THE THIRD CANON OF THOUGHT
A KEY TO THE ENIGMAS OF THE WORLD
What can we learn about the fourth dimension by studying geometrical relationships within our space? What should be the relationship of a three-dimensional body to a four-dimensional one? A four-dimensional body as the trace of the movement of a three-dimensional body in a direction not contained in it. A four-dimensional body as composed of an infinite number of three-dimensional bodies. A three-dimensional body as a section of a four-dimensional one. Parts of bodies and whole bodies in three and in four dimensions. Incommensurability of a three-dimensional and a fourdimensional body. A material atom as a section of a four-dimensional line.
If we examine the profound difference that exists between a point and a line, between a line and a surface, between a surface and a solid, i.e. the difference between the laws which govern a point and a line, a line and a surface and so on, and the difference of phenomena which are possible in a point, a line, a surface, we shall realize how many things, new and incomprehensible for us, lie in the fourth dimension.
As within a point it is impossible to visualize a line and the laws of the line, as within a line it is impossible to visualize a surface and the laws of a surface, as within a surface it is impossible to visualize a solid and understand the laws of a solid, so within our space it is impossible to visualize a body possessing more than three dimensions and impossible to understand the laws of the existence of such a body.
But, by studying the mutual relations between a point, a line, a surface and a solid we begin to learn something about the fourth dimension, i.e. about four-dimensional space. We begin to learn what it can be as compared with our three-dimensional space, and what it cannot be.
This last we learn first of all. And it is especially important, because it frees us from a great many deep-rooted illusions, which are very harmful for right knowledge. We learn what cannot be in four-dimensional space, and this enables us to establish what can be there.
In his book, The Fourth Dimension, Hinton makes an interesting remark in connection with the method which helps us to approach the question of higher dimensions. He says:
Space itself bears within it relations of which we can determine it as related to other [higher] space.
For within space are given the conceptions of point and line, line and plane, plane and solid, which really involve the relation of space to a higher space.*
Let us try to examine these relations within our space and see what conclusions may be drawn from a study of them. We know that our geometry regards a line as the trace of the movement of a point; a surface, as the trace of the movement of a line; and a solid as the trace of the movement of a surface. On this basis we mayask ourselves the question: is it not possible to regard a 'four-dimensional body' as the trace of the movement of a three-dimensional body?
What then is this movement and in what direction?
A point, moving in space and leaving the trace of its motion in the form of a line, moves in a direction not contained in itself, for in a point there is no direction.
A line, moving in space and leaving the trace of its motion in the form of a surface, moves in a direction not contained in itself, because should it move in a direction contained in itself, it would always remain a line.
A surface, moving in space and leaving the trace of its motion in the form of a solid, also moves in a direction not contained in itself. If it should move in one of the directions contained in itself, it would always remain a surface. In order to leave a trace of its motion in the form of a 'solid' or a threedimensional figure, it must move away from itself, move in a direction which does not exist within it.
By analogy with all this, a solid, in order to leave the trace of its motion in the form of a four-dimensional figure, must also move in a direction not contained in itself; in other words, a solid must get out of itself, away from itself. Later, it will be established how we should understand this.
In the meantime we may say that the direction of motion in the fourth dimension lies outside all those directions which are possible in a threedimensional figure.
We regard a line as an infinite number of points; a surface as an infinite number of lines; a solid as an infinite number of surfaces.
By analogy with this it is possible to assume that a four-dimensional bodyshould be regarded as an infinite number of three-dimensional bodies, and four-dimensional space as an infinite number of three-dimensional spaces.
* C. H. Hinton, The Fourth Dimension, London, 1912, reprinted Arno Press, New York, 1976, p. 3.
Further, we know that a line is limited by points, a surface is limited bylines, a solid is limited by surfaces.
It is possible, therefore, that four-dimensional space is limited by threedimensional bodies.
We may say that a line is the distance between points; a surface, the distance between lines; a solid, the distance between surfaces.
Or we can put it this way: a line separates two or several points from one another (a straight line is the shortest distance between two points); a surface separates two or more lines from one another; a solid separates several surfaces from one another. Thus, a cube separates six flat surfaces, which we call its sides, from one another.
A line binds several points into a certain whole (a straight, a curved, an irregular line); a surface binds several lines into a certain whole (a square, a triangle); a solid binds several surfaces into a certain whole (a cube, a pyramid).
It is more than possible that four-dimensional space is the distance between a number of solids, separating yet at the same time binding into some incomprehensible whole, those solids which to us appear to be separate from one another.
Moreover, we regard a point as a section of a line; a line as a section of a surface; a surface as a section of a solid.
By analogy with this it may be possible to regard a solid (a cube, a sphere, a pyramid) as a section of a four-dimensional body; and the whole of threedimensional space as a section of four-dimensional space.
If every three-dimensional body is the section of a four-dimensional one, then every point of a three-dimensional body is the section of a fourdimensional line. An 'atom' of a physical body may be regarded, not as something material, but as the intersection of a four-dimensional line by the plane of our consciousness.
The view of a three-dimensional body as a section of a four-dimensional one leads us to the thought that many three-dimensional bodies, which appear separate for us, may be sections or parts of one four-dimensional body.
A simple example will illustrate this idea. If we imagine a horizontal plane, intersecting the top of a tree in a direction parallel to the earth, then on this plane the sections of the branches will appear separate and quite unconnected with one another. And yet in our space, from our point of view, these are sections of the branches of one tree, together forming one top, fed by one common root and casting one shadow.
Or again, another interesting example illustrating the same idea is given by the theosophical writer, C. W. Leadbeater, in one of his books. If we touch the surface of a table with our five fingertips of one hand, there will be then on the surface of the table only five circles, and on this surface it is impossible to have any idea either of the hand or of the man to whom the hand belongs. There will be five separate circles on the table's surface. How, from these, is it possible to picture a man, with all the richness of his physical and psychological life? It is impossible. Our relation to the four-dimensional world may be exactly the same as the relationship between that consciousness which sees the five circles on the table and the man. We see only 'fingertips'; that is why the fourth dimension is incomprehensible for us.
In addition, we know that it is possible to draw an image of a threedimensional body on a plane, that it is possible to draw a cube, a polyhedron, or a sphere. But it will not be a real cube or a real sphere, but only the projection of a cube or a sphere on a plane. So it may be that we are justified in thinking that the three-dimensional bodies we see in our space are images, so to speak, of four-dimensional bodies, incomprehensible for us.