ith 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
_representation_ of the higher in the lower. This is a process with
which he is already acquainted, for he employs it every time he makes
a perspective drawing, which is the representation of a solid on
a plane. All that is required is an extension of the method: a
hyper-solid can be represented in a figure of three dimensions, and
this in turn can be projected on a plane. The achieved result will
constitute a perspective of a perspective--the representation of a
representation.

This may sound obscure to the uninitiated, and it is true that the
plane projection of some of the regular hyper-solids are staggeringly
intricate affairs, but the author is so sure that this matter lies so
well within the compass of the average non-mathematical mind that he
is willing to put his confidence to a practical test.

It is proposed to develop a representation of the tesseract or
hyper-cube on the paper of this page, that is, on a space of two
dimensions. Let us start as far back as we can: with a point.
This point, a, [Figure 14] is conceived to move in a direction w,
developing the line a b. This line next moves in a direction at right
angles to w, namely, x, a distance equal to its length, forming
the square a b c d. Now for the square to develop into a cube by a
movement into the third dimension it would have to move in a direction
at right angles to both w and x, that is, out of the plane of the
paper--away from it altogether, either up or down. This is not
possible, of course, but the third direction can be _represented_ on
the plane of the paper.

[Illustration: Figure 14. TWO PROJECTIONS OF THE HYPERCUBE OR
TESSERACT, AND THEIR TRANSLATION INTO ORNAMENT.]

Let us represent it as diagonally d