A Theory of the Mechanism of Survival: The Fourth Dimension and Its Applications

"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."

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 O_b_, 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.

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´.

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.