f introduction to the subject Prof. James Byrnie Shaw, in an
article in the _Scientific Monthly_, has this to say:

Up to the period of the Reformation algebraic equations of
more than the third degree were frowned upon as having no
real meaning, since there is no fourth power or dimension.
But about one hundred years ago this chimera became an actual
existence, and today it is furnishing a new world to physics,
in which mechanics may become geometry, time be co-ordinated
with space, and every geometric theorem in the world is a
physical theorem in the experimental world in study in the
laboratory. Startling indeed it is to the scientist to be told
that an artificial dream-world of the mathematician is
more real than that he sees with his galvanometers,
ultra-microscopes, and spectroscopes. It matters little that
he replies, "Your four-dimensional world is only an analytic
explanation of my phenomena," for the fact remains a fact,
that in the mathematician's four-dimensional space there is
a space not derived in any sense of the term as a residue of
experience, however powerful a distillation of sensations or
perceptions be resorted to, for it is not contained at all in
the fluid that experience furnishes. It is a product of the
creative power of the mathematical mind, and its objects are
real in exactly the same way that the cube, the square, the
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 w