Journey to the Land of the Fourth Dimension

by Gaston de Pawlowski

  More disturbing and bewildering is the observation that one can achieve a possible abstraction of space solely by an effort of will. Moreover, it must be said, all our contemporary effort has been tending for a long time towards a similar result, and we are already beginning to understand that progress can, in large measure, be realized by continually increasing the speed of our actions.

  For a long time, economists have considered the sum total of the capital in circulation within a county as representative of its wealth. That element is, however, insignificant by comparison with the qualitative element: the rapidity of labor and traffic. Indeed, whether it is a matter of capital or means of transportation, what it is necessary to obtain, before anything else, is a better return of labor, an increase in speed—and social life is increased 365 times over when one accomplishes in a single day what our ancestors, with the same amount of capital and the same individual energy, could only realize in a year. It is for that reason that, in certain countries that are highly advanced from an industrial viewpoint—America, for example—special engineers called speed-merchants6 are only occupied with one thing: indefinitely increasing the speed of work, without any corresponding increase in general expenses (quite the contrary).

  To take a commonplace example of this extraordinary transformation, it is sufficient to reflect momentarily on what simple day’s journey once amounted to, in a humble diligence.7

  To increase the services that could be rendered by a postal service thus conceived, it was necessary to multiply fantastically the number of vehicles. On the other hand, merely by perfecting the quality of the traffic, increasing the speed of the old diligence by replacing it with an automobile, it was possible for a single vehicle to cover 50 times the distance in the same day, and the service presumably works 50 times better without any need to increase the number of vehicles.

  Now, imagine that speed increased in an infinite fashion, and you will establish logically that, if such an increase in speed were possible, the same single vehicle would end up presenting itself at every stop along its route at every moment of the day. That seems unrealizable in practical terms because our material forces are insufficient and we can only conceive of movement in a three-dimensional space—which is to say, as a succession of situations. On the other hand, as soon as we have a total conception of the four-dimensional universe, that which was hitherto absurd becomes easily realizable, and we understand clearly that the same vehicle can find itself simultaneously in all the different situations at every moment of the day, since speed suppresses time on this occasion.

  Our mind, which reasons in a four-dimensional space itself, is not astonished when it performs an analogous operation every day in making an abstraction from various situations and grasping at a single stroke the idea of the route as such, or of absolute speed. If we hesitate to apply these abstractions to the material world, it is because our natural weakness causes us to distinguish and categorize in time that which we call a memory and a present vision. A little reflection will suffice, however, to make us understand that, if our mind had the force necessary to evoke a memory as a whole, it would have as much effective reality as our present vision.

  Every day, our four-dimensional mind incites us, in spite of ourselves, to disencumber ourselves of the material obligations of the three-dimensional world. Why do we not do, for our material actions, that which we do for intellectual reasoning? Why retake a road previously traveled? Why repeat an itinerary whose details we know in advance? It becomes an obsession when one runs the same familiar course every day. Why must we submit to the administrative formality that constrains us to retake the same steps we have already taken, to follow the same routes already traversed, to end up at a point at which we know in advance that we are bound to end up? Is there no new procedure that will permit us to escape this base and material obligation?

  Already certain modern thinkers have done justice to the prejudice of the straight line. It has been demonstrated, for example, that in a world in which the size of its inhabitants increases and decreases the nearer they approach its center, the shortest route between two points on the spherical surface is a curved line passing via the equator, not the straight line that pierces a tunnel from one point on the spherical surface to another. Can one not conceive, equally, that in addition to these geometrical conditions of transport from one point to another, there exists a more direct process of abstraction, permitting the emancipation of the body and its abstraction from space, in the same fashion in which the mind acts, moving without physical displacement from one idea to another in four-dimensional space?

  This idea was, for me, only a striking suggestion—until the day when, happening to be living in a village, I succeeded, solely by the desire of my mind, in catching the local diligence anywhere I happened to be, at any hour of the day, according to the caprice of my will acting in four-dimensional space. The phenomenon produced itself for me spontaneously, without rational explanation, and it was a long time afterwards that I realized that it was materially realized by what I call, for want of a better term, a transmutation of special atoms.

  IV. The Horizontal Staircase

  These initial steps in the discovery of the fourth dimension were particularly painful for me. They occurred, in fact, in direct contradiction to the geometrical notions, full of logic and common sense, which were familiar to me. After the first advertisement of the untied ribbon and the innumerable diligence, however, the notion of the fourth dimension had to materialize itself to me in an even more precise fashion, in a form that I had not foreseen, and which seemed at first to be nightmarish.

  At very close intervals and in almost identical mental conditions, I found myself in the presence of staircases that were not constructed in a geometrical fashion—and, to begin with, nothing was more revolting to me than the negotiation of those sorts of staircases. Other people might, perhaps, not have been affected to the same degree. There are, in fact, people who—despite being well-educated or very intelligent—are insensitive to visual constructions and the equilibrium of things, and for whom every mechanical or architectural problem remains a closed book. They conceive the facts psychologically with their brains, but they do not seek to represent the events or ideas that they conceive materially to themselves. This is the case with writers who do not feel an imperious need, when they are analyzing a state of mind or a character, to do it in a graphical or musical fashion. It is evident, however, that this research is essential for the realization of a work of art. In our intelligence, properly speaking, there is no music, painting or literature; there are only obscure and silent impressions, and these impressions, all similar, are strictly self-contained. It seems difficult, in consequence, to have a complete artistic sensation if we neglect to examine the subject we are studying in all its aspects, according to all the information that our senses can give us. It is, however, a fragmentary fashion of envisaging things that is habitual to many people.

  When, on the contrary, one experiences one of the greatest pleasures of the mind in discovering the universal harmony of entities in all these aspects, nothing is more painful, morally, than to see certain material constructions not realized in accordance with the eternal logic of things.

  Now, among the architectural considerations that best symbolize our ideas, none is more seductive, albeit more complicated in its apparent simplicity, than the establishment of a staircase. The architects of yesteryear understood this very well and succeeded in working wonders in this respect. On the one hand, there are, as at Chambord, two intertwining staircases, which do not permit a person going up to encounter a person coming down; on the other hand, there are curious Gothic staircases whose savant helices seem to resolve all the problems of transcendental geometry. There are also, sometimes, and more simply, complicated staircases like those which still exist in certain old provincial houses; they intersect cleverly, each requiring several determinate landings. When one mistakenly takes the wrong staircase, one does not end up on
the intended floor; one finds oneself above it or below it, and it is necessary to make a certain effort of the imagination to rediscover the general design of the labyrinth.

  All this, however, is rapidly explicable, as soon as one gives it a little attention, and one soon realizes the reasons for the apparent illogic in the superposition of constructions of different eras, brought together over the centuries.

  Much more disturbing is the problem of the staircase which, after an undeniable succession of steps, brings you to the floor from which you started. They are things at which one smiles the first time, believing it a temporary error, but they are problems which become frightening when one persists in seeking the problem according to the primitive principles of three-dimensional Euclidean geometry.8

  And I admit, for my part, that I experienced a real relief on the day when I realized that if such staircases could exist, that their possibility could only be conceived in four-dimensional space, and only that this recourse sufficed to give a definitive explanation of the problem. Soon, I was even able to take a strange pleasure in passing through a few of these invisible residences, conceived by transcendental geometry, in which the floors are confused, where the first is not necessarily below the fourth, nor the third above the ground floor.

  IV. Spatial Abstractions

  One generally forms a completely false idea of the fourth dimension, wishing to describe it in terms of the givens furnished by the vision of a three-dimensional world. One ends up, in consequence, with impossibilities—and, by definition, irreducible absurdities. One is often similarly mistaken, as I have said, in simply wishing to add the fourth dimension to the other three, as if it were only a matter of creating a supplementary dimension, rendering possible the extension, to infinity, of further “dimensions” completing length, breadth and height.

  There again, without realizing it, one submits transcendental geometry to Euclidean definition; one renders all explanation impossible or absurd in advance. The fact is that Euclidean geometry, like all contemporary science, only operates in quantities, in numbers that divide our vision of the world into slices, cutting nature up into classes and categories. As soon as we want to begin on the most elevated research, we sense that this quantitative procedure is purely artificial, and that it cannot take account of the world entire. We know this because our consciousness, in contrast to our senses, is not constructed according to the vision of the three-dimensional world, but reveals to us, on the contrary, the fourth dimension—which is merely, in sum, the necessary complement of a total comprehension of the universe.

  It is, therefore, above the quantities excised by science, that our mind perpetually reveals to us those qualities which know no scientific measurement, and which translate themselves materially to our eyes by virtue of the existence of works of art.

  It is, therefore, grossly mistaken to think that the vision of a non-Euclidean world is opposed to our current vision of phenomena; in fact, it completes it.

  The external world appears to us initially, according to our retinal sensations, in a visual plane of two dimensions; then the muscular sensations of convergence and accommodation permit us to distinguish the distance of objects and to conceive the third dimension. Only the mind, which possesses a divine spark superior to the senses, permits us to understand that, above the world of appearances and scientific constructions, a complete and continuous vision of the universe exists. Thus, we can, without any great effort, realize at every instant the abstraction of time, associating very different ideas with one another. We can avoid recommencing a reasoning already completed, and set forth on a new mental path already designed to return us to the same mental location.

  Beneath the habitual three-dimensional vision, other, simpler ones can similarly be conceived. Yes, certainly, Euclidean geometry is for us presently the most comfortable means of grasping the universe, given the construction of our bodies and our centuries-old habits, but that does not make it a universal and indispensable form of sensation, Modern writers have tested this prejudice. Plane beings moving on a spherical surface would quite naturally conceive a geometry in which the sum of the angles of a triangle would always be greater than the sum of two right angles. Similarly, in a world derived of solids, our geometry might experience some difficulty in becoming manifest. Henri Poincaré has written some highly perceptively pages on this subject.

  We can divide up volumes by means of surfaces; we can divide up surfaces by means of lines; we can divide up lines by means of points—but when it comes to defining the point, our Euclidean science fails and vanishes. When it is necessary for us to take account of the physical continuum, our powerlessness is extreme. We come to understand that science is nothing but a conventional language, which permits us to catalogue and classify certain fractions of phenomena that we detach from one another artificially, according to their qualities, but we begin to sense that science, like language, is incapable of translating the continuity that belongs to the world of qualities and cannot be defined by numbers.

  The observations that we have just made immediately provoke an objection. If our continuous consciousness alone reveals to us the real existence of qualities—which is to say, the four dimensions united—how is it that our senses, developed according to the needs of the mind, do not perceive that fourth dimension with the same facility? Why must we have recourse to the numerical analyses of science and divide up the universe in three dimensions to render it intelligible?

  The answer to this question is easy. Our world is, for us, in perpetual transformation—which is to say, in perpetual progress. Now, the vision of the continuous universe is opposed to any idea of motion or change. Our motionless consciousness participates in the universality of things; it has no need to resort to the fractionation of the universe, but it is not the same for the body. The mind, which only conceives absolute unity, employs an admirable artifice to create the world in its image, and multiplies it to infinity. It reflects itself in numbers; it parcels out its entire personality to fractions of the universe that it wishes to analyze and comprehend. Thus, for the mind, number, beyond the figure 1 is nothing but a mirage—but a useful mirage. It permits it to create artificial individualities, which it distinguishes purely as new qualities of the eternal unity.

  It is impossible to understand space and the universe in an absolute fashion without being condemned, by the same token, to the divine immobility of consciousness. Just as the human mind creates gods in its own image, so it creates lines and numbers—but that is no more than a means of analysis, a scientific procedure of purely transitory demonstration.

  Human activity is only possible with the vision of the three-dimensional world, which renders the world mobile for us; but that suffices to give us a better understanding of the necessary existence of a fourth dimension that completes its unity and renders it immobile in the vulgar sense of the word—because, when man attains unity, by virtue of that very fact, he kills the illusory contradictions of life.

  As soon as one elevates oneself above the world of three dimensions—as soon as the mind, detached from the suggestions of sensation, recovers its integral power in the land of the fourth dimension—the activity of the three-dimensional world ceases, apparent mobility disappears and the abstractions of space and time become as natural in reality as they are in reasoning.

  It was initially in automobiles, on very long journeys, that I succeeded in realizing my first abstractions of distance, of which I have long conserved the memory. The first time, coming back from Florence to Paris via Aoste, I completely forgot the fragment of the route between Ambérieu and Tournos. On another occasion, on the Route d’Espagne, it was the immediate surroundings of Tours that I omitted to traverse.

  These material abstractions, on familiar journeys, were initially revealed to me by the veritable remorse that I felt, immediately afterwards, in observing my forgetfulness. It was as if all my atavistic senses were in revolt, as if in a protest of traditional logic, and I immediately
tried to find the indispensable rational explanation that would have liberated my senses. Doubtless, being very familiar with the route, I must have confused an old memory with present reality. I thought I had forgotten a journey, when in reality I had accomplished it while thinking about something else. Certain items of irrefutable material evidence—the consumption of fuel, the indications of a kilometric counter and those of a watch—proved to me that it was not so.

  Naturally, I still tried to think that there was nothing there but a collection of purely material coincidences and that I was the victim of an illusion. I experienced, moreover, a veritable cerebral fatigue in registering such facts and I tried not to think about them any longer, if only to escape those painful and material fits of remorse that I mentioned—fits of physical remorse much more discomfiting, when one is not accustomed to the mysteries of the fourth dimension, than all the moral remorse that one can experience in the ordinary life of three dimensions.

  VI. The Instantaneous Voyage

  Interesting as they were, the first abstractions of distance that I succeeded in realizing in the course of automobile journeys were, for me, only elementary indications of the possibility of a journey to the land of the fourth dimension. Only the abstraction of time, however, could give me conclusive results.

  It is curious to observe, in this respect, how many centuries-old prejudices are rooted in our minds when it comes to calculating the time necessary to accomplish an action or to conceive an idea. Fatally, we take for our base the average duration of human life. We estimate that this average time is necessary for the complete development of our personality. Furthermore, we cannot imagine that an important action or an idea of genius can become manifest without long preparation, without a historical series of successive actions—and we estimate the necessary time of gestation arbitrarily, according to the result obtained.