CHAPTER VITHE HIGHER WORLD

It is indeed strange, the manner in which we must begin to think about the higher world.

Those simplest objects analogous to those which are about us on every side in our daily experience such as a door, a table, a wheel are remote and uncognisable in the world of four dimensions, while the abstract ideas of rotation, stress and strain, elasticity into which analysis resolves the familiar elements of our daily experience are transferable and applicable with no difficulty whatever. Thus we are in the unwonted position of being obliged to construct the daily and habitual experience of a four-dimensional being, from a knowledge of the abstract theories of the space, the matter, the motion of it; instead of, as in our case, passing to the abstract theories from the richness of sensible things.

What would a wheel be in four dimensions? What the shafting for the transmission of power which a four-dimensional being would use.

The four-dimensional wheel, and the four-dimensional shafting are what will occupy us for these few pages. And it is no futile or insignificant enquiry. For in the attempt to penetrate into the nature of the higher, to grasp within our ken that which transcends all analogies, because what we know are merely partial views of it, the purely material and physical path affords a means of approachpursuing which we are in less likelihood of error than if we use the more frequently trodden path of framing conceptions which in their elevation and beauty seem to us ideally perfect.

For where we are concerned with our own thoughts, the development of our own ideals, we are as it were on a curve, moving at any moment in a direction of tangency. Whither we go, what we set up and exalt as perfect, represents not the true trend of the curve, but our own direction at the present—a tendency conditioned by the past, and by a vital energy of motion essential but only true when perpetually modified. That eternal corrector of our aspirations and ideals, the material universe draws sublimely away from the simplest things we can touch or handle to the infinite depths of starry space, in one and all uninfluenced by what we think or feel, presenting unmoved fact to which, think it good or think it evil, we can but conform, yet out of all that impassivity with a reference to something beyond our individual hopes and fears supporting us and giving us our being.

And to this great being we come with the question: “You, too, what is your higher?”

Or to put it in a form which will leave our conclusions in the shape of no barren formula, and attacking the problem on its most assailable side: “What is the wheel and the shafting of the four-dimensional mechanic?”

In entering on this enquiry we must make a plan of procedure. The method which I shall adopt is to trace out the steps of reasoning by which a being confined to movement in a two-dimensional world could arrive at a conception of our turning and rotation, and then to apply an analogous process to the consideration of the higher movements. The plane being must be imagined as no abstract figure, but as a real body possessing all threedimensions. His limitation to a plane must be the result of physical conditions.

We will therefore think of him as of a figure cut out of paper placed on a smooth plane. Sliding over this plane, and coming into contact with other figures equally thin as he in the third dimension, he will apprehend them only by their edges. To him they will be completely bounded by lines. A “solid” body will be to him a two-dimensional extent, the interior of which can only be reached by penetrating through the bounding lines.

Now such a plane being can think of our three-dimensional existence in two ways.

First, he can think of it as a series of sections, each like the solid he knows of extending in a direction unknown to him, which stretches transverse to his tangible universe, which lies in a direction at right angles to every motion which he made.

Secondly, relinquishing the attempt to think of the three-dimensional solid body in its entirety he can regard it as consisting of a number of plane sections, each of them in itself exactly like the two-dimensional bodies he knows, but extending away from his two-dimensional space.

A square lying in his space he regards as a solid bounded by four lines, each of which lies in his space.

A square standing at right angles to his plane appears to him as simply a line in his plane, for all of it except the line stretches in the third dimension.

He can think of a three-dimensional body as consisting of a number of such sections, each of which starts from a line in his space.

Now, since in his world he can make any drawing or model which involves only two dimensions, he can represent each such upright section as it actually is, and can represent a turning from a known into the unknown dimension as a turning from one to another of his known dimensions.

To see the whole he must relinquish part of that which he has, and take the whole portion by portion.

Fig. 34.

Consider now a plane being in front of a square,fig. 34. The square can turn about any point in the plane—say the pointA. But it cannot turn about a line, asAB. For, in order to turn about the lineAB, the square must leave the plane and move in the third dimension. This motion is out of his range of observation, and is therefore, except for a process of reasoning, inconceivable to him.

Rotation will therefore be to him rotation about a point. Rotation about a line will be inconceivable to him.

The result of rotation about a line he can apprehend. He can see the first and last positions occupied in a half-revolution about the lineAC. The result of such a half revolution is to place the squareABCDon the left hand instead of on the right hand of the lineAC. It would correspond to a pulling of the whole bodyABCDthrough the lineAC, or to the production of a solid body which was the exact reflection of it in the lineAC. It would be as if the squareABCDturned into its image, the lineABacting as a mirror. Such a reversal of the positions of the parts of the square would be impossible in his space. The occurrence of it would be a proof of the existence of a higher dimensionality.

Fig. 35.

Let him now, adopting the conception of a three-dimensional body as a series of sections lying, each removed a little farther than the preceding one, in direction at right angles to his plane, regard a cube,fig. 36, as a series of sections, each like the square which forms its base, all rigidly connected together.

If now he turns the square about the pointAin the plane ofxy, each parallel section turns with the square he moves. In each of the sections there is a point at rest, that vertically overA. Hence he would conclude that in the turning of a three-dimensional body there is one line which is at rest. That is a three-dimensional turning in a turning about a line.

In a similar way let us regard ourselves as limited to a three-dimensional world by a physical condition. Let us imagine that there is a direction at right angles to every direction in which we can move, and that we are prevented from passing in this direction by a vast solid, that against which in every movement we make we slip as the plane being slips against his plane sheet.

We can then consider a four-dimensional body as consisting of a series of sections, each parallel to our space, and each a little farther off than the preceding on the unknown dimension.

Fig. 36.

Take the simplest four-dimensional body—one which begins as a cube,fig. 36, in our space, and consists of sections, each a cube likefig. 36, lying away from our space. If we turn the cube which is its base in our space about a line, if,e.g., infig. 36we turn the cube about the lineAB, not only it but each of the parallel cubes moves about a line. The cube we see moves about the lineAB, the cube beyond it about a line parallel toABand so on. Hence the whole four-dimensional body moves about a plane, for the assemblage of these lines is our way of thinking about the plane which, starting from the lineABin our space, runs off in the unknown direction.

In this case all that we see of the plane about which the turning takes place is the lineAB.

But it is obvious that the axis plane may lie in our space. A point near the plane determines with it a three-dimensional space. When it begins to rotate round the plane it does not move anywhere in this three-dimensional space, but moves out of it. A point can no more rotate round a plane in three-dimensional space than a point can move round a line in two-dimensional space.

We will now apply the second of the modes of representation to this case of turning about a plane, building up our analogy step by step from the turning in a plane about a point and that in space about a line, and so on.

In order to reduce our considerations to those of the greatest simplicity possible, let us realise how the plane being would think of the motion by which a square is turned round a line.

Let,fig. 34,ABCDbe a square on his plane, and represent the two dimensions of his space by the axesAxAy.

Now the motion by which the square is turned over about the lineACinvolves the third dimension.

He cannot represent the motion of the whole square in its turning, but he can represent the motions of parts of it. Let the third axis perpendicular to the plane of the paper be called the axis ofz. Of the three axesx,y,z, the plane being can represent any two in his space. Let him then draw, infig. 35, two axes,xandz. Here he has in his plane a representation of what exists in the plane which goes off perpendicularly to his space.

In this representation the square would not be shown, for in the plane ofxzsimply the lineABof the square is contained.

The plane being then would have before him, infig. 35, the representation of one lineABof his square and two axes,xandz, at right angles. Now it would be obviousto him that, by a turning such as he knows, by a rotation about a point, the lineABcan turn roundA, and occupying all the intermediate positions, such asAB1, come after half a revolution to lie asAxproduced throughA.

Again, just as he can represent the vertical plane throughAB, so he can represent the vertical plane throughA´B´,fig. 34, and in a like manner can see that the lineA´B´can turn about the pointA´till it lies in the opposite direction from that which it ran in at first.

Now these two turnings are not inconsistent. In his plane, ifABturned aboutA, andA´B´aboutA´, the consistency of the square would be destroyed, it would be an impossible motion for a rigid body to perform. But in the turning which he studies portion by portion there is nothing inconsistent. Each line in the square can turn in this way, hence he would realise the turning of the whole square as the sum of a number of turnings of isolated parts. Such turnings, if they took place in his plane, would be inconsistent, but by virtue of a third dimension they are consistent, and the result of them all is that the square turns about the lineACand lies in a position in which it is the mirror image of what it was in its first position. Thus he can realise a turning about a line by relinquishing one of his axes, and representing his body part by part.

Let us apply this method to the turning of a cube so as to become the mirror image of itself. In our space we can construct three independent axes,x,y,z, shown infig. 36. Suppose that there is a fourth axis,w, at right angles to each and every one of them. We cannot, keeping all three axes,x,y,z, representwin our space; but if we relinquish one of our three axes we can let the fourth axis take its place, and we can represent what lies in the space, determined by the two axes we retain and the fourth axis.

Fig. 37.

Let us suppose that we let theyaxis drop, and that we represent thewaxis as occupying its direction. We have in fig. 37 a drawing of what we should then see of the cube. The squareABCD, remains unchanged, for that is in the plane ofxz, and we still have that plane. But from this plane the cube stretches out in the direction of theyaxis. Now theyaxis is gone, and so we have no more of the cube than the faceABCD. Considering now this faceABCD, we see that it is free to turn about the lineAB. It can rotate in thextowdirection about this line. Infig. 38it is shown on its way, and it can evidently continue this rotation till it lies on the other side of thezaxis in the plane ofxz.

Fig. 38.

We can also take a section parallel to the faceABCD, and then letting drop all of our space except the plane of that section, introduce thewaxis, running in the oldydirection. This section can be represented by the same drawing,fig. 38, and we see that it can rotate about the line on its left until it swings half way round and runs in the opposite direction to that which it ran in before. These turnings of the different sections are not inconsistent, and taken all together they will bring the cube from the position shown infig. 36to that shown infig. 41.

Since we have three axes at our disposal in our space, we are not obliged to represent thewaxis by any particular one. We may let any axis we like disappear, and let the fourth axis take its place.

Fig. 39.

Fig. 40.

Fig. 41.

Infig. 36suppose thezaxis to go. We have thensimply the plane ofxyand the square base of the cubeACEG,fig. 39, is all that could be seen of it. Let now thewaxis take the place of thezaxis and we have, infig. 39again, a representation of the space ofxyw, in which all that exists of the cube is its square base. Now, by a turning ofxtow, this base can rotate around the lineAE, it is shown on its way infig. 40, and finally it will, after half a revolution, lie on the other side of theyaxis. In a similar way we may rotate sections parallel to the base of thexwrotation, and each of them comes to run in the opposite direction from that which they occupied at first.

Thus again the cube comes from the position offig. 36. to that offig. 41. In thisxtowturning, we see that it takes place by the rotations of sections parallel to the front face about lines parallel toAB, or else we may consider it as consisting of the rotation of sections parallel to the base about lines parallel toAE. It is a rotation of the whole cube about the planeABEF. Two separate sections could not rotate about two separate lines in our space without conflicting, but their motion is consistent when we consider another dimension. Just, then, as a plane being can think of rotation about a line as a rotation about a number of points, these rotations not interfering as they would if they took place in his two-dimensional space, so we can think of a rotation about aplane as the rotation of a number of sections of a body about a number of lines in a plane, these rotations not being inconsistent in a four-dimensional space as they are in three-dimensional space.

We are not limited to any particular direction for the lines in the plane about which we suppose the rotation of the particular sections to take place. Let us draw the section of the cube,fig. 36, throughA,F,C,H, forming a sloping plane. Now since the fourth dimension is at right angles to every line in our space it is at right angles to this section also. We can represent our space by drawing an axis at right angles to the planeACEG, our space is then determined by the planeACEG, and the perpendicular axis. If we let this axis drop and suppose the fourth axis,w, to take its place, we have a representation of the space which runs off in the fourth dimension from the planeACEG. In this space we shall see simply the sectionACEGof the cube, and nothing else, for one cube does not extend to any distance in the fourth dimension.

Fig. 42.

If, keeping this plane, we bring in the fourth dimension, we shall have a space in which simply this section of the cube exists and nothing else. The section can turn about the lineAF, and parallel sections can turn about parallel lines. Thus in considering the rotation about a plane we can draw any lines we like and consider the rotation as taking place in sections about them.

To bring out this point more clearly let us take two parallel lines,AandB, in the space ofxyz, and letCDandEFbe two rods running above and below the plane ofxy, from these lines. If weturn these rods in our space about the linesAandB, as the upper end of one,F, is going down, the lower end of the other,C, will be coming up. They will meet and conflict. But it is quite possible for these two rods each of them to turn about the two lines without altering their relative distances.

To see this suppose theyaxis to go, and let thewaxis take its place. We shall see the linesAandBno longer, for they run in theydirection from the pointsGandH.

Fig. 43.

Fig. 43 is a picture of the two rods seen in the space ofxzw. If they rotate in the direction shown by the arrows—in theztowdirection—they move parallel to one another, keeping their relative distances. Each will rotate about its own line, but their rotation will not be inconsistent with their forming part of a rigid body.

Now we have but to suppose a central plane with rods crossing it at every point, likeCDandEFcross the plane ofxy, to have an image of a mass of matter extending equal distances on each side of a diametral plane. As two of these rods can rotate round, so can all, and the whole mass of matter can rotate round its diametral plane.

This rotation round a plane corresponds, in four dimensions, to the rotation round an axis in three dimensions. Rotation of a body round a plane is the analogue of rotation of a rod round an axis.

In a plane we have rotation round a point, in three-space rotation round an axis line, in four-space rotation round an axis plane.

The four-dimensional being’s shaft by which he transmits power is a disk rotating round its centralplane—the whole contour corresponds to the ends of an axis of rotation in our space. He can impart the rotation at any point and take it off at any other point on the contour, just as rotation round a line can in three-space be imparted at one end of a rod and taken off at the other end.

A four-dimensional wheel can easily be described from the analogy of the representation which a plane being would form for himself of one of our wheels.

Suppose a wheel to move transverse to a plane, so that the whole disk, which I will consider to be solid and without spokes, came at the same time into contact with the plane. It would appear as a circular portion of plane matter completely enclosing another and smaller portion—the axle.

This appearance would last, supposing the motion of the wheel to continue until it had traversed the plane by the extent of its thickness, when there would remain in the plane only the small disk which is the section of the axle. There would be no means obvious in the plane at first by which the axle could be reached, except by going through the substance of the wheel. But the possibility of reaching it without destroying the substance of the wheel would be shown by the continued existence of the axle section after that of the wheel had disappeared.

In a similar way a four-dimensional wheel moving transverse to our space would appear first as a solid sphere, completely surrounding a smaller solid sphere. The outer sphere would represent the wheel, and would last until the wheel has traversed our space by a distance equal to its thickness. Then the small sphere alone would remain, representing the section of the axle. The large sphere could move round the small one quite freely. Any line in space could be taken as an axis, and round this line the outer sphere could rotate, while the inner sphere remained still. But in all these directions ofrevolution there would be in reality one line which remained unaltered, that is the line which stretches away in the fourth direction, forming the axis of the axle. The four-dimensional wheel can rotate in any number of planes, but all these planes are such that there is a line at right angles to them all unaffected by rotation in them.

An objection is sometimes experienced as to this mode of reasoning from a plane world to a higher dimensionality. How artificial, it is argued, this conception of a plane world is. If any real existence confined to a superficies could be shown to exist, there would be an argument for one relative to which our three-dimensional existence is superficial. But, both on the one side and the other of the space we are familiar with, spaces either with less or more than three dimensions are merely arbitrary conceptions.

In reply to this I would remark that a plane being having one less dimension than our three would have one-third of our possibilities of motion, while we have only one-fourth less than those of the higher space. It may very well be that there may be a certain amount of freedom of motion which is demanded as a condition of an organised existence, and that no material existence is possible with a more limited dimensionality than ours. This is well seen if we try to construct the mechanics of a two-dimensional world. No tube could exist, for unless joined together completely at one end two parallel lines would be completely separate. The possibility of an organic structure, subject to conditions such as this, is highly problematical; yet, possibly in the convolutions of the brain there may be a mode of existence to be described as two-dimensional.

We have but to suppose the increase in surface and the diminution in mass carried on to a certain extent to find a region which, though without mobility of theconstituents, would have to be described as two-dimensional.

But, however artificial the conception of a plane being may be, it is none the less to be used in passing to the conception of a greater dimensionality than ours, and hence the validity of the first part of this objection altogether disappears directly we find evidence for such a state of being.

The second part of the objection has more weight. How is it possible to conceive that in a four-dimensional space any creatures should be confined to a three-dimensional existence?

In reply I would say that we know as a matter of fact that life is essentially a phenomenon of surface. The amplitude of the movements which we can make is much greater along the surface of the earth than it is up or down.

Now we have but to conceive the extent of a solid surface increased, while the motions possible tranverse to it are diminished in the same proportion, to obtain the image of a three-dimensional world in four-dimensional space.

And as our habitat is the meeting of air and earth on the world, so we must think of the meeting place of two as affording the condition for our universe. The meeting of what two? What can that vastness be in the higher space which stretches in such a perfect level that our astronomical observations fail to detect the slightest curvature?

The perfection of the level suggests a liquid—a lake amidst what vast scenery!—whereon the matter of the universe floats speck-like.

But this aspect of the problem is like what are called in mathematics boundary conditions.

We can trace out all the consequences of four-dimensional movements down to their last detail. Then, knowingthe mode of action which would be characteristic of the minutest particles, if they were free, we can draw conclusions from what they actually do of what the constraint on them is. Of the two things, the material conditions and the motion, one is known, and the other can be inferred. If the place of this universe is a meeting of two, there would be a one-sideness to space. If it lies so that what stretches away in one direction in the unknown is unlike what stretches away in the other, then, as far as the movements which participate in that dimension are concerned, there would be a difference as to which way the motion took place. This would be shown in the dissimilarity of phenomena, which, so far as all three-space movements are concerned, were perfectly symmetrical. To take an instance, merely, for the sake of precising our ideas, not for any inherent probability in it; if it could be shown that the electric current in the positive direction were exactly like the electric current in the negative direction, except for a reversal of the components of the motion in three-dimensional space, then the dissimilarity of the discharge from the positive and negative poles would be an indication of a one-sideness to our space. The only cause of difference in the two discharges would be due to a component in the fourth dimension, which directed in one direction transverse to our space, met with a different resistance to that which it met when directed in the opposite direction.

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