Hinton - The Fourth Dimension.pdf

THE SIMPLEST FOUR-DIMENSIONAL SOLID 173
The number of squares is found thus—round the cube
are six squares, these will give six squares in their initial
and six in their final positions. Then each of the twelve
lines of a cube trace out a square in the motion in the
fourth dimension. Hence there will be altogether
12 + 12 = 24 squares.
If we look at any one of these squares we see that it
is the meeting surface of two of the cubic sides. Thus,
the red line by its movement in the fourth dimension
traces out a purple square—this is common to two
cubes, one of which is traced out by the pink square
moving in the fourth dimension, and the other is
traced out by the orange square moving in the same
way. To take another square, the light yellow one, this
is common to the ochre cube and the light green cube.
The ochre cube comes from the light yellow square
by moving it in the up direction, the light green cube
is made from the light yellow square by moving it in
the fourth dimension. The number of lines is thirty-
two, for the twelve lines of the cube give twelve lines
of the tesseract in their initial position, and twelve in
their final position, making twenty-four, while each of
the eight points traces out a line, thus forming thirty-
two lines altogether.
The lines are each of them common to three cubes, or
to three square faces; take, for instance, the red line.
This is common to the orange face, the pink face, and
that face which is formed by moving the red line in the
fourth dimension, namely, the purple face. It is also
common to the ochre cube, the pale purple cube, and the
brown cube.
The points are common to six square faces and to four
cubes; thus, the null point from which we start is common
to the three square faces—pink, light yellow, orange, and
to the three square faces made by moving the three lines