The fourth dimension

REMARKS ON THE FIGURES 193

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

BOD

Null bo Fig. 118.

the light purple cube, this space going off from the pink face is fixed.

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

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