The Fourth Dimension

by Charles Howard Hinton

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 we are 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 is no 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.