Methods of investigation of the problem of higher dimensions. The analogy between imaginary worlds of different dimensions. The one-dimensional world on a line. “Space” and “time” of a one-dimensional being. The two-dimensional world on a plane. “Space” and “time,” “ether,” “matter” and “motion” of a two- dimensional being. Reality and illusion on a plane. The impossibility of seeing an “angle.” An angle as motion. The incomprehensibility to a two-dimensional being of the functions of things in our world. Phenomena and noumena of a two-dimensional being. How could a plane being comprehend the third dimension?
A SERIES of analogies and comparisons are used for the definition of that which can be, and that which cannot be, in the region of the higher dimension.
We imagine “worlds” of one, and of two dimensions, and out of the relations of lower-dimensional worlds to higher ones we deduce possible relations of our world to one of four dimensions; just as out of the relations of points to lines, of lines to surfaces, and of surfaces to solids we deduce the relations of our solids to four-dimensional ones.
Let us try to investigate everything that this method of analogy can yield.
Let us imagine a world of one dimension.
It will be a line. Upon this line let us imagine living beings. Upon this line, which represents the universe for them, they will be able to move forward and backward only, and these beings will be as the points, or segments of a line. Nothing will exist for them outside their line—and they will not be aware of the line upon which they are living and moving. For there will exist only two points, ahead and behind, or may be just one point ahead. Noticing the change in states of these points, the one-dimensional being will call these changes phenomena. If we suppose the line upon which the one-dimensional being lives to be passing through the different objects of our world, then of all these objects the one-dimensional being will perceive one point only; if different bodies intersect his line, the one-dimensional being will sense them only as the appearance, the more or less prolonged existence, and the disappearance of a point. This appearance, existence, and disappearance of a point will constitute a phenomenon. Phenomena, according to the character and properties of passing objects and the velocity and properties of their motions, for the one-dimensional being will be constant or variable, long or short timed, periodical or unperiodical. But the one-dimensional being will be absolutely unable to understand or explain the constancy or variability, the duration or brevity, the periodicity or unperiodicity of the phenomena of his world, and will regard these simply as properties of such phenomena. The solids intersecting his line may be different, but for the one-dimensional being all phenomena will be absolutely identical—just the appearance or the disappearance of a point—and phenomena will differ only in duration and in greater or less periodicity.
Such strange monotony and similarity of the diverse and heterogeneous phenomena of our world will be the characteristic peculiarity of the one-dimensional world.
Moreover, if we assume that the one-dimensional being possesses memory, it is clear that recalling all the points seen by him as phenomena, he will refer them to time. The point which was: this is the phenomenon already non-existent, and the point which may appear tomorrow: this is the phenomenon which does not exist as yet. All of our space except one line will be in the category of time, i.e., something wherefrom phenomena come and into which they disappear.
And the one-dimensional being will declare that the idea of time arises for him out of the observation of motion, that is to say, out of the appearance and disappearance of points. These will be considered as temporal phenomena, beginning at that moment when they become visible, and ending—ceasing to exist—at that moment when they become invisible. The one-dimensional being will not be in a position to imagine that the phenomenon goes on existing somewhere, though invisibly to him; or he will imagine it as existing somewhere on his line, far ahead of him.
We can imagine this one-dimensional being more vividly. Let us take an atom hovering in space, or simply a particle of dust, carried along by the air, and let us imagine that this atom or particle of dust possesses a consciousness, i.e., separates himself from the outside world, and is conscious only of that which lies in the line of his motion, and with which he himself comes in contact. He will then be a one-dimensional being in the full sense of the word. He can fly and move in all directions, but it will always seem to him that he is moving upon a single line; outside of this line will be for him only a great Nothingness—the whole universe will appear to him as one line. He will feel none of the turns and angles of his line, for to feel an angle it is necessary to be conscious of that which lies to right or left, above or below. In all other respects such a being will be absolutely identical with the before-described imaginary being living upon the imaginary line. Everything that he comes in contact with, that is, everything that he is conscious of, will seem to him to be emerging from time, i.e., from nothing, vanishing into time, i.e., into nothing. This nothing will be all our world. All our world except one line will be called time and will be counted as actually non-existent.
Let us next consider the two-dimensional world, and the being living on a plane. The universe of this being will be one great plane. Let us imagine beings on this plane having the shape of points, lines, and flat geometrical figures. The objects and “solids” of that world will have the shape of flat geometrical figures too.
In what manner will a being living on such a plane universe cognize his world?
First of all we can affirm that he will not feel the plane upon which he lives. He will not do so because he will feel the objects, i.e., figures which are on this plane. He will feel the lines which limit them, and for this reason he will not feel his plane, for in that case he would not be in a position to discern the lines. The lines will differ from the plane in that they produce sensations; therefore they exist. The plane does not produce sensations; therefore it does not exist. Moving on the plane, the two-dimensional being, feeling no sensations, will declare that nothing now exists. After having encountered some figure, having sensed its lines, he will say that something appeared. But gradually, by a process of reasoning, the two-dimensional being will come to the conclusion that the figures he encounters exist on something, or in something. Thereupon he may name such a plane (he will not know, indeed, that it is a plane) the “ether.” Accordingly he will declare that the “ether” fills all space, but differs in its qualities from “matter.” By “matter” he will mean lines. Having come to this conclusion the two-dimensional being will regard all processes as happening in his “ether,” i.e., in his space.
He will not be in a position to imagine anything outside of this ether, that is, out of his plane. If anything, proceeding out of his plane, comes in contact with his consciousness, then he will either deny it, or regard it as something subjective, the creation of his own imagination; or else he will believe that it is proceeding right on the plane, in the ether, as are all other phenomena.
Sensing lines only, the plane being will not sense them as we do. First of all, he will see no angle. It is extremely easy for us to verify this by experiment. If we will hold before our eyes two matches, inclined one to the other in a horizontal plane, then we shall see one line. To see the angle we shall have to look from above. The two-dimensional being cannot look from above and therefore cannot see the angle. But measuring the distance between the lines of different “solids” of his world, the two-dimensional being will come continually in contact with the angle, and he will regard it as a strange property of the line, which is sometimes manifest and sometimes is not. That is, he will refer the angle to time; he will regard it as a temporary, evanescent phenomenon, a change in the state of a “solid,” or as motion. It is difficult for us to understand this. It is difficult to imagine how the angle can be regarded as motion. But it must be absolutely so, and cannot be otherwise. If we try to represent to ourselves how the plane being studies the square, then certainly we shall find that for the plane being the square will be a moving body. Let us imagine that the plane being is opposite one of the angles of the square. He does not see the angle—before him is a line, but a line possessing very curious properties. Approaching this line, the two-dimensional being observes that a strange thing is happening to the line. One point remains in the same position, and other points are withdrawing back from both sides. We repeat, that the two-dimensional being has no idea of an angle. Apparently the line remains the same as it was, yet something is happening to it, without a doubt. The plane being will say that the line is moving, but so rapidly as to be imperceptible to sight. If the plane being goes away from the angle and follows along a side of the square, then the side will become immobile. When he comes to the angle, he will notice the motion again. After going around the square several times, he will establish the fact of regular, periodical motions of the line. Quite probably in the mind of the plane being the square will assume the form of a body possessing the property of periodical motions, invisible to the eye, but producing definite physical effects (molecular motion)—or it will remain there as a perception of periodical moments of rest and motion in one complex line, and still more probably it will seem to be a rotating body.
Quite possibly the plane being will regard the angle as his own subjective perception, and will doubt whether any objective reality corresponds to this subjective perception. Nevertheless he will reflect that if there is action, yielding to measurement, so must there be the cause of it, consisting in the change of the state of the line, i.e., in motion.
The lines visible to the plane being he may call matter, and the angles—motion. That is, he may call the broken line with an angle, moving matter. And truly to him such a line by reason of its properties will be quite analogous to matter in motion.
If a cube were to rest upon the plane upon which the plane being lives, then this cube will not exist for the two-dimensional being, but only the square face of the cube in contact with the plane will exist for him—as a line, with periodical motions. Correspondingly, all other solids lying outside of his plane., in contact with it, or passing through it, will not exist for the plane being. The planes of contact or cross-sections of these bodies will alone be sensed. But if these planes or sections move or change, then the two-dimensional being will think, indeed, that the cause of the change or motion is in the bodies themselves, i.e., right there on his plane.
As has been said, the two-dimensional being will regard the straight lines only as immobile matter; irregular lines and curves will seem to him as moving. So far as really moving lines are concerned, that is, lines limiting the cross-sections or planes of contact passing through or moving along the plane, these will be for the two-dimensional being something inconceivable and incommensurable. It will be as though there were in them the presence of something independent, depending upon itself only, animated. This effect will proceed from two causes: He can measure the immobile angles and curves, the properties of which the two-dimensional being calls motion, for the reason that they are immobile; moving figures, on the contrary, he cannot measure, because the changes in them will be out of his control. These changes will depend upon the properties of the whole body and its motion, and of that whole body the two-dimensional being will know only one side or section. Not perceiving the existence of this body, and contemplating the motion pertaining to the sides and sections he probably will regard them as living beings. He will affirm that there is something in them which differentiates them from other bodies: vital energy, or even soul. That something will be regarded as inconceivable, and really will be inconceivable to the two-dimensional being, because to him it is the result of an incomprehensible motion of inconceivable solids.
If we imagine an immobile circle upon the plane, then for the two-dimensional being it will appear as a moving line with some very strange and to him inconceivable motions.
The two-dimensional being will never see that motion. Perhaps he will call such motion molecular motion, i.e., the movement of minutest invisible particles of “matter.”
Moreover, a circle rotating around an axis passing through its center, for the two-dimensional being will differ in some inconceivable way from the immobile circle. Both will appear to be moving, but moving differently.
For the two-dimensional being a circle or a square, rotating around its centre, on, account of its double motion will be an inexplicable and incommensurable phenomenon, like a phenomenon of life for a modern physicist.
Therefore, for a two-dimensional being, a straight line will be immobile matter; a broken or a curved line—matter in motion; and a moving line—living matter.
The centre of a circle or a square will be inaccessible to the plane being, just as the centre of a sphere or of a cube made of solid matter is inaccessible to us—and for the two-dimensional being even the idea of a centre will be incomprehensible, since he possesses no idea of a centre.
Having no idea of phenomena proceeding outside of the plane—that is, out of his “space”—the plane being will think of all phenomena as proceeding on his plane as has been stated. And all phenomena which he regards as proceeding on his plane, he will consider as being in causal interdependence one with another: that is, he will think that one phenomenon is the effect of another which has happened right there, and the cause of a third which will happen right on the same plane.
If a multi-colored cube passes through the plane, the plane being will perceive the entire cube and its motion as a change in color of lines lying in the plane. Thus, if a blue line replaces a red one, then the plane being will regard the red line as a past event. He will not be in a position to realize the idea that the red line is still existing somewhere. He will say that the line is single, but that it becomes blue as a consequence of certain causes of a physical character. If the cube moves backward so that the red line appears again after the blue one, then for the two-dimensional being this will constitute a new phenomenon. He will say that the line became red again.
For the being living on a plane, everything above and below (if the plane be horizontal), and on the right or left (if the plane be vertical) will be existing in time, in the past and in the future: that which in reality is located outside of the plane will be regarded as non-existent, either as that which is already past, i.e., as something which has disappeared, ceased to be, will never return; or as in the future, i.e., as not existent, not manifested, as a thing in potentiality.
Let us imagine that a wheel with the spokes painted different colors is rotating through the plane upon which the plane being lives. To such a being all the motion of the wheel will appear as a variation of the color of the line of intersection of the wheel and the plane. The plane being will call this variation of the color of the line a phenomenon, and observing these phenomena he will notice in them a certain succession. He will know that the black line is followed by the white one, the white by the blue, the blue by the red, and so on. If simultaneously with the appearance of the white line some other phenomenon occurs—say the ringing of a bell—the two-dimensional being will say that the white line is the cause of that ringing. The change of the color of the lines, in the opinion of the two-dimensional being, will depend on causes lying right in his plane. Any pre-supposition of the possibility of the existence of causes lying outside of the plane he will characterize as fantastic and entirely unscientific. It will seem so to him because he will never be in a position to represent the wheel to himself, i.e., the parts of the wheel on both sides of the plane. After a rough study of the color of the lines, and knowing the order of their sequence, the plane being, perceiving one of them, say the blue one, will think that the black and the white ones have already passed, i.e., disappeared, ceased to exist, gone into the past; and that those lines which have not as yet appeared—the yellow, the green, and so on, and the new white and black ones still to come—do not yet exist, but lie in the future.
Therefore, though not conceiving the form of his universe, and regarding it as infinite in all directions, the plane being will nevertheless involuntarily think of the past as situated somewhere at one side of all, and of the future as somewhere at the other side of this totality. In such manner will the plane being conceive of the idea of time. We see that this idea arises because the two-dimensional being senses only two out of three dimensions of space; the third dimension he senses only after its effects become manifest upon the plane, and therefore he regards it as something different from the first two dimensions of space, calling it time.
Now let us imagine that through the plane upon which the two-dimensional being lives, two wheels with multi-colored spokes are rotating and are rotating in opposite directions. The spokes of one wheel come from above and go below; the spokes of the other come from below and go above.
The plane being will never notice it.
He will never notice that where for one line (which he sees) there lies the past, for another line there lies the future. This thought will never even come into his head, because he will conceive of the past and the future very confusedly, regarding them as concepts, not as actual facts. But at the same time he will be firmly convinced that the past goes in one direction, and the future in another. Therefore it will seem to him a wild absurdity that on one side something past and something future can lie together, and on another side—and also beside these two—something future and something past. To the plane being the idea that some phenomena come whence others go, and vice versa, will seem equally absurd. He will tenaciously think that the future is that wherefrom everything comes, and the past is that whereto everything goes and wherefrom nothing returns. He will be totally unable to understand that events may arise from the past just as they do from the future.
Thus we see that the plane being will regard the changes of color of the lines lying on the plane very naively. The appearance of different spokes he will regard as the change of color of one and the same line, and the repeated appearance of the same colored spoke he will regard every time as a new appearance of a given color.
But nevertheless, having noticed periodicity in the change of the color of the lines upon the surface, having remembered the order of their appearance, and having learned to define the “time” of the appearance of certain spokes in relation to some other more constant phenomenon, the plane being will be in a position to foretell the change of the line from one color to another. Thereupon he will say that he has studied this phenomenon, that he can apply to it “the mathematical method”—can “calculate” it.
If we ourselves enter the world of plane beings, then its inhabitants will sense the lines limiting the sections of our bodies. These sections will be for them living beings; they will not know from whence they appear, why they alter, or whither they disappear in such a miraculous manner. So also, the sections of all our inanimate but moving objects will seem independent living beings.
If the consciousness of a plane being should suspect our existence, and should come into some sort of communion with our consciousness, then to him we would appear as higher, omniscient, possibly omnipotent, but above all incomprehensible beings of a quite inconceivable category.
We could see his world just as it is, and not as it seems to him. We could see the past and the future; could foretell, direct, and even create events.
We could know the very substance of things—could know what “matter” (the straight line) is, what “motion” (the broken line, the curve, the angle) is. We could see an angle, and we could see a centre. All this would give us an enormous advantage over the two-dimensional being.
In all the phenomena of the world of the two-dimensional being we could see considerably more than he sees—or could see quite other things than he.
And we could tell him very much that was new, amazing, and unexpected about the phenomena of his world, provided indeed that be could hear us and understand us.
First of all we could tell him that what he regards as phenomena—angles and curves, for instance—are properties of higher figures; that other “phenomena” of his world are not phenomena, but only “parts” or “sections” of phenomena; that what he calls “solids” are only sections of solids—and many things besides.
We would be able to tell him that on both sides of his plane (i.e., of his space or ether) lies infinite space (which the plane being calls time); and that in this space lie the causes of all his phenomena, and the phenomena themselves, the past as well as the future ones; moreover, we might add that “phenomena” themselves are not something happening and then ceasing to be, but combinations of properties of higher solids.
But we should experience considerable difficulty in explaining anything to the plane being; and it would be very difficult for him to understand us. First of all it would be difficult because he would not have the concepts corresponding to our concepts. He would lack “necessary words.”
For instance, “section”—this would be for him a quite new and inconceivable word; then “angle”—again an inconceivable word; “centre”—still more inconceivable; the third perpendicular—something incomprehensible, lying outside of his geometry.
The fallacy of his conception of time would be the most difficult thing for the plane being to understand. He could never understand that that which has passed and that which is to be are existing simultaneously on the lines perpendicular to his plane. And he could never conceive the idea that the past is identical with the future, because phenomena come from both sides and go in both directions.
But the most difficult thing for the plane being would be to conceive the idea that “time” includes in itself two ideas: the idea of space, and the idea of motion upon this space.
We have shown that what the two-dimensional being living on the plane calls motion has for us a quite different aspect.
In his book The Fourth Dimension, under the heading “The First Chapter in the History of Four-space,” Hinton writes:
Parmenides, and the Asiatic thinkers with whom he is in close affinity, propound a theory of existence which is in close accord with a conception of a possible relation between a higher and lower dimensional space. . . . It is one which in all ages has had a strong attraction for pure intellect, and is the natural mode of thought for those who refrain from projecting their own volition into nature under the guise of causality.
According to Parmenides of the school of Elea the all is one, unmoving and unchanging. The permanent amid the transient—that foothold for thought, that solid ground for feeling, on the discovery of which depends all our life—is no phantom; it is the image amidst deception of true being, the eternal, the unmoved, the one. Thus says Parmenides.
But how is it possible to explain the shifting scene, these mutations of things? “Illusion,” answers Parmenides. Distinguishing between truth and error, he tells of the true doctrine of the one—the false opinion of a changing world. He is no less memorable for the manner of his advocacy than for the cause he advocates.
Can the mind conceive a more delightful intellectual picture than that of Parmenides pointing to the one, the true, the unchanging, and yet on the other hand ready to discuss all manner of false opinion! . . .
In support of the true opinion he proceeded by the negative way of showing the self-contradictions in the ideas of change and motion. . . . To express his doctrine in the ponderous modern way we must make the statement that motion is phenomenal, not real.
Let us represent his doctrine.
Imagine a sheet of still water into which a slanting stick is being lowered with a motion vertically downward. Let 1, 2, 3 (Fig.1) be three consecutive positions of the stick. A, B, C will be three connective positions of the meeting of the stick with the surface of the water. As the stick passes down, the meeting will move from A on to B and C.
Suppose now all the water to be removed except a film. At the meeting of the film and the stick there will be an interruption of the film. If we suppose the film to have a property, like that of a soap bubble, of closing up round any penetrating object, then as the stick goes vertically down-ward the interruption in the film will move on. If we pass a spiral through the film the intersection will give a point moving in a circle (shown by the dotted lines in Fig.2). For the plane being such a point, moving in a circle in its plane, would probably constitute a cosmic phenomenon, something like the motion of a planet in its orbit.
Suppose now the spiral to be still and the film to move vertically upward, the whole spiral will be represented in the film in the consecutive positions of the point of intersection.
If instead of one spiral we take a complicated construction consisting of spirals, inclined and straight lines, broken and curved lines, and if the film move vertically upward we shall have an entire universe of moving points the movements of which will appear to the plane being as original.
The plane being will explain these movements as depending one upon another, and indeed he will never happen to think that these movements are fictitious and are dependent upon the spirals and other lines lying outside his space.
Returning to the plane being and his perception of the world, and analyzing his relations to the three-dimensional world, we see that for the two-dimensional or plane being it will be very difficult to understand all the complexity of the phenomena of our world, as it appears to us. He (the plane being) is accustomed to perceive the world as being too simple.
Taking into consideration the sections of figures instead of the figures themselves, the plane being will compare them in relation to their length and their greater or lesser curvature, i.e., their for him more or less rapid motion.
The differences between the objects of our world, as they exist for us he would not understand. The functions of the objects of our world would be completely mysterious to his mind incomprehensible, “supernatural.”
Let us imagine that a coin, and a candle the diameter of which is equal to that of the coin, are on the plane upon which the two-dimensional being lives. To the plane being they will seem two equal circles, i.e., two moving, and absolutely identical lines; he will never discover any difference between them. The functions of the coin and of the candle in our world—these are for him absolutely a terra incognita. If we try to imagine what an enormous evolution the plane being must pass through in order to understand the function of the coin and of the candle and the difference between these functions, we shall understand the nature of the division between the plane world and the world of three dimensions, and the complete impossibility of even imagining, on the plane, anything at all like the three-dimensional world, with its manifoldness of function.
The properties of the phenomena of the plane world will be extremely monotonous; they will differ by the order of their appearance, their duration, and their periodicity. Solids, and the things of this world will be flat and uniform, like shadows, i.e., like the shadows of quite different solids, which seem to us uniform. Even if the plane being could come in contact with our consciousness, he would never be in a position to understand all the manifoldness and richness the phenomena of our world and the variety of function of the things of that world.
Plane beings would not be in a position to master our most ordinary concepts.
It would be extremely difficult for them to understand that phenomena, identical for them, are in reality different; and on the other hand, that phenomena quite separate for them are in reality parts of one great phenomenon, and even of one object or one being.
This last will be one of the most difficult things for the plane being to understand. If we imagine our plane to inhabit a horizontal plane, intersecting the top of a tree, and parallel to the surface of the earth, then for such a being each of the various sections of the branches will appear as a quite separate phenomenon or object. The idea of the tree and its branches will never occur to him.
Generally speaking, the understanding of the most fundamental and simple things of our world will be infinitely long and difficult to the plane being. He would have to entirely reconstruct his concepts of space and time. This would be the first step. Unless it is taken, nothing is accomplished. Until the plane being shall imagine our entire universe as existing in time, i.e., until he refers to time everything lying on both sides of his plane, he will never understand anything. In order to begin to understand “the third dimension” the inhabitant of the plane must conceive of his time concepts spatially, that is, translate his time into space.
To achieve even the spark of a true understanding of our world he will have to reconstruct completely all his ideas—to revaluate all values, to revise all concepts, to dissever the uniting concepts, to unite those which are dissevered; and, what is most important, to create an infinite number of new ones.
If we put down the five fingers of one hand on the plane of the two-dimensional being they will be for him five separate phenomena.
Let us try to imagine what an enormous mental evolution he would have to undergo in order to understand that these five separate phenomena on his plane are the finger-tips of the hand of a large, active and intelligent being—man.
To make out, step by step, how the plane being would attain to an understanding of our world, lying in the region of the to him mysterious third dimension—i.e., partly in the past, partly in the future—would be interesting in the highest degree. First of all, in order to understand the world of three dimensions, he must cease to be two-dimensional—he must become three-dimensional himself, or in other words he must feel an interest in the life of three-dimensional space.
After having felt the interest of this life, he will by so doing transcend his plane, and will never be in a position thereafter to return to it. Entering more and more within the circle of ideas and concepts which were entirely incomprehensible to him before, he will have already become, not two-dimensional, but three-dimensional. But all along the plane being will have been essentially three-dimensional, that is, he will have had the third dimension, without his being conscious of it himself. To become three-dimensional he must be three-dimensional. Then as the end of ends he can address himself to the self-liberation from the illusion of the two-dimensionality of himself and the world, and to the apprehension of the three-dimensional world.
The Problem of Higher Dimensions is an extract from Tertium Organum: The Third Canon of Thought by by P. D. Ouspensky