The Knights of the Limits

Book 16: Gateway Essentials

by Barrington J. Bayley

Genres: Science Fiction; Sci-fi Action & Adventure; Military Sci-Fi

Synopsis

Nine brilliant stories of infinite space and alien consciousness, suffused with a sense of wonder . . . The Exploration of Space The Bees of Knowledge Exit from City Me and my Antronoscope All the King's Men An Overload Mutation Planet The Problem of Morley's Emission The Cabinet of Oliver Naylor

Release date: September 29, 2011

Publisher: Gateway

Print pages: 220

Sample

The physical space in which we and the worlds move and have our being may easily be presumed to be a necessary and absolute condition of existence, the only form of the universe that is possible or even conceivable. Mathematicians may invent fictitious spaces of higher dimensions than our own but these, our intuition tells us, are no more than idealistic inventions which could nowhere be translated into reality and do not therefore properly deserve the designation ‘space’. The space we know, having the qualities of symmetry and continuity, is intimately and automatically the concomitant of any universe containing things and events, and therefore is inevitable; without space as we know it there could be no existence. The commonplace mind accepts this notion without question; thoughtful philosophers have spoken of the symmetrical, continuous space of three dimensions as an a priori world principle whose contradiction would remain a contradiction even in the mind of God. Yet not only has this belief no axiomatic justification but, as I shall attempt to show presently, it is untrue.

I had just smoked my second pipe of opium and was settling into a pleasant reverie. The opium smell, a sweet, cloying and quite unique odour, still hung in the air of my study, mingling with the aroma of the polished mahogany bookcases and the scent of flowers from the garden. Through the open window I could see that garden, with its pretty shrubs and crazy pathways, and beyond, the real ball of the far-off sun sinking through strata of pink and blue clouds.

My attention, however, was on the chessboard before me. Perhaps I should say a few words about myself. I believe that my brief participation in ‘orthodox’ experimental research may permit me to call myself a man of science, although these days my studies are more mathematical and deductive. It will surprise some that my main interest throughout my life has been alchemy. I have myself practised the Hermetic Art with some assiduity, if only to feel for myself the same numinousness experienced by my alchemistic forebears in manipulating the chemical constituents of the world. Hence I have known what it is to search for the prima materia (which others call the Philosopher’s stone, being the root of transformation); and I have pondered long and deeply on that profoundly basic manual, the Emerald Table of Hermes Trismegistus.

Unlike most contemporary men I am not inclined to the belief that alchemy has been rendered obsolete by modern science, but rather that its inadequate techniques and theories have been temporarily outstripped, while the essence of the Art remains unapproached. In the not too distant future the reverent search for prima materia may once again be conducted with the full charisma of symbology, but employing the best of particle accelerators. If the outlook I am displaying seems to run counter to the spirit of inductive science, let me admit that my thoughts do sometimes wander, for good or ill, outside the pale wherein dwell the more active members of the scientific community. There is value, I believe, in looking back over the history of science as well as forward to future expectations. I am not, for instance, hypnotised by the success of atomic theory, as are practically all of my colleagues. If I may be permitted to say so, the objections to the atomic view of nature listed by Aristotle have never been answered. These objections are still valid, and eventually they will have to be answered – or vindicated – on the level of subnuclear physics.

Opium has the happy conjunction of both inducing a feeling of relaxation and well-being and of opening the inner doors of the mind to a realm of colourful creativity. By opium, it is conjectured, Coleridge glimpsed the poem Kubla Khan, only fragments of which he managed to remember. By opium I met my new, though sadly soon-departed, friend, the Chessboard Knight.

A chessboard, to recapitulate the obvious, consists of 8×8 locations, or ‘squares’ arranged in a rectilinear grid. To us, the chessboard represents a peculiarly restricted world. The entities, or ‘pieces’ of this world are distinguished from one another only by their power of movement: a pawn can only move forward, one square at a time; a castle can move longitudinally for up to eight squares, a bishop likewise diagonally, and a knight can move to the opposite corner of a 2×3 rectangle. For all pieces movement is always directly from square to square, with no locations existing between the squares: none of them possesses the power of continuous, non-discrete movement we enjoy in our own world. On the other hand none of us possesses the power of simultaneous transition from location to location enjoyed by chessmen, particularly by the knight, who is unimpeded by intervening obstacles.

A rapid succession of similar thoughts was passing through my head as I gazed at the chessboard, though ostensibly to study the game laid out thereon, which I was playing by letter with a distant correspondent. As sometimes happens when smoking opium, time suddenly slowed down and thoughts seemed to come with incredible speed and clarity. Normally, I mused, one would unhesitatingly suppose our real physical world to be the superior of the chessboard world, because no limitation is placed upon the number of locations we may occupy. No arbitrary laws restrict me from moving in any way I please about my study, my garden, or the countryside beyond. But is that so important? The significance of chess lies not in its very simplified space-time environment but in the relation the pieces hold to one another. By this latter criterion our own degree of freedom undergoes a drastic reduction: the number of stances I can hold in relation to my wife, to my friends or to my employer (though being retired, I have no employer) is by no means greater; insignificant, in fact, when compared to the infinite number of relationships that would obtain by mathematically permutating all possible locations in our continuum of physical space. Is it an unfounded presumption, then, that our own work of continuous consecutive motions is logically any more basic to nature, or any richer in content, than one based on the principle of the chessboard, comprising discrete transitions between non-continuous locations?

I had reached thus far in my speeding express-train of thought when before my dazed eyes the chess pieces, like a machine that had suddenly been switched on, began flicking themselves around the board, switching from square to square with all the abruptness of the winking patterns of lights on a computer console. After this brief, flurried display they arranged themselves in a formation which left the centre of the board empty and were still – except for the White King’s Knight, who went flickering among them in his corner-turning manner, executing a dizzying but gracefully arabesque circuit of the board before finishing up in the centre, where he turned to me, bowed slightly, and lifted his head to speak to me in a distant, somewhat braying tone.

In my drugged state this happening did not induce in me the same surfeit of bewilderment and incredulity that would normally, I believe, have been my reaction. Astonished I certainly was. It is not every day that one’s chess set shows a life of its own, or that the pieces remain so true to their formal nature as laid down by the rules that they move from one position to another without bothering to traverse the spaces between. Not, let me add for the sake of the record, that the pieces showed any carelessness or laziness, or that they took short cuts. In order to move, say, from Qkt4 to Kr4, a castle was required to manifest himself in all the intervening squares so as to show that he came by a definite route and that the way was unimpeded – only the Knight flashed to his opposite corner unperturbed by whatever might surround him. These manifestations were, however, fleeting in the extreme, and nothing was ever seen of the castle in between adjacent squares – because, naturally, in a game of chess there is no ‘between adjacent squares’.

But I jump ahead of myself. My astonishment was so great that I missed the Knight’s first words and he was obliged to repeat himself. What he said was:

‘We enter your haven with gratitude.’

His voice, as I have said, was distant, with a resinous, braying quality. Yet not cold or unpleasant; on the contrary it was cordial and civilised. I replied:

‘I was not aware that you were in need of haven; but that being the case, you are welcome.’ In retrospect my words might appear to have received weighty consideration, but in fact they were flippant and extemporary, the only response my brain could form to an impossible situation. And so began my conversation with the Chessboard Knight, the strangest and most informative conversation I have ever held.

So total was my bemusement that I accepted with an unnatural calmness the Knight’s announcement that he was a space explorer. My sense of excitement returned, however, when he went on to explain that he was not a space explorer such as our imagination might conjure by the phrase, but that he was an explorer of alternative types of spatial framework of which, he assured me, there were a good number in the universe. What we are pleased to call the sidereal universe, that is, the whole system of space-time observable by us on Earth, is merely one among a vast range of various systems. Even more astounding, in the circumstances, was the revelation that the Knight hailed from a system of space identical to that which I had a moment before been contemplating! One analogous to a game of chess, where space, instead of being continuous and homogeneous as we know it, was made up of discrete locations, infinite or at any rate indefinite in number, and to which entities can address themselves instantaneously and in any order. There is no extended spatial framework in which these locations are ordered or arrayed and all locations are equally available from any starting point (provided they are not already occupied). An entity may, however, occupy only one location at a time and therein lies the principle of order in this well-nigh incomprehensible world. Structures, systems and events consist of convoluted arabesque patterns of successive occupations, and of the game-like relationships these manoeuvres hold to one another. The chess-people’s analogy of a long distance takes the form of a particularly difficult sequence of locations; alternatively the sequence could correspond to a particularly clever construct or device – the chess-people make little distinction between these two interpretations.

As do the occupants of a chessboard, the entities of this space (which I shall term locational-transitional space) vary in the range and ingenuity of their movements. Primitive organisms can do no more than transfer themselves slowly from one location to another, without pattern or direction, like pawns, while the most evolved intelligent species, like my friend the Knight, had advanced to dizzying achievements as laid down by the possibilities of such a realm. Their most staggering achievement was that of travel to other spaces; this was accomplished by a hazardous, almost infinitely long series of locations executed at colossal speed and comprising a pattern of such subtlety and complexity that my mind could not hope to comprehend it. Indeed, few even in the Knight’s spatial realm comprehended it and for their science it constituted a triumph comparable to our release of atomic energy from matter.

The discerning reader who has followed me this far might justly wonder at the coincidence which brought these bizarre travellers to my presence at the very moment when I had been theoretically contemplating something resembling their home space. This question was uppermost in my mind, also, but there was, the Knight told me, no coincidence involved at all. On entering our continuum (which the Knight and the companions under his command did indirectly, via other realms less weird to them) the space explorers had become confused and lost their bearings, seeming to wander in a sea of primeval chaos where no laws they could hypothesise, not even those garnered in their wide experience of spatial systems, seemed to obtain. Then, like a faint beacon of light in the uncognisable limbo, they had sighted a tiny oasis of ordered space, and with great expertise and luck had managed to steer their ship towards it.

That oasis was my chessboard. Not the board alone, of course – tens of thousands of chess games in progress at the same moment failed to catch their attention – but the fact that it had been illumined and made real by the thoughts I had entertained while gazing upon it, imbuing it with conceptions that approached, however haltingly, the conditions of their home world. Hence I owed the visitation to a lesser, more credible coincidence: chess and opium. At any rate, having landed their ship upon the board and thus bringing it under the influence of that vessel’s internally maintained alien laws, they had carried out simple manipulations of the pieces in order to signal their presence and establish communication – the real ship and its occupants not being visible or even conceivable to me, since they did not have contiguous spatial extension.

My reader, still suspicious of my truthfulness, will also want to know how it was that the Knight spoke to me in English. The appalling difficulties offered by any other explanation have tempted me to decide that we did not really speak at all, but only telepathically from mind to mind. And yet my grosser, more stubborn recollection belies this evasion: we did speak, the air vibrated and brought to me the thin, resinous tones of the Knight’s voice. His own remarks on the matter were off-hand and baffling. There was scarcely a language in the universe that could not be mastered in less than a minute, he said, provided it was of the relational type, which they nearly all were. He seemed to find my own mystification slightly disconcerting. The only comment I can contribute, after much reflection, is that for a locational-transitional being what he says may well be the case. Language, as he pointed out, largely concerns relations between things and concepts. To the Knight relations are the stuff of life, and he would find our own comprehension of them far below the level of imbecility. In our world to have but one fraction of his appreciation of relations, which to us are so important but so difficult to manage, would make us past masters of strategy and I believe no power would be able to withstand such knowledge.

But here lies the antinomy: the Knight and his crew were coming to me for help. They found the conditions of four three-dimensional continuum as incomprehensible and chaotic as we would find their realm. They had not even been able to ascertain what manner of space it was, and begged me to explain its laws to them in order that they might be able to find their way out of it.

There was a certain irony in being asked to describe the world I knew when I yearned to question the Knight as to his world. (Indeed my imagination was exploding – were there galaxies, stars and planets in the locational-transitional space? No, of course there could not be: such things were products of continuous space. What, then, was there? Some parallel to our phenomena there must be, but try as I might I could not picture what.) However, a cry of distress cannot go unanswered and I launched into an exposition.

It was quite a test of the intellect to have to describe the utterly familiar to a being whose conceptions are absolutely different from one’s own. At first I had great difficulty in explaining the rules and limitations by which we stereo beings (that is the phrase I have decided upon to describe our spatial characteristics) are obliged to order our lives. In particular it was hard to convey to the Knight that to get from point A to point B the basic strategy is to proceed in a straight line. To give them credit, the chessman crew had already experimented with the idea that continuous motion of some kind might be needed, but they had conceived the natural form of motion to be in a circle. When sighting my chessboard they had proceeded in the opposite direction and approached it by executing a perfect circle of a diameter several times that of the galaxy. I could not help but admire the mathematical expertise that had put both their starting point and their destination on the circumference of this circle.

After a number of false starts the Knight successfully mastered the necessary concepts and was able to identify the class of spaces to which ours belongs, a class some other members of which had been explored previously. They were regarded as dangerous but none, he informed me, had so far proved as hazardous and weird as our own, nor so difficult to move in. He still could not visualise our space, but I had apparently given him enough information for the ship’s computer to chart a course homeward (computers, theirs as well as ours, are notoriously untroubled by the limitations of imagination).

During the conversation I had naturally enough sought his opinion on various contemporary theories of the space we inhabit: on Riemannian space, Poincaré space, special and general relativity. Is our space positively or negatively curved? Spherical, parabolic or saddle-shaped – or is it curved at all? Is it finite or infinite? I acquainted him with the equation for the general theory of gravitation and invited his comments:

R1k – 1/2g1k R = T1k

His reply to all this was discouraging. The only definitive datum he would give me was that our space is infinite. As for Einstein’s equation, he said that it merely gave an approximate, superficial description of behaviour and did not uncover any law. He told me that in our continuum motion depends on a set of expansion.*

Our whole idea of analysing space by means of dimensions is inadequate and artificial, the Knight advised. The notion is an internally generated side-effect, and to anyone from outside, e.g. from another kind of space, it is neither meaningful nor descriptive. The essence of a spatial structure is more often expressed by a plain maxim that might appear to be ad hoc and rule-of-thumb, but that actually contains the nub of its specific law. At this I could not refrain from interrupting with the boast that privately I had once reached the same conclusion; and that if I had to state the basic physical law of our space (which I then thought of as the universe) it would be that in moving towards any one thing one is necessarily moving away from some other thing. The Knight complimented me on my insight; his ship’s computer was at that very moment grinding out the implications of a formulation quite close to the one I had come up with.

Following this, the Knight expressed his gratitude and announced his intention to leave. I begged him to stay a while; but he replied that to continue meshing the spatial laws of the ship (i.e: locational-transitional laws) with the pieces on the chessboard was proving to be a drain on the power unit. Guiltily, I confess that I allowed selfishness to come to the fore here. Did he not owe me something for the help I had given him, I argued? Could he and his crew not spend a little more energy, and would it truly endanger their lives? My unethical blackmail was prompted solely by my burning desire to learn as much as I could while the opportunity remained. I think he understood my feelings for, after a brief hesitation, he agreed to remain and discourse with me for a short time, or at least until the power drain approached a critical level.

Eagerly I besought him to tell me as much as he could of this vast universe of divers space-times to which he had access but I had not. To begin with, where did the Knight’s own spatial realm lie? Was it beyond the boundaries of our own space (beyond infinity!) or was it at right angles to it in another dimension? (I babbled carelessly, forgetting his former objection to the term.) Or was it, perhaps, co-extensive with our continuum, passing unnoticed because its own mode of existence is so unutterably different from it? To all these hasty suggestions the Knight replied by chiding me gently for my naivety. I would never know the answer while I persisted in thinking in a such a way, he said, for the simple reason that there was no answer and no question. While I was still capable of asking this non-existent question the non-answer would never be apparent to me.

Somewhat abashed, I asked a more pertinent question: was each space-time unique, or was each type duplicated over and over? As far as was known, the Knight said, each was unique, but they were classified by similarities and some differed only in details or in the quantitative value of some physical constant. It was to be expected, for instance, that there would be a range of stereo space-times resembling our own but with different values on the velocity of light. To my next request, that he describe some alien space-times to me, he explained that many would be totally inconceivable to me and that there was no way to express them in my language, mathematical or spoken. The majority of the spaces that were known to the chess-people were variations on the locational-transitional theme. There was a theory in his home world that locational-transitional (or chessboard) space was the basic kind of space in the universe and that all others were permutations and variants of it; but he agreed with me that this theory could be suspected of special pleading and that deeper penetration into the universe by the chess-people’s spaceships might well bring home a different story. He would not bore me, he added, by describing meaningless variations on locational-transitional space, but felt that I would be more entertained by those spaces whose qualities made striking comparisons with the qualities of my own realm.

There was, for instance, a space that, though continuous, was not symmetrical in all directions but was hung between two great poles like a magnetic field. Motion along the direction of the axis between the poles was as easy as it is for us, but transverse movement was an altogether different phenomenon that required a different type of energy and a different name. This polarisation continued down into every event and structure, which was invariably positioned. . .