The fourth dimension

THE SIMPLEST FOUR-DIMENSIONAL SOLID 173

The number of squares is found thus—round the cube are six squares, these wil] give six squares in their initial and six in their final positions. Then each of the twelve lines of the cube trace out a square in the motion in the fourth dimension. Hence there will be altogether 12 + 12 = 24 squares.

If we look at any one of these squares we see that it is the meeting surface of two of the cubic sides. Thus, the red line by its movement in the fourth dimension, traces out a purple square—this is common to two cubes, one of which is traced out by the pink square moving in the fourth dimension, and the other is traced out by the orange square moving in the same way. To take another square, the light yellow one, this is common to the ochre cube and the light green cube. The ochre cube comes from the light yellow square by moving it in the up direction, the light green cube is made from the light yellow square by moving it in the fourth dimension. The number of lines is thirtytwo, for the twelve lines of the cube give twelve lines of the tesseract in their initial position, and twelve in their final position, making twenty-four, while each of the eight points traces out a line, thus forming thirtytwo lines altogether.

The lines are each of them common to three cubes, or to three square faces; take, for instance, the red line. This is common to the orange face, the pink face, and that face which is formed by moving the red line in the sixth dimension, namely, the purple face. It is also common to the ochre cube, the pale purple cube, and the brown cube.

The points are common to six square faces and to four cubes ; thus, the null point from which we start is common to the three square faces—pink, light yellow, orange, and to the three square faces made by moving the three lines