Thus the space traced out by the pink face, if it is moved equally in the yellow and blue directions, is represented by the set of planes delineated in Fig. 118, pink face or 0, then 1, 2, 3, and finally pink face or 4. This solid is a diagonal solid of the tesseract, running from a pink face to a pink face. Its length is the length of the diagonal of a square, its side is a square. Let us now consider the unlimited space which springs from the pink face extended. This space, if it goes off in the yellow direction, gives us in it the ochre cube of the tesseract. Thus, if we have the pink face given and a point in the ochre cube, we have determined this particular space. Similarly going off from the pink face in the blue direction is another space, which gives us the light purple cube of the tesseract in it. And any point being taken in the light purple cube, this space going off from the pink face is fixed. [Illustration: Fig. 118.] The space we are speaking of can be conceived as swinging round the pink face, and in each of its positions it cuts out a solid figure from the tesseract, one of which we have seen represented in fig. 118. Each of these solid figures is given by one position of the swinging space, and by one only. Hence in each of them, if one point is taken, the particular one of the slanting spaces is fixed. Thus we see that given a plane and a point out of it a space is determined. Now, two points determine a line. Again, think of a line and a point outside it. Imagine a plane rotating round the line. At some time in its rotation it passes through the point. Thus a line and a point, or three points, determine a plane. And finally four points determine a space. We have seen that a plane and a point determine a space, and that three points determine a plane; so four points will determine a space. These four points may be any points, and we can take, for instance, the four points at the extremities of the red, white, yellow, blue axes, in the tesseract. These will determine a space slanting with regard to the section spaces we have been previously considering. This space will cut the tesseract in a certain figure. One of the simplest sections of a cube by a plane is that in which the plane passes through the extremities of the three edges which meet in a point. We see at once that this plane would cut the cube in a triangle, but we will go through the process by which a plane being would most conveniently treat the problem of the determination of this shape, in order that we may apply the method to the determination of the figure in which a space cuts a tesseract when it passes through the 4 points at unit distance from a corner. We know that two points determine a line, three points determine a plane, and given any two points in a plane the line between them lies wholly in the plane. [Illustration: Fig. 119.]