CHAPTER IITHE ANALOGY OF A PLANE WORLD
At the risk of some prolixity I will go fully into the experience of a hypothetical creature confined to motion on a plane surface. By so doing I shall obtain an analogy which will serve in our subsequent enquiries, because the change in our conception, 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.
A piece of paper on a smooth table affords a ready image of a two-dimensional existence. If we suppose the being represented by the piece of paper to have no knowledge of the thickness by which he projects above the surface of the table, it is obvious that he can have no knowledge of objects of a similar description, except by the contact with their edges. His body and the objects in his world have a thickness of which however, he has no consciousness. Since the direction stretching up from the table is unknown to him he will think of the objects of his world as extending in two dimensions only. Figures are to him completely bounded by their lines, just as solid objects are to us by their surfaces. He cannot conceive of approaching the centre of a circle, except by breaking through the circumference, for the circumference encloses the centre in the directions in which motion is possible tohim. The plane surface over which he slips and with which he is always in contact will be unknown to him; there are no differences by which he can recognise its existence.
But for the purposes of our analogy this representation is deficient.
A being as thus described has nothing about him to push off from, the surface over which he slips affords no means by which he can move in one direction rather than another. Placed on a surface over which he slips freely, he is in a condition analogous to that in which we should be if we were suspended free in space. There is nothing which he can push off from in any direction known to him.
Let us therefore modify our representation. Let us suppose a vertical plane against which particles of thin matter slip, never leaving the surface. Let these particles possess an attractive force and cohere together into a disk; this disk will represent the globe of a plane being. He must be conceived as existing on the rim.
Fig. 4.
Fig. 4.
Let 1 represent this vertical disk of flat matter and 2 the plane being on it, standing upon its rim as we stand on the surface of our earth. The direction of the attractive force of his matter will give the creature a knowledge of up and down, determining for him one direction in his plane space. Also, since he can move along the surface of his earth, he will have the sense of a direction parallel to its surface, which we may call forwards and backwards.
He will have no sense of right and left—that is, of the direction which we recognise as extending out from the plane to our right and left.
The distinction of right and left is the one that we must suppose to be absent, in order to project ourselves into the condition of a plane being.
Let the reader imagine himself, as he looks along the plane,fig. 4, to become more and more identified with the thin body on it, till he finally looks along parallel to the surface of the plane earth, and up and down, losing the sense of the direction which stretches right and left. This direction will be an unknown dimension to him.
Our space conceptions are so intimately connected with those which we derive from the existence of gravitation that it is difficult to realise the condition of a plane being, without picturing him as in material surroundings with a definite direction of up and down. Hence the necessity of our somewhat elaborate scheme of representation, which, when its import has been grasped, can be dispensed with for the simpler one of a thin object slipping over a smooth surface, which lies in front of us.
It is obvious that we must suppose some means by which the plane being is kept in contact with the surface on which he slips. The simplest supposition to make is that there is a transverse gravity, which keeps him to the plane. This gravity must be thought of as different to the attraction exercised by his matter, and as unperceived by him.
At this stage of our enquiry I do not wish to enter into the question of how a plane being could arrive at a knowledge of the third dimension, but simply to investigate his plane consciousness.
It is obvious that the existence of a plane being must be very limited. A straight line standing up from the surface of his earth affords a bar to his progress. An object like a wheel which rotates round an axis would be unknown to him, for there is no conceivable way in which he can get to the centre without going through the circumference. He would have spinning disks, but could not get to the centre of them. The plane being can represent the motion from any one point of his spaceto any other, by means of two straight lines drawn at right angles to each other.
Fig. 5.
Fig. 5.
LetAXandAYbe two such axes. He can accomplish the translation fromAtoBby going alongAXtoC, and then fromCalongCBparallel toAY.
The same result can of course be obtained by moving toDalongAYand then parallel toAXfromDtoB, or of course by any diagonal movement compounded by these axial movements.
By means of movements parallel to these two axes he can proceed (except for material obstacles) from any one point of his space to any other.
Fig. 6.
Fig. 6.
If now we suppose a third line drawn out fromAat right angles to the plane it is evident that no motion in either of the two dimensions he knows will carry him in the least degree in the direction represented byAZ.
The linesAZandAXdetermine a plane. If he could be taken off his plane, and transferred to the planeAXZ, he would be in a world exactly like his own. From every line in his world there goes off a space world exactly like his own.
Fig. 7.
Fig. 7.
From every point in his world a line can be drawn parallel toAZin the direction unknown to him. If we suppose the square infig. 7to be a geometrical square from every point of it, inside as well as on the contour, a straight line can be drawn parallel toAZ. The assemblage of these lines constitute a solid figure, of which the square in the plane is the base. If we consider the square to represent an object in the planebeing’s world then we must attribute to it a very small thickness, for every real thing must possess all three dimensions. This thickness he does not perceive, but thinks of this real object as a geometrical square. He thinks of it as possessing area only, and no degree of solidity. The edges which project from the plane to a very small extent he thinks of as having merely length and no breadth—as being, in fact, geometrical lines.
With the first step in the apprehension of a third dimension there would come to a plane being the conviction that he had previously formed a wrong conception of the nature of his material objects. He had conceived them as geometrical figures of two dimensions only. If a third dimension exists, such figures are incapable of real existence. Thus he would admit that all his real objects had a certain, though very small thickness in the unknown dimension, and that the conditions of his existence demanded the supposition of an extended sheet of matter, from contact with which in their motion his objects never diverge.
Analogous conceptions must be formed by us on the supposition of a four-dimensional existence. We must suppose a direction in which we can never point extending from every point of our space. We must draw a distinction between a geometrical cube and a cube of real matter. The cube of real matter we must suppose to have an extension in an unknown direction, real, but so small as to be imperceptible by us. From every point of a cube, interior as well as exterior, we must imagine that it is possible to draw a line in the unknown direction. The assemblage of these lines would constitute a higher solid. The lines going off in the unknown direction from the face of a cube would constitute a cube starting from that face. Of this cube all that we should see in our space would be the face.
Again, just as the plane being can represent any motion in his space by two axes, so we can represent any motion in our three-dimensional space by means of three axes. There is no point in our space to which we cannot move by some combination of movements on the directions marked out by these axes.
On the assumption of a fourth dimension we have to suppose a fourth axis, which we will callAW. It must be supposed to be at right angles to each and every one of the three axesAX,AY,AZ. Just as the two axes,AX,AZ, determine a plane which is similar to the original plane on which we supposed the plane being to exist, but which runs off from it, and only meets it in a line; so in our space if we take any three axes such asAX,AY, andAW, they determine a space like our space world. This space runs off from our space, and if we were transferred to it we should find ourselves in a space exactly similar to our own.
We must give up any attempt to picture this space in its relation to ours, just as a plane being would have to give up any attempt to picture a plane at right angles to his plane.
Such a space and ours run in different directions from the plane ofAXandAY. They meet in this plane but have nothing else in common, just as the plane space ofAXandAYand that ofAXandAZrun in different directions and have but the lineAXin common.
Omitting all discussion of the manner on which a plane being might be conceived to form a theory of a three-dimensional existence, let us examine how, with the means at his disposal, he could represent the properties of three-dimensional objects.
Fig. 8.
Fig. 8.
There are two ways in which the plane being can think of one of our solid bodies. He can think of the cube,fig. 8, as composed of a number of sections parallel tohis plane, each lying in the third dimension a little further off from his plane than the preceding one. These sections he can represent as a series of plane figures lying in his plane, but in so representing them he destroys the coherence of them in the higher figure. The set of squares,A,B,C,D, represents the section parallel to the plane of the cube shown in figure, but they are not in their proper relative positions.
The plane being can trace out a movement in the third dimension by assuming discontinuous leaps from one section to another. Thus, a motion along the edge of the cube from left to right would be represented in the set of sections in the plane as the succession of the corners of the sectionsA,B,C,D. A point moving fromAthroughBCDin our space must be represented in the plane as appearing inA, then inB, and so on, without passing through the intervening plane space.
In these sections the plane being leaves out, of course, the extension in the third dimension; the distance between any two sections is not represented. In order to realise this distance the conception of motion can be employed.
Fig. 9.
Fig. 9.
Letfig. 9represent a cube passing transverse to the plane. It will appear to the plane being as a square object, but the matter of which this object is composed will be continually altering. One material particle takes the place of another, but it does not come from anywhere or go anywhere in the space which the plane being knows.
The analogous manner of representing a higher solid in our case, is to conceive it as composed of a number ofsections, each lying a little further off in the unknown direction than the preceding.
Fig. 10.
Fig. 10.
We can represent these sections as a number of solids. Thus the cubesA,B,C,D, may be considered as the sections at different intervals in the unknown dimension of a higher cube. Arranged thus their coherence in the higher figure is destroyed, they are mere representations.
A motion in the fourth dimension fromAthroughB,C, etc., would be continuous, but we can only represent it as the occupation of the positionsA,B,C, etc., in succession. We can exhibit the results of the motion at different stages, but no more.
In this representation we have left out the distance between one section and another; we have considered the higher body merely as a series of sections, and so left out its contents. The only way to exhibit its contents is to call in the aid of the conception of motion.
Fig. 11.
Fig. 11.
If a higher cube passes transverse to our space, it will appear as a cube isolated in space, the part that has not come into our space and the part that has passed through will not be visible. The gradual passing through our space would appear as the change of the matter of the cube before us. One material particle in it is succeeded by another, neither coming nor going in any direction we can point to. In this manner, by the duration of the figure, we can exhibit the higher dimensionality of it; a cube of our matter, under the circumstances supposed, namely, that it has a motion transverse to our space, would instantly disappear. A higher cube would last till it had passed transverse to our space by its whole distance of extension in the fourth dimension.
As the plane being can think of the cube as consisting of sections, each like a figure he knows, extending away from his plane, so we can think of a higher solid as composed of sections, each like a solid which we know, but extending away from our space.
Thus, taking a higher cube, we can look on it as starting from a cube in our space and extending in the unknown dimension.
Fig. 12.
Fig. 12.
Take the faceAand conceive it to exist as simply a face, a square with no thickness. From this face the cube in our space extends by the occupation of space which we can see.
But from this face there extends equally a cube in the unknown dimension. We can think of the higher cube, then, by taking the set of sectionsA,B,C,D, etc., and considering that from each of them there runs a cube. These cubes have nothing in common with each other, and of each of them in its actual position all that we can have in our space is an isolated square. It is obvious that we can take our series of sections in any manner we please. We can take them parallel, for instance, to any one of the three isolated faces shown in the figure. Corresponding to the three series of sections at right angles to each other, which we can make of the cube in space, we must conceive of the higher cube, as composed of cubes starting from squares parallel to the faces of the cube, and of these cubes all that exist in our space are the isolated squares from which they start.