Now this light brown higher solid has for boundaries: first, the ochre cube in its initial position, second, the same cube in its final position, 1 and 3, fig. 103. Each of the squares which bound the cube, moreover, by movement in this new direction traces out a cube, so we have from the front pink faces of the cube, third, a pink blue or light purple cube, shown as a light purple face on cube 2 in fig. 103, this cube standing for any number of intermediate sections; fourth, a similar cube from the opposite pink face; fifth, a cube traced out by the orange face—this is coloured brown and is represented by the brown face of the section cube in fig. 103; sixth, a corresponding brown cube on the right hand; seventh, a cube starting from the light yellow square below; the unknown dimension is at right angles to this also. This cube is coloured light yellow and blue or light green; and, finally, eighth, a corresponding cube from the upper light yellow face, shown as the light green square at the top of the section cube. The tesseract has thus eight cubic boundaries. These completely enclose it, so that it would be invisible to a four-dimensional being. Now, as to the other boundaries, just as the cube has squares, lines, points, as boundaries, so the tesseract has cubes, squares, lines, points, as boundaries. The number of squares is found thus—round the cube are six squares, these will 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 thirty-two, 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 thirty-two 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.