The Fourth Dimension

by Charles Howard Hinton

Fig. 107.

In this figure the tesseract is shown in five stages distant from our space: first, zero; second, 1/4 in.; third, 2/4 in.; fourth, 3/4 in.; fifth, 1 in.; which are called b0, b1, b2, b3, b4, because they are sections taken at distances 0, 1, 2, 3, 4 quarter inches along the blue line. All the regions can be named from the first cube, the b0 cube, as before, simply by remembering that transference along the b axis gives the addition of blue to the colour of the region in the ochre, the b0 cube. In the final cube b4, the colouring of the original b0 cube is repeated. Thus the red line moved along the blue axis gives a red and blue or purple square. This purple square appears as the three purple lines in the sections b1, b2, b3, taken at 1/4, 2/4, 3/4 of an inch in the fourth dimension. If the tesseract moves transverse to our space we have then in this particular region, first of all a red line which lasts for a moment, secondly a purple line which takes its place. This purple line lasts for a minute—that is, all of a minute, except the moment taken by the crossing our space of the initial and final red line. The purple line having lasted for this period is succeeded by a red line, which lasts for a moment; then this goes and the tesseract has passed across our space. The final red line we call red bl., because it is separated from the initial red line by a distance along the axis for which we use the colour blue. Thus a line that lasts represents an area duration; is in this mode of presentation equivalent to a dimension of space. In the same way the white line, during the crossing our space by the tesseract, is succeeded by a light blue line which lasts for the inside of a minute, and as the tesseract leaves our space, having crossed it, the white bl. line appears as the final termination.

Take now the pink face. Moved in the blue direction it traces out a light purple cube. This light purple cube is shown in sections in b₁, b₂, b₃, and the farther face of this cube in the blue direction is shown in b₄—a pink face, called pink b because it is distant from the pink face we began with in the blue direction. Thus the cube which we colour light purple appears as a lasting square. The square face itself, the pink face, vanishes instantly the tesseract begins to move, but the light purple cube appears as a lasting square. Here also duration is the equivalent of a dimension of space—a lasting square is a cube. It is useful to connect these diagrams with the views given in the coloured plate.

Take again the orange face, that determined by the red and yellow axes; from it goes a brown cube in the blue direction, for red and yellow and blue are supposed to make brown. This brown cube is shown in three sections in the faces b₁, b₂, b₃. In b₄ is the opposite orange face of the brown cube, the face called orange b, for it is distant in the blue direction from the orange face. As the tesseract passes transverse to our space, we have then in this region an instantly vanishing orange square, followed by a lasting brown square, and finally an orange face which vanishes instantly.

Now, as any three axes will be in our space, let us send the white axis out into the unknown, the fourth dimension, and take the blue axis into our known space dimension. Since the white and blue axes are perpendicular to each other, if the white axis goes out into the fourth dimension in the positive sense, the blue axis will come into the direction the white axis occupied, in the negative sense.

Fig. 108.

Hence, not to complicate matters by having to think of two senses in the unknown direction, let us send the white line into the positive sense of the fourth dimension, and take the blue one as running in the negative sense of that direction which the white line has left; let the blue line, that is, run to the left. We have now the row of figures in [fig. 108]. The dotted cube shows where we had a cube when the white line ran in our space—now it has turned out of our space, and another solid boundary, another cubic face of the tesseract comes into our space. This cube has red and yellow axes as before; but now, instead of a white axis running to the right, there is a blue axis running to the left. Here we can distinguish the regions by colours in a perfectly systematic way. The red line traces out a purple square in the transference along the blue axis by which this cube is generated from the orange face. This purple square made by the motion of the red line is the same purple face that we saw before as a series of lines in the sections b₁, b₂, b₃. Here, since both red and blue axes are in our space, we have no need of duration to represent the area they determine. In the motion of the tesseract across space this purple face would instantly disappear.

From the orange face, which is common to the initial cubes in [fig. 107] and [fig. 108], there goes in the blue direction a cube coloured brown. This brown cube is now all in our space, because each of its three axes run in space directions, up, away, to the left. It is the same brown cube which appeared as the successive faces on the sections b₁, b₂, b₃. Having all its three axes in our space, it is given in extension; no part of it needs to be represented as a succession. The tesseract is now in a new position with regard to our space, and when it moves across our space the brown cube instantly disappears.

In order to exhibit the other regions of the tesseract we must remember that now the white line runs in the unknown dimension. Where shall we put the sections at distances along the line? Any arbitrary position in our space will do: there is no way by which we can represent their real position.