That is, as the plane being conceives himself as having a very small thickness in the third dimension, of which he is not aware in his sense experience, so we must suppose ourselves as having a very small thickness in the fourth dimension, and, being thus four-dimensional beings, to be prevented from realising that we are such beings by a constraint which keeps us always in contact with a vast solid sheet, which stretches on in every direction. We are against that sheet, so that, if we had the power of four-dimensional movement, we should either go away from it or through it; all our space movements as we know them being such that, performing them, we keep in contact with this solid sheet.
Now consider the exposition a plane being would make for himself as to the question of the enclosure of a square, and of a cube.
He would say the square A, in Fig. 96, is completely enclosed by the four squares, A far, A near, A above, A below, or as they are written An, Af, Aa, Ab.
Fig. 96.
If now he conceives the square A to move in the, to him, unknown dimension it will trace out a cube, and the bounding squares will form cubes. Will these completely surround the cube generated by A? No; there will be two faces of the cube made by A left uncovered; the first, that face which coincides with the square A in its first position; the next, that which coincides with the square A in its final position. Against these two faces cubes must be placed in order to completely enclose the cube A. These may be called the cubes left and right or Al and Ar. Thus each of the enclosing squares of the square A becomes a cube and two more cubes are wanted to enclose the cube formed by the movement of A in the third dimension.
Fig. 97.
The plane being could not see the square A with the squares An, Af, etc., placed about it, because they completely hide it from view; and so we, in the analogous case in our three-dimensional world, cannot see a cube A surrounded by six other cubes. These cubes we will call A near An, A far Af, A above Aa, A below Ab, A left Al, A right Ar, shown in [fig. 97]. If now the cube A moves in the fourth dimension right out of space, it traces out a higher cube—a tesseract, as it may be called. Each of the six surrounding cubes carried on in the same motion will make a tesseract also, and these will be grouped around the tesseract formed by A. But will they enclose it completely?
All the cubes An, Af, etc., lie in our space. But there is nothing between the cube A and that solid sheet in contact with which every particle of matter is. When the cube A moves in the fourth direction it starts from its position, say Ak, and ends in a final position An (using the words “ana” and “kata” for up and down in the fourth dimension). Now the movement in this fourth dimension is not bounded by any of the cubes An, Af, nor by what they form when thus moved. The tesseract which A becomes is bounded in the positive and negative ways in this new direction by the first position of A and the last position of A. Or, if we ask how many tesseracts lie around the tesseract which A forms, there are eight, of which one meets it by the cube A, and another meets it by a cube like A at the end of its motion.