However, as the brown cube comes off from the orange face to the left, let us put these successive sections to the left. We can call them wh0, wh1, wh2, wh3, wh4, because they are sections along the white axis, which now runs in the unknown dimension.
Running from the purple square in the white direction we find the light purple cube. This is represented in the sections wh1, wh2, wh3, wh4, [fig. 108]. It is the same cube that is represented in the sections b1, b2, b3: in [fig. 107] the red and white axes are in our space, the blue out of it; in the other case, the red and blue are in our space, the white out of it. It is evident that the face pink y, opposite the pink face in [fig. 107], makes a cube shown in squares in b1, b2, b3, b4, on the opposite side to the l purple squares. Also the light yellow face at the base of the cube b0, makes a light green cube, shown as a series of base squares.
The same light green cube can be found in [fig. 107]. The base square in wh0 is a green square, for it is enclosed by blue and yellow axes. From it goes a cube in the white direction, this is then a light green cube and the same as the one just mentioned as existing in the sections b0, b1, b2, b3, b4.
The case is, however, a little different with the brown cube. This cube we have altogether in space in the section wh0, [fig. 108], while it exists as a series of squares, the left-hand ones, in the sections b0, b1, b2, b3, b4. The brown cube exists as a solid in our space, as shown in [fig. 108]. In the mode of representation of the tesseract exhibited in [fig. 107], the same brown cube appears as a succession of squares. That is, as the tesseract moves across space, the brown cube would actually be to us a square—it would be merely the lasting boundary of another solid. It would have no thickness at all, only extension in two dimensions, and its duration would show its solidity in three dimensions.
It is obvious that, if there is a four-dimensional space, matter in three dimensions only is a mere abstraction; all material objects must then have a slight four-dimensional thickness. In this case the above statement will undergo modification. The material cube which is used as the model of the boundary of a tesseract will have a slight thickness in the fourth dimension, and when the cube is presented to us in another aspect, it would not be a mere surface. But it is most convenient to regard the cubes we use as having no extension at all in the fourth dimension. This consideration serves to bring out a point alluded to before, that, if there is a fourth dimension, our conception of a solid is the conception of a mere abstraction, and our talking about real three-dimensional objects would seem to a four-dimensional being as incorrect as a two-dimensional being’s telling about real squares, real triangles, etc., would seem to us.
The consideration of the two views of the brown cube shows that any section of a cube can be looked at by a presentation of the cube in a different position in four-dimensional space. The brown faces in b1, b2, b3, are the very same brown sections that would be obtained by cutting the brown cube, wh0, across at the right distances along the blue line, as shown in [fig. 108]. But as these sections are placed in the brown cube, wh0, they come behind one another in the blue direction. Now, in the sections wh1, wh2, wh3, we are looking at these sections from the white direction—the blue direction does not exist in these figures. So we see them in a direction at right angles to that in which they occur behind one another in wh0. There are intermediate views, which would come in the rotation of a tesseract. These brown squares can be looked at from directions intermediate between the white and blue axes. It must be remembered that the fourth dimension is perpendicular equally to all three space axes. Hence we must take the combinations of the blue axis, with each two of our three axes, white, red, yellow, in turn.
In [fig. 109] we take red, white, and blue axes in space, sending yellow into the fourth dimension. If it goes into the positive sense of the fourth dimension the blue line will come in the opposite direction to that in which the yellow line ran before. Hence, the cube determined by the white, red, blue axes, will start from the pink plane and run towards us. The dotted cube shows where the ochre cube was. When it is turned out of space, the cube coming towards from its front face is the one which comes into our space in this turning. Since the yellow line now runs in the unknown dimension we call the sections y0, y1, y2, y3, y4, as they are made at distances 0, 1, 2, 3, 4, quarter inches along the yellow line. We suppose these cubes arranged in a line coming towards us—not that that is any more natural than any other arbitrary series of positions, but it agrees with the plan previously adopted.
Fig. 109.
The interior of the first cube, y0, is that derived from pink by adding blue, or, as we call it, light purple. The faces of the cube are light blue, purple, pink. As drawn, we can only see the face nearest to us, which is not the one from which the cube starts—but the face on the opposite side has the same colour name as the face towards us.