The fourth dimension

194 THE ¥OURTH DIMENSION

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 @ 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 eut 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.

Let now the plane being study the section made by a plane passing through the null 7, 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 7 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

Fig. 119.

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