Methods of investigating the problem of higher dimensions. Analogy between imaginary worlds of different dimensions. One-dimensional world on a line. 'Space' and 'time' of a one-dimensional being. Two-dimensional world on a plane. 'Space' and 'time', 'ether', 'matter' and 'motion' of a two-dimensional being. Reality and illusion on a plane. Impossibility of seeing an 'angle'. An 'angle' as motion. Incomprehensibility, for a two-dimensional being, of the functions of the objects of our world. Phenomena and noumena of a two-dimensional being. How could a plane being understand the third dimension?
In order to determine what the domain of higher dimensions could be and what it could not be, a series of analogies and comparisons are generally used.
The usual way is to imagine 'worlds' of one and two dimensions and, from the relationship between the lower worlds and the higher worlds to deduce the possible relation of our world to the four-dimensional world in the same way as from the relations of points to line, of lines to surfaces, of surfaces to solids, we deduce the relationship of our solids to four-dimensional bodies.
Let us examine all that this method of analogies has to offer.
Let us imagine a one-dimensional world.
It will be a line. On this line let us imagine living beings. They will only be able to move backwards and forwards along this line which represents their universe, and they themselves will have the aspect of points or sections of the line. Nothing outside this line will exist for them, neither will they be conscious of the line itself on which they live and move. Only two points will exist for them -ahead and behind; or maybe only one point, ahead. Observing changes in the state of these points the one-dimensional being will call these changes phenomena. If we suppose that the line on which the onedimensional being lives, passes through various objects of our world, then, in all these objects the one-dimensional being will see only one point. If his line is intersected by different bodies, 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 be a phenomenon. For the one-dimensional being phenomena will be constant or variable, of long or short duration, periodical or not periodical, according to the character and qualities and the rate and nature of the motion of objects passing through the line. But the one-dimensional being will be totally unable to explain the constancy or variability, the long or short duration, the periodicity or non-periodicity of the phenomena of his world, and will simply regard these as attributes inherent in the phenomena. Bodies intersecting the line may be very different, but for the one-dimensional being all phenomena will be absolutely identical -only the appearance and disappearance of a point - and all phenomena will differ from one another only in duration and greater or lesser periodicity.
This curious monotony and homogeneity of phenomena, which, from our point of view, are so diverse and heterogeneous, will be the characteristic peculiarity of the onedimensional world.
Then, if we suppose that the one-dimensional being possesses memory, we shall see that, calling all the points he has seen phenomena, he will refer them all to time. The point which was is a phenomenon no longer existing, and the point which may appear tomorrow is a phenomenon not yet existing. The whole of our space, with the exception of one line, will be called time, i.e. something whence phenomena come and whither they go. And the one-dimensional being will say that he got the idea of time from the observation of motion, i.e. from the appearance and disappearance of poults. Points will be regarded as time-phenomena, i.e. as phenomena coming into being at that moment when they become visible, and disappearing -ceasing to exist -at that moment when they become invisible. It is impossible for a one-dimensional being to imagine that a phenomenon can exist somewhere and yet be invisible; or he will imagine it as existing somewhere on his line, far ahead of him.
We can imagine this one-dimensional being still more realistically. Let us take an atom floating in space, or simply a speck of dust driven by the wind, and let us suppose that this atom or speck of dust possesses consciousness, i.e. that it differentiates between itself and the surrounding world and is conscious of that which lies on the line of its motion, that with which it comes into direct contact. This will be a onedimensional being in the full sense of the word. He may move and fly in all directions, but it will always seem to him that he moves on one line; outside this line only a vast Nothing will exist for him -the whole universe will appear to him as one line. He will neither feel nor represent to himself any of the turnings of his line, that is, none of the angles, because to feel an angle, one must be aware of what lies to the right and the left, or above and below. In all other respects this being will be absolutely identical with the imaginary being living on the imaginary line I have just described. Everything he comes into contact with, i.e. everything he is conscious of, will seem to him to be emerging from time, i.e. out of nothing, and vanishing into time, i.e. into nothing. This nothing will be all our world. Apart from one line, the whole of our world will be called time and will be regarded as having no real existence.
Now let us consider the two-dimensional world and a being living on a plane. For this being the universe will be one vast plane. On this plane let us imagine beings in the shape of points, lines and flat geometrical figures. The objects and 'bodies' of this world will also have the shape of flat geometrical figures. How will a being living on this plane universe perceive his world? We can say, first of all, that he will not sense the plane on which he lives. He will sense the objects, i.e. the figures lying on this plane; he will sense the lines which bound them, and for that very reason he will not sense his own plane because if he did, he would be unable to distinguish these lines. The lines will differ from the plane by the fact that they produce sensations, consequently they exist. The plane does not produce sensations; consequently it does not exist. Moving along the plane and not experiencing any sensations, the two-dimensional being will say that at the moment there is nothing there. Approaching some figure and getting the sensation of its lines, he will say that something has appeared. But gradually, through reasoning, the two-dimensional being will come to the conclusion that the figures he meets with exist on something or in something. So he may call this plane - 'ether' (of course, he will not know that it is actually a plane). Then he will say that 'ether' fills all space, but differs in its properties from 'matter'. So he will call lines -'matter'. As a result, the two-dimensional being will regard everything that happens as happening in his 'ether', that is, in his space. He will not be able to imagine anything as being outside this ether, i.e. outside his plane. If something happening outside his plane reaches his consciousness, he will either deny it, taking it as subjective, i.e. as a creation of his own imagination, or he will think of it as he thinks of all other phenomena, as happening on that very plane, in ether.
Sensing the lines only, the plane being will sense them quite differently from us. First of all, he will not sense an angle. It is very easy to verify this in practice. If we hold on a level with our eyes two matches placed on a horizontal surface at an angle to one another, we will see one line. To see the angle we must look from above. The two-dimensional being cannot look from above, and therefore cannot see an angle. But by measuring the distance between the lines of the different 'solids' of his world, the two-dimensional being will be constantly confronted with angles and will regard the angle as a strange property of the line which at times appears and at others does not appear. In other words, he will refer the angle to time, will regard it as a transitory temporal phenomenon - a change in the state of the 'solid' - or as motion. It is difficult for us to understand this, difficult to imagine how an angle can be taken as motion. But it must necessarily be so and cannot be otherwise. If we try to visualize how a plane being will study a square, we shall see that for a plane being the square must necessarily be a moving body. Let us imagine a plane being faced with one of the angles of the square. He does not see the angle -in front of him there is a line, but a line possessing very strange properties. As he comes nearer to this line, the twodimensional being will see a strange thing happening to the line. One pointwill remain in its place, but the other points, on both sides, will recede backwards. I repeat: the two-dimensional being has no idea of an angle. In its outward appearance the line will remain the same as it was; and yet, something will undoubtedly be happening to it. The plane being will say that the line moves, but so rapidly that it appears to be motionless. If the plane being draws away from the angle and moves along a side of the square, this line will become motionless. Reaching an angle, he will again notice motion. If he makes the circuit of the square several times, he will establish the fact that there are regular periodical movements of this line. It is probable that for the mind of the plane being, the square will be his conception of a body possessing the property of periodical movements, unnoticeable to the eye but producing definite physical effects (molecular motion), or the idea of periodical moments of rest and motion in one complex line; and still more probably the square will appear to him as a rotating body.
Very likely, the plane being will regard the angle as his own subjective representation and will doubt whether any objective reality corresponds to this subjective representation. But all the same, he will think that so long as an action capable of being measured exists, it must have a cause, and this cause must lie in the changing states of the line, i.e. in motion.
The plane being may call the lines he sees -matter, and the angles motion. Thus, the plane being will call an irregular line with an angle -moving matter. And indeed for him, because of its properties, such a line will be completely analogous to matter in motion.
If a cube is placed on the plane on which the plane being lives, the whole cube will not exist for the two-dimensional being, but only the square surface of it which is in contact with the plane, that is to say, the cube will exist as a line with periodical movements. In the same way, all other bodies lying outside his plane, touching his plane or passing through it, will not exist for the two-dimensional being. He will be able to sense only their surfaces of contact or their sections. But if these surfaces or sections move or change, quite naturally, the two-dimensional being will think that the cause of change or motion lies in themselves, i.e. is also there, on his plane.
It has already been said that the two-dimensional being will regard only straight lines as motionless matter, irregular lines or curves will appear to him to be moving. As regards the really moving lines, i.e. those lines which bind the sections or the surfaces of contact of the bodies moving through the plane or along the plane, these will contain something incomprehensible for a twodimensional being, something impossible to measure. They will seem to have in them something self-existing, self-dependent, animated. There are two reasons for this: the two-dimensional being can measure motionless angles and curves, whose properties he calls motion, for the very reason that they are motionless; but he cannot measure moving figures because the changes in them are outside his control. These changes will depend on the properties of the whole body and its motion, whereas the two-dimensional being knows only its section, only one side of the whole body. Having no idea of the existence of that body and regarding its motion as inherent in the sides and sections, he will probably regard them as living beings. He will credit them with the possession of something which is absent in ordinary bodies -vital energy, or even soul. This something will be regarded as unknowable for a two-dimensional being, since it is the result of an incomprehensible motion of incomprehensible bodies.
If we imagine a stationary circle lying on the plane, for a two-dimensional being this circle will appear as a moving line, possessing very strange and incomprehensible motion.
The plane being will never see this motion. He may possibly call it molecular motion, i.e. the movement of minute, invisible particles of 'matter'.
For a two-dimensional being, a circle rotating round a central axis will, in some incomprehensible way, appear different from a stationary circle. Both will seem to be moving, but moving differently.
Owing to its double movement, a circle or a square lying on the plane and rotating round its centre, will be, for a two-dimensional being, anincomprehensible and unmeasurable phenomenon, somewhat similar to the phenomenon of life for the modern physicist.
Thus, for a two-dimensional being, a straight line will be motionless matter; an irregular line or a curve will be matter in motion; and a movingline will be 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 a cube made of solid matter is inaccessible to us. Moreover, the two-dimensional being will be incapable of even understanding about a centre, since he will have no idea of what a centre means.
It has already been said that, having no conception of any phenomena occurring outside the plane, i.e. outside his space, the plane being will regardall phenomena as taking place on his plane. And all these phenomena, supposedly taking place on his plane, he will regard as being in causal interdependence one with another, that is, he will think that one phenomenon is the effect of another which has also taken place there -on his plane -and the cause of a third which will take place there also.
If a multi-coloured cube passes through the plane, the whole cube and its motion will be perceived by the plane being as changes in the colour of the lines lying on the surface. So, if a blue line replaces a red one, the plane beingwill regard the red line as a past event. He will be unable to conceive of the red line still existing somewhere. He will say that the line is the same but that it has become blue owing to certain causes of a physical nature. If the cube starts moving backwards and the red line again replaces the blue line, it will be a new phenomenon for the plane being. He will say that the line has become red again.
Everything situated above and below, if the plane is horizontal, or to the right and left if the plane is vertical, will lie in time for a being living on that plane, that is, it will be in the past and the future. Everything that exists in reality outside the plane will be regarded as non-existent: either as already in the past, i.e. as something that has vanished, ceased to be, something that will never return; or in the future, i.e. as something not yet existing, not manifested but merely potential.
Let us imagine a wheel with multi-coloured spokes rotating through the plane on which lives a two-dimensional being. The movement of the spokes will appear to a two-dimensional being as changes in the colour of a line lying onthe surface. The plane being will call these changes phenomena and, observing these phenomena, he will notice a certain sequence in them. He will know that the black line is followed by a white one, the white by a blue, the blue by a pink. If something else is connected with the appearance of the white line - the ringing of a bell for instance -the two-dimensional being will say that the white line is the cause of the ringing. The changing colour of the lines will, in the opinion of the two-dimensional being, depend on some causes to be found there, on his plane. Any conjecture as to the possible existence of causes lying outside the plane he will dismiss as utterly fantastic and absolutely unscientific. And this will be so because he himself will never be able to visualize the wheel, i.e. the different parts of the wheel on each side of the plane. Having studied the changes in the colour of the lines and learnt their order, the plane being, on seeing one of them -say, the blue one -will think that the black and the white have already passed, i.e. have vanished, have ceased to exist, have receded into the past; whereas the lines which have not yet appeared -the yellow, the green and so on, and among them the new white and the new black which are to come - do not yet exist but lie in the future.
Thus, although not conscious of the form of his universe and regarding it as infinite in all directions, the plane being will involuntarily think of the past as lying somewhere on one side of everything, and of the future as lying somewhere on the other side of everything. This is how the two-dimensional being arrives at the idea of time. We see that this idea arises from the fact that, out of three dimensions of space, the twodimensional being is aware of only two; the third dimension he senses only through its effects on the plane; therefore he regards it as something distinct from the first two dimensions of space, and calls it time.
Now let us imagine two wheels with multi-coloured spokes rotating through the plane on which the two-dimensional being lives, and rotating in opposite directions. The spokes of one of them come from above and go below; the spokes of the other come from below and go above.
The plane being will never notice this.
He will never notice that in the direction in which for one line, visible to him, lies the past, for the other line lies the future. This thought will never even occur to him, because he will have a very nebulous idea of both the past and the future, and will regard them only as concepts, and not as concrete facts. At the same time he will be firmly convinced that the past proceeds in one direction and the future in another. For him it will seem a wild absurdity that on one side something past and something future may lie together, and on another side, also together, something future and something past. No less absurd will be the idea that some phenomena appear from where others disappear and vice versa. He will persist in thinking that the future is that from which everything comes and the past is that to which everything goes, and from which nothing returns. The plane being will be incapable of understanding that phenomena may proceed from the past as well as from the future.
Thus we see that the plane being will have a very naive view of the changing colour of the line lying on the surface. The appearance of different spokes he will regard as changes in the colour of one and the same line, and, for him, the recurring appearance of a spoke of the same colour will be, each time, a new appearance of the given colour.
Yet, having noticed a certain periodicity in the changes of the colour of the lines on the surface, having memorized the order of their appearance and learned to determine the 'time' of the appearance of certain spokes in relation to some other more permanent phenomenon, the plane being will be able to foretell the change of the line from one colour to another.
Then he will say that he has studied this phenomenon, i.e. that he can apply to it the 'mathematical method' - can 'calculate it'.
If we enter the world of the plane being, he will sense only the lines bounding the sections of our bodies. These sections, which will be livingbeings for him, will appear from nowhere, change for no apparent reason, and disappear somewhere in a miraculous manner. The sections of all our inanimate but moving objects will also be independent living beings for him.
If the consciousness of a plane being could have the faintest suspicion of our existence or enter into any kind of communication with our consciousness, we would be for him higher, omniscient, maybe omnipotent and, above all, unknowable beings of a totally incomprehensible category.
We would see his world as it is and not as it appears to him. We would see the past and the future; we would be able to foretell, direct and even create events.
We would know the essence of things. We would know what 'matter' (a straight line) is, what 'motion' (a curve, an irregular line, an angle) is. We would see the angle and see the centre. And this would give us an enormous advantage over a two-dimensional being.
In all the phenomena of the two-dimensional world we would see much more than the plane being does, or would see something quite different from what he sees.
We would be able to tell him many new, unexpected and striking things about the phenomena of his world - if he could hear and understand us.
First of all, we would be able to tell him that what he regards as phenomena, such as angles and curves, are the properties of higher bodies; that other 'phenomena' of his world are not phenomena at all but only parts or 'sections' of phenomena; that what he calls 'bodies' are only sections of bodies -and many other things besides.
We could tell him that on both sides of his plane (i.e. his space or his ether) there lies an infinite space (which the plane being calls time), and in that space lie not only the causes of all his 'phenomena' but the phenomena themselves, either of the past or the future. And we could add that a 'phenomenon' is not just something that happens and then ceases to be, but is a combination of the properties of higher bodies.
Nevertheless, we would find it very difficult to explain anything to a plane being, and he would find it very difficult to understand us. Above all, it would be difficult because he would have no concepts corresponding to our concepts. The necessary 'words' would be lacking.
For instance, section would be a completely new and incomprehensible word for him. Then, angle -again an incomprehensible word. Centre -still more incomprehensible. The third perpendicular -something unfathomable, lying outside his geometry.
The most difficult thing for the plane being to understand would be the error of his idea of time. He would never be able to imagine that what has passed and what is to come exist simultaneously on lines at right angles to his plane. He could never understand that the past is identical with the future, since phenomena can both come and go from either side.
But the most difficult thing of all for the plane being to understand would be that 'time' contains two ideas: the idea of space and the idea of motion in this space.

We have pointed out already that that which a two-dimensional being living on a plane calls motion, would bear quite a different aspect for us.
In his book, The Fourth Dimension, under the title '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 a lower dimensional space. This theory ... 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 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 downwards. Let 1, 2, 3 (Figure 1), be three consecutive positions of the stick. A, B, C, will be three consecutive 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 downwards 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 the figure (Figure 2).*
For the plane being such a point, moving in a circle on its surface will probably be a cosmic phenomenon in the nature of the motion of a planet in its orbit.
Suppose now the spiral to be still and the film to move vertically upwards, the circular motion of the point will continue until this motion stops.
If instead of one spiral we take a complicated structure of spirals, inclined lines, straight lines, irregular lines and curves, then, with

* C. H. Hinton, The Fourth Dimension, London, 1912, reprinted Arno Press, New York, 1976, pp. 23, 24, 25.

Relation of two- to three-dimensional perception

the movement of the film upwards, we shall have in the film a whole world of moving points, whose movements will appear independent to the plane being.
The plane being will naturally explain these movements as dependent upon one another, and the fictitious nature of this movement and its dependence on spirals and other lines lying outside his space will never occur to him.
If we examine the relationship of the plane being to the three-dimensional world we shall see that the two-dimensional plane being would find it very difficult to understand all the complexity of the phenomena of our world, as it appears to us. The plane being is accustomed to represent to himself too simple a world.
Taking sections of bodies for bodies, the plane being would compare them only as regards their length and their greater or lesser curvature, i.e. for him their greater or lesser speed of motion. Such differences as exist for us between the things of our world, could not exist for him. The functions of the objects of our world would be utterly beyond his understanding; they would be incomprehensible, 'supernatural'.
Imagine a coin and a candle, both of the same diameter, placed on the plane on which the two-dimensional being lives. For the plane being these would be two equal circles, i.e. two moving lines, absolutely identical; he would never discover any difference between them. The functions which the coin and the candle have in our world would be for him entirely terra incognita. If we try to imagine what a tremendous evolution the plane being would have to undergo in order to understand the functions of the coin and the candle and the difference between these functions, we should understand what it is that divides the plane world from the three-dimensional world.
Before anything else, they are divided because of the utter impossibility -on a plane - of even imagining anything like the three-dimensional world with all the variety of its functions.
The properties of the phenomena of the plane world will be extremely monotonous; phenomena will be distinguished by the order of their appearance, their duration, their periodicity. Bodies and objects of this world will be flat and uniform, like shadows, i.e. like the shadows of completelydifferent objects, which seem to us alike. Even if the consciousness of a plane being could enter into communication with our consciousness, he would still be unable ever to understand all the variety and richness of phenomena of our world and the variety of functions of our objects.
Plane beings would be unable to grasp any of our most ordinary concepts.
It would be very difficult for them to understand that phenomena which are the same for them are actually different and that, on the other hand, phenomena which are quite separate for them are actually parts of one bigphenomenon, or even parts of one object or one being.
This last would be one of the most difficult things for the plane being to understand. If we suppose our two-dimensional being to live on a horizontal plane, intersecting the top of a tree, but parallel to the earth, then for him the sections of branches will appear each as a completely independentphenomenon or object. The idea of a tree with its branches can never even occur to him.
Altogether, to understand even the most fundamental and simple things of our world will be, for the plane being, an infinitely long and difficult process. He will have to remodel his ideas of space and time. This must be the first step. Nothing can be achieved until this is done. So long as the plane beingvisualizes all our universe in time, i.e. refers to time everything that lies on both sides of his plane, he will never understand anything. In order to begin to understand the 'third dimension', the two-dimensional being living on the plane must visualize all his time-concepts spatially, i.e. translate his time into space.
To achieve even an inkling of a right conception of our world, he must completely reconstruct all his ideas of the world -revalue all his values, reexamine all his concepts; he must disunite all those concepts which unify and bring together those which disconnect and, above all, he must create an infinite number of new concepts.
If we place five fingertips on the plane of the two-dimensional being, this will represent for him five separate phenomena.
Let us try to imagine the enormous mental evolution the plane being must undergo to understand that the five separate phenomena on his plane are the fingertips of the hand of a large, active and intelligent being -man.
It would be extremely interesting to follow, step by step, the road the plane being must travel to come to the understanding of our world which, for him, lies in the region of the mysterious third dimension, i.e. partly in the past, partly in the future. In order to comprehend the three-dimensional world, the plane being must, first of all, cease to be two-dimensional, i.e. he must himself become three-dimensional; in other words, he must enter into the life interests of a three-dimensional space. If he feels the interests of that life he will, by this very fact, draw away from his own plane and will never be able to return there. Entering more and more into the orbit of ideas and concepts which previously were totally incomprehensible for him, he will no longer be a twodimensional being, but will become a three-dimensional one. But for this the plane being must really be three-dimensional, i.e. without being aware of it, he must possess a third dimension. A really two-dimensional being will never become threedimensional. In order to become three-dimensional, he must be three-dimensional. Then, in the end, he will be able to get free from the illusion of the two-dimensionality of the world and of himself, and feel the three-dimensional world.

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