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 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.