The fourth dimension

THE SIMPLEST FOUR-DIMENSIONAL SOLID 159

for himself as to the question of the enclosure of a square, and of a cube.

He would say the square 4, 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, Ad, Ab. as] If now he conceives the square A to move in the, to him, unknown jan | a [Ar dimension it will trace out a cube, and the bounding squares will form As 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, ES 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

moyement of a in the third dimension.

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 Ad, 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,

Fig. 97.