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.
Take the face A and 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 sections A, 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.
CHAPTER III
THE SIGNIFICANCE OF A FOUR-DIMENSIONAL EXISTENCE
Having now obtained the conception of a four-dimensional space, and having formed the analogy which, without any further geometrical difficulties, enables us to enquire into its properties, I will refer the reader, whose interest is principally in the mechanical aspect, to Chapters VI. and VII. In the present chapter I will deal with the general significance of the enquiry, and in the next with the historical origin of the idea.
First, with regard to the question of whether there is any evidence that we are really in four-dimensional space, I will go back to the analogy of the plane world.
A being in a plane world could not have any experience of three-dimensional shapes, but he could have an experience of three-dimensional movements.