Hinton - The Fourth Dimension.pdf

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.
Null
0
Pink
1
2
b0 b1 b2 b3 b4
Fig. 118.
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
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 space 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
3
4