We come here to a very curious thing. The whole solid cube A is to be looked on merely as a boundary of the tesseract.
Yet this is exactly analogous to what the plane being would come to in his study of the solid world. The square A ([fig. 96]), which the plane being looks on as a solid existence in his plane world, is merely the boundary of the cube which he supposes generated by its motion.
The fact is that we have to recognise that, if there is another dimension of space, our present idea of a solid body, as one which has three dimensions only, does not correspond to anything real, but is the abstract idea of a three-dimensional boundary limiting a four-dimensional solid, which a four-dimensional being would form. The plane being’s thought of a square is not the thought of what we should call a possibly existing real square, but the thought of an abstract boundary, the face of a cube.
Let us now take our eight coloured cubes, which form a cube in space, and ask what additions we must make to them to represent the simplest collection of four-dimensional bodies—namely, a group of them of the same extent in every direction. In plane space we have four squares. In solid space we have eight cubes. So we should expect in four-dimensional space to have sixteen four-dimensional bodies-bodies which in four-dimensional space correspond to cubes in three-dimensional space, and these bodies we call tesseracts.
Fig. 98.
Given then the null, white, red, yellow cubes, and those which make up the block, we notice that we represent perfectly well the extension in three directions (fig. 98). From the null point of the null cube, travelling one inch, we come to the white cube; travelling one inch away we come to the yellow cube; travelling one inch up we come to the red cube. Now, if there is a fourth dimension, then travelling from the same null point for one inch in that direction, we must come to the body lying beyond the null region.
I say null region, not cube; for with the introduction of the fourth dimension each of our cubes must become something different from cubes. If they are to have existence in the fourth dimension, they must be “filled up from” in this fourth dimension.
Now we will assume that as we get a transference from null to white going in one way, from null to yellow going in another, so going from null in the fourth direction we have a transference from null to blue, using thus the colours white, yellow, red, blue, to denote transferences in each of the four directions—right, away, up, unknown or fourth dimension.
