CHAPTER VIITHE EVIDENCES FOR A FOURTH DIMENSION
The method necessarily to be employed in the search for the evidences of a fourth dimension, consists primarily in the formation of the conceptions of four-dimensional shapes and motions. When we are in possession of these it is possible to call in the aid of observation, without them we may have been all our lives in the familiar presence of a four-dimensional phenomenon without ever recognising its nature.
To take one of the conceptions we have already formed, the turning of a real thing into its mirror image would be an occurrence which it would be hard to explain, except on the assumption of a fourth dimension.
We know of no such turning. But there exist a multitude of forms which show a certain relation to a plane, a relation of symmetry, which indicates more than an accidental juxtaposition of parts. In organic life the universal type is of right- and left-handed symmetry, there is a plane on each side of which the parts correspond. Now we have seen that in four dimensions a plane takes the place of a line in three dimensions. In our space, rotation about an axis is the type of rotation, and the origin of bodies symmetrical about a line as the earth is symmetrical about an axis can easily be explained. But where there is symmetry about a plane no simple physical motion, such as weare accustomed to, suffices to explain it. In our space a symmetrical object must be built up by equal additions on each side of a central plane. Such additions about such a plane are as little likely as any other increments. The probability against the existence of symmetrical form in inorganic nature is overwhelming in our space, and in organic forms they would be as difficult of production as any other variety of configuration. To illustrate this point we may take the child’s amusement of making from dots of ink on a piece of paper a lifelike representation of an insect by simply folding the paper over. The dots spread out on a symmetrical line, and give the impression of a segmented form with antennæ and legs.
Now seeing a number of such figures we should naturally infer a folding over. Can, then, a folding over in four-dimensional space account for the symmetry of organic forms? The folding cannot of course be of the bodies we see, but it may be of those minute constituents, the ultimate elements of living matter which, turned in one way or the other, become right- or left-handed, and so produce a corresponding structure.
There is something in life not included in our conceptions of mechanical movement. Is this something a four-dimensional movement?
If we look at it from the broadest point of view, there is something striking in the fact that where life comes in there arises an entirely different set of phenomena to those of the inorganic world.
The interest and values of life as we know it in ourselves, as we know it existing around us in subordinate forms, is entirely and completely different to anything which inorganic nature shows. And in living beings we have a kind of form, a disposition of matter which is entirely different from that shown in inorganic matter.Right- and left-handed symmetry does not occur in the configurations of dead matter. We have instances of symmetry about an axis, but not about a plane. It can be argued that the occurrence of symmetry in two dimensions involves the existence of a three-dimensional process, as when a stone falls into water and makes rings of ripples, or as when a mass of soft material rotates about an axis. It can be argued that symmetry in any number of dimensions is the evidence of an action in a higher dimensionality. Thus considering living beings, there is an evidence both in their structure, and their different mode of activity, of a something coming in from without into the inorganic world.
And the objections which will readily occur, such as those derived from the forms of twin crystals and the theoretical structure of chemical molecules, do not invalidate the argument; for in these forms too the presumable seat of the activity producing them lies in that very minute region in which we necessarily place the seat of a four-dimensional mobility.
In another respect also the existence of symmetrical forms is noteworthy. It is puzzling to conceive how two shapes exactly equal can exist which are not superposible. Such a pair of symmetrical figures as the two hands, right and left, show either a limitation in our power of movement, by which we cannot superpose the one on the other, or a definite influence and compulsion of space on matter, inflicting limitations which are additional to those of the proportions of the parts.
We will, however, put aside the arguments to be drawn from the consideration of symmetry as inconclusive, retaining one valuable indication which they afford. If it is in virtue of a four-dimensional motion that symmetry exists, it is only in the very minute particles of bodies that that motion is to be found, for there isno such thing as a bending over in four dimensions of any object of a size which we can observe. The region of the extremely minute is the one, then, which we shall have to investigate. We must look for some phenomenon which, occasioning movements of the kind we know, still is itself inexplicable as any form of motion which we know.
Now in the theories of the actions of the minute particles of bodies on one another, and in the motions of the ether, mathematicians have tacitly assumed that the mechanical principles are the same as those which prevail in the case of bodies which can be observed, it has been assumed without proof that the conception of motion being three-dimensional, holds beyond the region from observations in which it was formed.
Hence it is not from any phenomenon explained by mathematics that we can derive a proof of four dimensions. Every phenomenon that has been explained is explained as three-dimensional. And, moreover, since in the region of the very minute we do not find rigid bodies acting on each other at a distance, but elastic substances and continuous fluids such as ether, we shall have a double task.
We must form the conceptions of the possible movements of elastic and liquid four-dimensional matter, before we can begin to observe. Let us, therefore, take the four-dimensional rotation about a plane, and enquire what it becomes in the case of extensible fluid substances. If four-dimensional movements exist, this kind of rotation must exist, and the finer portions of matter must exhibit it.
Consider for a moment a rod of flexible and extensible material. It can turn about an axis, even if not straight; a ring of india rubber can turn inside out.
What would this be in the case of four dimensions?
Fig. 44.Axis of x running towards the observer.
Fig. 44.Axis of x running towards the observer.
Let us consider a sphere of our three-dimensional matter having a definite thickness. To represent this thickness let us suppose that from every point of the sphere infig. 44rods project both ways, in and out, likeDandF. We can only see the external portion, because the internal parts are hidden by the sphere.
In this sphere the axis ofxis supposed to come towards the observer, the axis ofzto run up, the axis ofyto go to the right.
Fig. 45.
Fig. 45.
Now take the section determined by thezyplane. This will be a circle as shown infig. 45. If we let drop thexaxis, this circle is all we have of the sphere. Letting thewaxis now run in the place of the oldxaxis we have the spaceyzw, and in this space all that we have of the sphere is the circle. Fig. 45 then represents all that there is of the sphere in the space ofyzw. In this space it is evident that the rodsCDandEFcan turn round the circumference as an axis. If the matter of the spherical shell is sufficiently extensible to allow the particlesCandEto become as widely separated as they would be in the positionsDandF, thenthe strip of matter represented byCDandEFand a multitude of rods like them can turn round the circular circumference.
Thus this particular section of the sphere can turn inside out, and what holds for any one section holds for all. Hence in four dimensions the whole sphere can, if extensible turn inside out. Moreover, any part of it—a bowl-shaped portion, for instance—can turn inside out, and so on round and round.
This is really no more than we had before in the rotation about a plane, except that we see that the plane can, in the case of extensible matter, be curved, and still play the part of an axis.
If we suppose the spherical shell to be of four-dimensional matter, our representation will be a little different. Let us suppose there to be a small thickness to the matter in the fourth dimension. This would make no difference infig. 44, for that merely shows the view in thexyzspace. But when thexaxis is let drop, and thewaxis comes in, then the rodsCDandEFwhich represent the matter of the shell, will have a certain thickness perpendicular to the plane of the paper on which they are drawn. If they have a thickness in the fourth dimension they will show this thickness when looked at from the direction of thewaxis.
Supposing these rods, then, to be small slabs strung on the circumference of the circle infig. 45, we see that there will not be in this case either any obstacle to their turning round the circumference. We can have a shell of extensible material or of fluid material turning inside out in four dimensions.
And we must remember that in four dimensions there is no such thing as rotation round an axis. If we want to investigate the motion of fluids in four dimensions we must take a movement about an axis in our space, andfind the corresponding movement about a plane in four space.
Now, of all the movements which take place in fluids, the most important from a physical point of view is vortex motion.
A vortex is a whirl or eddy—it is shown in the gyrating wreaths of dust seen on a summer day; it is exhibited on a larger scale in the destructive march of a cyclone.
A wheel whirling round will throw off the water on it. But when this circling motion takes place in a liquid itself it is strangely persistent. There is, of course, a certain cohesion between the particles of water by which they mutually impede their motions. But in a liquid devoid of friction, such that every particle is free from lateral cohesion on its path of motion, it can be shown that a vortex or eddy separates from the mass of the fluid a certain portion, which always remain in that vortex.
The shape of the vortex may alter, but it always consists of the same particles of the fluid.
Now, a very remarkable fact about such a vortex is that the ends of the vortex cannot remain suspended and isolated in the fluid. They must always run to the boundary of the fluid. An eddy in water that remains half way down without coming to the top is impossible.
The ends of a vortex must reach the boundary of a fluid—the boundary may be external or internal—a vortex may exist between two objects in the fluid, terminating one end on each object, the objects being internal boundaries of the fluid. Again, a vortex may have its ends linked together, so that it forms a ring. Circular vortex rings of this description are often seen in puffs of smoke, and that the smoke travels on in the ring is a proof that the vortex always consists of the same particles of air.
Let us now enquire what a vortex would be in a four-dimensional fluid.
We must replace the line axis by a plane axis. We should have therefore a portion of fluid rotating round a plane.
We have seen that the contour of this plane corresponds with the ends of the axis line. Hence such a four-dimensional vortex must have its rim on a boundary of the fluid. There would be a region of vorticity with a contour. If such a rotation were started at one part of a circular boundary, its edges would run round the boundary in both directions till the whole interior region was filled with the vortex sheet.
A vortex in a three-dimensional liquid may consist of a number of vortex filaments lying together producing a tube, or rod of vorticity.
In the same way we can have in four dimensions a number of vortex sheets alongside each other, each of which can be thought of as a bowl-shaped portion of a spherical shell turning inside out. The rotation takes place at any point not in the space occupied by the shell, but from that space to the fourth dimension and round back again.
Is there anything analogous to this within the range of our observation?
An electric current answers this description in every respect. Electricity does not flow through a wire. Its effect travels both ways from the starting point along the wire. The spark which shows its passing midway in its circuit is later than that which occurs at points near its starting point on either side of it.
Moreover, it is known that the action of the current is not in the wire. It is in the region enclosed by the wire, this is the field of force, the locus of the exhibition of the effects of the current.
And the necessity of a conducting circuit for a current isexactly that which we should expect if it were a four-dimensional vortex. According to Maxwell every current forms a closed circuit, and this, from the four-dimensional point of view, is the same as saying a vortex must have its ends on a boundary of the fluid.
Thus, on the hypothesis of a fourth dimension, the rotation of the fluid ether would give the phenomenon of an electric current. We must suppose the ether to be full of movement, for the more we examine into the conditions which prevail in the obscurity of the minute, the more we find that an unceasing and perpetual motion reigns. Thus we may say that the conception of the fourth dimension means that there must be a phenomenon which presents the characteristics of electricity.
We know now that light is an electro-magnetic action, and that so far from being a special and isolated phenomenon this electric action is universal in the realm of the minute. Hence, may we not conclude that, so far from the fourth dimension being remote and far away, being a thing of symbolic import, a term for the explanation of dubious facts by a more obscure theory, it is really the most important fact within our knowledge. Our three-dimensional world is superficial. These processes, which really lie at the basis of all phenomena of matter, escape our observation by their minuteness, but reveal to our intellect an amplitude of motion surpassing any that we can see. In such shapes and motions there is a realm of the utmost intellectual beauty, and one to which our symbolic methods apply with a better grace than they do to those of three dimensions.