Hinton - The Fourth Dimension.pdf

CHAPTER X
A FOUR-DIMENSIONAL FIGURE
THE method used in the preceding chapter to illustrate
the problem of Kant’s critique, gives a singularly easy
and direct mode of constructing a series of important
figures in any number of dimensions.
We have seen that to represent our space a plane being
must give up one of his axes, and similarly to represent
the higher shapes we must give up one amongst our
three axes.
But there is another kind of giving up which reduces
the construction of higher shapes to a matter of the
utmost simplicity.
Ordinarily we have on a straight line any number of
positions. The wealth of space in position is illimitable,
while there are only three dimensions.
I propose to give up this wealth of positions, and to
consider the figures obtained by taking just as many
positions as dimensions.
In this way I consider dimensions and positions as two
“kinds,” and applying the simple rule of selecting every
one of one kind with every other of every other kind,
get a series of figures which are noteworthy because
they exactly fill space of any number of dimensions
(as the hexagon fills a plane) by equal repetitions of
themselves.
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