_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.]