn, nor that one might rotate them, and I’d suffered another attack of daydreaming. But I did notice that the teacher had drawn on a piece of large card what the solid body would look like should this rotation be performed.

  The product was an impossible object: a solid with a disk of infinite radius, and two hyperbolic spires above and below the disk, each of which narrowed towards an infinitely distant point.

  It was a mathematical abstraction, and held for me then as much interest as such an item should.

  But that mathematical impossibility was not taught to us for no reason, and the teacher had not without reason attempted to draw it for us. In the indirect manner of all our education, that day I had seen the shape of the world on which I lived.

  5

  Denton and I rode through the woodland at the bottom of the range of hills…and there ahead of us was the pass.

  Involuntarily, I drew back on the reins and halted the horse.

  “The city!” I said. “Where is it?”

  “Still by the river I should imagine.”

  “Then it must have been destroyed!”

  There could be no other explanation. Had the city not moved in all those thirty days, only another attack could have delayed it. By now the city should at least be in its new position in the pass.

  Denton was watching me, an amused expression on his face.

  “Is this the first time you’ve been so far north of optimum?” he said.