circle, the sphere or the straight line. We are enabled to see
with the penetrating vision of the mathematical insight that
no less real and no more real are these fantastic forms of the
world of relativity than those supposed to be uncreatable or
indestructible in the play of the forces of nature.

These "fantastic forms" alone need concern the artist. If by some
potent magic he can precipitate them into the world of sensuous images
so that they make music to the eye, he need not even enter into the
question of their reality, but in order to achieve this transmutation
he should know something, at least, of the strange laws of their
being, should lend ear to a fairy-tale in which each theorem is a
paradox, and each paradox a mathematical fact.

He must conceive of a space of four mutually independent directions; a
space, that is, having a direction at right angles to every direction
that we know. We cannot point to this, we cannot picture it, but we
can reason about it with a precision that is all but absolute. In such
a space it would of course be possible to establish four axial lines,
all intersecting at a point, and all mutually at right angles with one
another. Every hyper-solid of four-dimensional space has these four
axes.

The regular hyper-solids (analogous to the Platonic solids of
three-dimensional space) are the "fantastic forms" which will prove
useful to the artist. He should learn to lure them forth along them
axis lines. That is, let him build up his figures, space by space,
developing them from lower spaces to higher. But since he cannot enter
the fourth dimension, and build them there, nor even the third--if he
confines himself to a sheet of paper--he must seek out some form of