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THE MEANING OF FOUR-DIMENSIONAL SPACE.

The main line of thought developed in these pages has no claims to originality. Professor Zöllner of Leipsic was an ardent exponent of the theory in the "seventies" and some authors hold that even the ancient writings of the East contain attempts to express Four-Dimensional concepts.

Whether this is actually so is open to doubt but it must be remembered that in the days when these writings were produced mathematical knowledge was itself in its infancy and that there was, therefore, no terminology available in which the Higher Space concepts could be suitably expressed even supposing that the ancient philosophers had them in mind.

It is only through accumulated knowledge, especially the work of Gauss, Lobatschewsky, Bolyai, Riemann, and others that modern mathematicians are able to deal easily with space of more than three dimensions.

It may be noted that Kant says:

"If it be possible that there are developments of other dimensions of space, it is very probable that God has somewhere produced them. For His works have all the grandeur and glory that can be comprised."

According to Mr. G.R.S. Mead similar ideas are to be found in certain of the Gnostic cosmogonies.

(Fragments of a Faith forgotten, p. 318.)

But a detailed historical review would be out of place here and I will therefore proceed at once to a discussion of what is meant by the term "fourth dimension" and will try to explain how it is that we can determine some of the necessary properties of four-dimensional space, even although we cannot picture it to ourselves.

At this point I would urge the reader to try to believe that the subject is not one of great difficulty. As a matter of fact it is really exceptionally straightforward if only one faces it and does not allow oneself to be frightened.

I know that it is impossible to form any clear mental picture of four-dimensional conditions, but that does not matter. The ideas involved are admittedly unprecedented in our experience, but they are not contrary to reason and I do not ask more than a formal and intellectual assent to the propositions and analogies concerned.

Let me start, then, by defining what is meant by a Dimension. The best definition I can think of is to say that, in the sense in which the word is used here, a Dimension means "An independent direction in space."

I must amplify this by saying that, "Two directions in space are to be considered as independent when they are so related that no movement, however great, along one of them will result in the slightest movement along, or parallel to, the other. That is to say, at right angles, or perpendicular to one another."

Fig. 1

Thus in Fig. 1 AOA´ and BOB´ are independent directions. One might move for ever along OA or OA´ and yet one would not have moved in the very least in the direction of OB or of OB´.

Now on a flat surface, such as a sheet of paper, it is not possible to draw more than two such directions. Any other line that can be drawn, XOX´ for instance, is in a compound direction, so to speak. That is to say it is partly in the direction AOA´ and partly in the direction BOB´ and it is possible to reach any point in it, Y for example, by moving along OA´ to a and then moving in the direction of OB´ a distance equal to Ob, or vice versa or by doing the two simultaneously.

For the benefit of those who are absolutely ignorant of the rudiments of Geometrical knowledge, I would point out that Parallel lines are said to point, in fact do point, in the same direction.

Fig. 2

Thus, in Fig. 2, the direction of the line ZZ´ is the same as that of AOA´ and the direction of the line PP´ is the same as that of XOX´.

Thus we see that in a flat surface we find only two dimensions and consequently we can refer to a flat surface as "Space of two dimensions" or "Two-dimensional space."

But if we refuse to be restricted to a flat surface we find that it is possible to draw a third line through O which is quite "independent" of the directions of the two lines we have previously drawn. We can do this by drawing it vertically, that is to say, perpendicular to the plane of the paper. Call this line COC´.

Fig. 3

I have shown it in perspective in Fig. 3. This line fulfils the definition we gave of an independent direction in space for it is at right angles both to AOA´ and to BOB´. But we have now exhausted our resources. Try as we will we are unable to draw a fourth line which shall be at right angles to AOA´, BOB´, and COC´ simultaneously.

On other words—In the space we know we find only three dimensions and consequently we can refer to it as "Space of three dimensions" or "Three-dimensional space."

Now the idea of a fourth dimension of space is simply this: That, whereas in three-dimensional space, we can draw, through any point in it, three, and only three, lines mutually at right angles: in four-dimensional space, it would be possible to draw, through any point in it, four, and only four, lines mutually at right angles.

Extending the idea to "Higher space" in general, we may say that,—In space of "n" dimensions we can draw, through any point in it, "n," and only "n," lines mutually at right angles.

Now I admit, that, at first sight, the idea that it might be possible, under any circumstances, to draw more than three such lines through a point, seems utterly staggering and inconceivable. And indeed the more one thinks of it and the more thoroughly one grasps what it means, the more absolutely impossible does it appear.

All the same, as I hope to show very soon, it is, as a matter of fact, quite possible that there may be another independent direction fulfilling the prescribed conditions, in spite of the fact that we are at present ignorant of it.

This we can only realize by a consideration of the time-honoured but indispensable analogy of a two-dimensional world, or "Flatland."

This analogy I propose to examine in some detail in the paragraphs which follow.

But before doing so I wish to point out, and I do not think it will be necessary to do more, that a "line" which has length, but neither breadth nor thickness, can be correctly described as "One-dimensional space" i.e.:—space having only one dimension.

A mathematical "point," which has only position and neither length nor breadth nor thickness, can similarly be called space of no dimensions or "Zero-dimensional space." Also I wish to take the opportunity of defining one or two words which I may have occasion to use and have the merit of brevity.

(1) Lines which are drawn through a point for the sake of determining direction are called in Geometrical parlance, "Axes."

Thus in Fig. 1 AOA´ and BOB´ are axes. The former would be known as "the axis of A," the latter as "the axis of B." Similarly in Fig. 3 COC´ is "the axis of C."

(2) The point in which two or more axes meet, is called the "Origin" and is commonly denoted by the letter O.

(3) When convenient, I shall use the terms, "Two space," "Three space," "Four space," etc., instead of writing "Two-dimensional space," "Three-dimensional space," "Four-dimensional space," etc. in full each time.

THE ANALOGY OF A TWO-DIMENSIONAL WORLD.

The consideration of the analogy of a two dimensional world is necessary because, as Mr. C.H. Hinton says in his book, "The Fourth Dimension," p. 6.

"The change in our conceptions, which we make in passing from the shapes and motions in two dimensions to those in three, affords a pattern by which we can pass on still further to the conception of an existence in four-dimensional space."

Let us start then by imagining a very large, flat and perfectly smooth surface; such for instance as the top of a highly polished table or the surface of a sheet of still liquid.

We have seen that such a surface constitutes space of two dimensions, because through any point in it we can only draw two lines at right angles to one another. In order to draw a third such line we must get out of the surface altogether and draw the line perpendicular to it.

Next we must try to imagine that this surface is populated by a race of beings of an extraordinary thinness.

In order to grasp the analogy properly we must imagine them to be so constituted that they are incapable of realising any direction in space which does not lie in the aforementioned flat surface on which they live.

We can imagine this by supposing that their thickness, i.e.:—their extension in the third dimension perpendicular to their surface,—is so small as to be invisible to them and also that their "nerve endings" all lie on their periphery. This last is equivalent to saying that they have no "sense organs" facing the third dimension and that therefore they cannot receive impressions, or respond to any stimuli that come to them from that direction.

It follows, therefore, that unless they develope special sense organs which face the third dimension they will be acquainted only with such objects and events as lie, or take place, in their surface.

It is of course inconceivable that they should be truly "plane" beings in the mathematical sense and possess no thickness at all. But if we suppose that their thickness is of the same order as the diameter of a chemical "Atom"—that they are "one atom thick" so to speak,—the conditions laid down as to their limitation will be fulfilled.

Now we have supposed the flat surface in our analogy to be perfectly smooth in the true sense of the word. That is to say of such a nature as to offer no resistance whatever to the passage of objects over it.

This means that plane beings will not be sensible of any opposition to their movement as far as the surface is concerned. Also, as we have supposed that they have no nerve endings facing it, it follows that they cannot feel any pressure from it. In short they will be totally unaware of its existence.

But for the purpose of strict analogy this is insufficient, because a being placed on such a surface would be as incapable of movement as we should be if we were freely suspended in infinite space, remote from all the material objects we know. There would be nothing, in any direction known to him, from which he could "push off." We must therefore further suppose that the force of gravity operates in his world in a manner similar to that which we know,—every particle of matter attracting every other particle.

This will mean two things; first, that every particle on the surface will be held against that surface and that plane beings will, therefore, never be able to move away from it; and, second, that matter on the surface will tend to collect together in a manner precisely analogous to what we observe in our space.

Finally, we may suppose that these hypothetical beings whom we are considering live on the rim of a very large disc of plane matter, which has collected and is held together by the action of gravity, just as we live on the surface of a very large sphere of solid matter. They will be kept up against the rim of the disc by the force of gravity, which will attract them towards its centre, in the same way that we are kept against the surface of the earth.

It is easy to realise that the existence of such a plane being will be very limited indeed. He will be conscious of two directions only. One will be "up and down" that is to say, towards or away from the centre of his plane earth: the other will be "forwards and backwards" along its rim. Again any object, that projects beyond the rim of the disc on which he lives, will be for him an obstacle, which can only be passed by climbing over or burrowing under it. He cannot go round it, because that would mean coming out of the flat surface, which he is unable to do. Thus in Fig. 4, if the curved line AB represents a portion of the rim of the disc or "plane earth," and C a plane being, then he can only pass from A to B by "climbing over" any intervening object such as D, i.e.:—by following the path indicated by the dotted line. Otherwise he would have to get out of the plane of the paper, which is impossible for him.

Fig. 4

Now that I have described in outline the strict analogy of a race of plane beings inhabiting a smooth surface, I shall take the liberty, in the course of developing the idea more fully, of treating it in a slightly less rigid fashion. That is to say I shall assume that the reader has grasped the main idea and I shall not trouble about the "Plane earth" etc., unless it is desirable to do so for the sake of bringing out some special point; and I shall substitute for the foregoing somewhat elaborate representation the simpler one of a thin object free to slide on a smooth surface lying in front of us.

But before doing so I would point out that already we begin to see our way a little. We can understand for instance that the fact of a Fourth dimension of space being unknown and inconceivable to us, is no proof that it does not exist. We have seen that a Third dimension would be equally unknown and inconceivable to a being limited in the manner described above; although we know that a third dimension does exist.

We have only to suppose that analogous limitations obtain in our own case to see that a Fourth dimension might well exist of which we would still be unaware.

We must, for instance, suppose that we have no sense organs facing that way and that we are prevented from moving in that direction by some circumstance analogous to the smooth sheet on which we supposed the plane being to live. The plane being would think that he could see all round his plane objects although we know that he could not really do so, and similarly our conviction that we can see all round our solid objects may be an illusion.

Thus we are already in a position to appreciate the fact that our inability to perceive or imagine Four-dimensional space or objects in it, is no argument against its existence. There is, therefore, no 'a priori' reason for supposing that four dimensional space is not a reality. It is a point which must be settled by an appeal to the evidence.

If, in the course of our investigation, we find that there are in our space phenomena, which closely resemble those which would in "two space" indicate the existence of a third dimension, then we shall be entitled to say that these phenomena indicate the probable existence of a fourth dimension.

We can now proceed with our consideration of a two dimensional world, remembering that,—

Shapes and events in four space bear to shapes and events in three space, the same relation that those in three space bear to those in two space.

Fig. 5[a] Fig. 5[b]

The very small three-dimensional thickness which we have supposed to exist in all the objects of our plane world is imperceptible to the plane beings which inhabit it and the objects which they perceive they will accordingly think of as geometrical figures and of their boundaries as geometrical lines, having length but no breadth. A circle will appear to a plane being as a completely closed space. He will, as he thinks, be able to go all round it without being able to find any opening in its bounding line. It will in fact be to him what a sphere is to us. A two space room will be a thing like the figure shown in Fig. 5a. He will be able to get into or out of it by the gap in the wall which is shown and which corresponds to the door. But he will not be able to conceive of any other mode of entry or exit, although we can see that from the direction of the third dimension it is not closed at all. Similarly, if Fig. 5b represents a closed two-dimensional box, we see that this is absolutely open to us, who are three dimensional beings, though appearing to be closed on all sides to a plane being. If we took advantage of this fact we could play all sorts of tricks on him for we could put things into the box or take them out of it, by way of the third dimension, while to the plane being the box would appear to be tightly closed the whole time. It will be noticed that as the path of an object in transference would lie wholly outside the plane being's space he would not be able to form any conception of the nature of the process involved. If he tried to understand it at all he would probably imagine that the object has been disintegrated into particles inside the box, passed in this condition through the minute interstices which he might suppose to exist in its walls, and reintegrated on the other side. Whereas the true explanation is far simpler. The very great importance of this will become apparent when we come to consider the question of the positive evidences for the existence of a fourth dimension.

It is because of this importance that I have dwelt on a point which to many readers will have been obvious as soon as stated.

Similarly we could make things appear "from nowhere" and disappear equally mysteriously simply by putting them down on to his flat surface and picking them up again.

I may as well repeat here that I do not for a moment expect that the reader will have been able to visualise four-dimensional space. But I do hope that he will have seen the force of the analogy and will be prepared to admit that so far as we have gone at present four dimensional space is by no means inconceivable though it may not be distinctly imaginable.

The foregoing is really all that is necessary on the mathematical or theoretical side for the understanding of the basic ideas with which I am dealing but for the benefit of those readers who like that sort of thing I have added a few simple propositions and extensions of the analogy in the form of an appendix.

The only other question that need really concern us here is that of the phenomena of change in a two-dimensional world.

We have already seen that a cube laid on a flat surface will present to a plane being, in that surface, the appearance of a square. It is also clear that if it is pushed through the surface it will continue to present the same appearance until it has passed right through, when it will suddenly vanish away.

He would be unconscious of any movement on the part of the cube unless there was some difference between the first and last sections which he perceived.

If, for instance, the bottom face was red and the top face blue he would be conscious of a colour change on the part of the square which he perceived. It would start by being red and would pass through various shades of purple till, just before its final disappearance, it would be pure blue. But now suppose that it was pressed through his surface not "normally" but corner wise as indicated in Fig. 6—that is to say with one of its corners leading and one of its diagonals vertical. The plane being would then see quite a different set of figures. First would be a point; this would grow into a triangle which would increase in size until it reached a certain maximum when it would begin to develope three new sides at its corners which would grow, at the expense of the original sides, until a regular hexagon was produced when the reverse process would set in and the hexagon gradually change back into a triangle which in turn would dwindle away and disappear. It is easy to work out what would happen in the case of other solids, e.g., Sphere, Cone, Tetrahedron, etc. All such changes would appear very mysterious to the plane being if he had formed no conception of three-dimensional space or the shapes of bodies therein.

Fig. 6

Let us now extend this idea rather further.

Suppose we were to take a series of cinematograph pictures of the two-dimensional world, from the direction of the third dimension. We should obtain a succession of pictures each representing the precise state of affairs at some given moment in the two space world. Every thing in it would be represented in each. There would be no question of one thing being hidden by another because we are regarding them all from the direction of the third dimension in which they have an inappreciable extension. If we imagine the two space world to be very small or our camera to be very large there is no difficulty in supposing that each of our pictures includes the whole of the two space universe,—plane beings, earth, sun, planets, etc., all complete.

Imagine further that these pictures are reproduced, as cinematograph films actually are, on a transparent substance and then let us superimpose these successive pictures on one another in order so as to form a block. By this means we can represent the disposition of all the objects in a two space system at a number of successive instants in one single three space figure. For instance, the motion of a two space planet round its sun would become a part of a helix or spiral. If we now cut away from our block all the blank material which intervenes between the representations of the various two space objects we shall have a complete synthesis in three space of a succession of two space arrangements. If we were now to pass this three space object through a penetrable two space surface, e.g., a soap film, we should exactly reproduce for the two space beings in it the changes which we had originally recorded.

By analogy we can see that it would be possible to account for all the changes in our three-dimensional space by supposing them to be due to the passage through it of suitably shaped and arranged four-dimensional solids, of which we only perceive at any moment a section whose extension in the fourth dimension is imperceptibly small.

It will appear later that I do not think that this is literally the case. The point I want to make here is that the phenomena of change or successive arrangement in space of a given dimensionality are capable of explanation in terms of forms in the next space higher, which latter do not change within themselves.

The precise import of this will appear when we come to consider the bearing of the higher space theory on the problem of the nature of Time.