The Fourth Dimension

by Charles Howard Hinton

Fig. 119.

Let now the plane being study the section made by a plane passing through the null r, null wh, and null y points, [fig. 119]. Looking at the orange square, which, as usual, we suppose to be initially in his plane, he sees that the line from null r to null y, which is a line in the section plane, the plane, namely, through the three extremities of the edges meeting in null, cuts the orange face in an orange line with null points. This then is one of the boundaries of the section figure.

Let now the cube be so turned that the pink face comes in his plane. The points null r and null wh are now visible. The line between them is pink with null points, and since this line is common to the surface of the cube and the cutting plane, it is a boundary of the figure in which the plane cuts the cube.

Again, suppose the cube turned so that the light yellow face is in contact with the plane being’s plane. He sees two points, the null wh and the null y. The line between these lies in the cutting plane. Hence, since the three cutting lines meet and enclose a portion of the cube between them, he has determined the figure he sought. It is a triangle with orange, pink, and light yellow sides, all equal, and enclosing an ochre area.

Let us now determine in what figure the space, determined by the four points, null r, null y, null wh, null b, cuts the tesseract. We can see three of these points in the primary position of the tesseract resting against our solid sheet by the ochre cube. These three points determine a plane which lies in the space we are considering, and this plane cuts the ochre cube in a triangle, the interior of which is ochre ([fig. 119] will serve for this view), with pink, light yellow and orange sides, and null points. Going in the fourth direction, in one sense, from this plane we pass into the tesseract, in the other sense we pass away from it. The whole area inside the triangle is common to the cutting plane we see, and a boundary of the tesseract. Hence we conclude that the triangle drawn is common to the tesseract and the cutting space.

Fig. 120.

Now let the ochre cube turn out and the brown cube come in. The dotted lines show the position the ochre cube has left ([fig. 120]).

Here we see three out of the four points through which the cutting plane passes, null r, null y, and null b. The plane they determine lies in the cutting space, and this plane cuts out of the brown cube a triangle with orange, purple and green sides, and null points. The orange line of this figure is the same as the orange line in the last figure.

Now let the light purple cube swing into our space, towards us, [fig. 121].