Hinton - The Fourth Dimension.pdf

REMARKS ON THE FIGURES 187
As the red line now runs in the fourth dimension, the
successive sections can be called r0, r1, r2, r3, r4, these
letters indicating that at distances 0, 1⁄4. 1⁄2. 3⁄4, 1 inch along
the red axis we take all of the tesseract that can be found
in a three-dimensional space, this three-dimensional space
extending not at all in the fourth dimension, but up and
down, right and left, far and near.
We can see what should replace the light yellow face of
r0, when the section r1 comes in, by looking at the cube
b0, fig 107. What is distant in it one-quarter of an inch
from the light yellow face in the red direction? It is an
ochre section with orange and pink lines and red points;
see also fig. 103.
This square then forms the top square of r1. Now we
can determine the nomenclature of all the regions of r1 by
considering what would be formed by the motion of this
square along a blue axis.
But we can adopt another plan. Let us take a horizontal
section of r0 and finding that section in the figures,
of fig. 107 or fig. 103, from them determine what will
replace it, going in the red direction.
A section of the r0 cube has green, light blue, green,
light blue sides and blue points.
Now this square occurs on the base of each of the
section figures, b1, b2, etc. In them we see that 1⁄4 inch in
the red direction from it lies a section with brown and
light purple lines and purple corners, the interior being
of light brown. Hence this is the nomenclature of the
section which in r1 replaces the section of r0 made from a
point along the blue axis.
Hence the colouring as given can be derived.
We have thus obtained a perfectly named group of
tesseracts. We can take a group of eighty-one of them
3 × 3 × 3 × 3, in four dimensions, and each tesseract will
have its name null, red, white, yellow, blue, etc., and