The figures included here may give the reader some idea of the different types of form which the principle of duality or polarity provides. In figure 5, the ellipse was created projectively as a manifold of lines rather than points, without reference to any center or focus and with no preconceived measures. The tangents completely envelop it, interweaving in the infinite area around the hollow form. Figure 6 shows a pattern of lines and points in a plane, and in figure 7 the attempt is made to awaken the idea of planes and lines in a point. The planes in both figures must, however, be thought of as infinite in extent; they continue indefinitely on all sides, and it would be impossible to draw a complete picture of them. Just as the lines and points in figure 6 are parts of members of the plane, so, too, the lines and planes in figure 7 are considered to be parts or members of the point. In the intensive world of the point, the part would seem to be greater than the whole; something which can never be said of forms in the familiar world of the plane.

The Austrian philosopher Rudolf Steiner, taking up the methods of scientific investigation begun by Goethe, was the first to underline the importance of projective geometry for a methodic approach to the phenomena of life. He held that it is not enough to return to vague, traditional ideas concerning phenomena, the full secrets of which do not readily open themselves to modern scientific methods. He pointed rather to the necessity to redefine in modern terms the realm and nature of the forces that shape living forms. The kind of training in thinking which projective geometry provides makes it possible to conceptualize such forces and their movements. Steiner called for a mathematical approach to the laws of the living world, and assigned to this realm the very qualities one Would expect to find in the kind of space which is mathematically the exact opposite of Euclidean space, wherein the physical-mechanical forces of nature have their field of action.

The first to work out in detail the properties of such a “negative space” was George Adams, who published essays simultaneously in German and English in 1933 on what he called “Physical and Ethereal Spaces.” He and Louis Locher-Ernst, working independently, thought out and formulated the forms and laws of the negative or counter-Euclidean type of space, which Locher-Ernst described in 1940 in a book entitled Raum und Gegenraum (“Space and Counterspace”).

This kind of space—the polar counterpart or, in a sense, the “negative” of Euclidean space—has indeed been conceived, at least as a possibility, by geometricians from time to time.1But from a physical point of view, its properties appeared too paradoxical, while in the purely formal sense it promised nothing new, being to the space of Euclid, so to speak, as the mould is to the cast in every detail. So far as we are aware, no one has taken the trouble to investigate it further. Scientists, interested in the interpretation of the real world on merely physical terms, have paid little or no attention to this other type of space. Yet if we allow the gestures of form in the living kingdoms to speak to us, and are not too exclusively biased in the direction of quasi-physical or atomist explanations, we awake to the fact that precisely this type of space-formation confronts us throughout the living world. Indeed, the transformations taking place in the forms of living organisms speak eloquently of that type of transformation which we now see is possible between positive (Euclidean) and negative type (counter-Euclidean) spaces through the mediation of projective transformations in the archetypal space which includes them both.

In this connection, Prof. H. V. Turnbull, who was a close friend of D’Arcy Thompson and editor of Newton’s correspondence, suggested that in the realm of growth and form the planewise and not only the pointwise approach should be significant. He wrote, “In the realm of growth and form, both analyses are significant. The seed, the stem and the leaf of a plant suggest two ways of studying the three-dimensional shape, the one point-wise microscopically and the other planewise.” He also drew attention to the fact that the relative completedness of a pointwise analysis, reached at a certain scientific stage, neither excludes nor is vitiated by the polar opposite aspect which may still be awaiting discovery. “This mathematical duality is not a case of competing theories, where one is right and the other is wrong. . . . The characteristic description of their relationship is that of in and through but not of for or against.”4

4 “Mathematics in the Larger Context,” Research, Vol. 3, No. 5, 1950.

Let us now try to picture the properties of the negative or counter-Euclidean type of space. The first thing to observe is that such a space is determined by a point-at-in-finity (the counterpart of the plane-at-infinity on which Euclidean space depends). A point-at-infinity is, then, the Absolute of this space, by which is meant a point functioning mathematically as infinitely distant—but not necessarily (and this is important) in the infinitely distant plane of ordinary Euclidean space. Conceivably, no doubt, the point-at-infinity of a negative space might also be infinitely distant in the space of Euclid, but it need not be so; above all, it will not be so in our present context, where this geometry is related to the living, germinating processes which develop on the Earth.

In this connection, it is interesting to note that in a short article published in 1910 (Proceedings of the Edinburgh Mathematical Society, Vol. 28), Professor D. M. Y. Sommerville enumerated no less than twenty-seven conceivable geometries of three-dimensional space. Among them are the Euclidean and the two well-known non-Euclidean geometries. One of the twenty-four others is the geometry of “anti-space.” Somewhere in mathematical literature there may be further developments in this direction: I have not found them. Interest has generally centered on such spaces as are more nearly in accord with the conditions of physical imagination; or else, alternatively, the geometry of abstract spaces of any number of dimensions has been worked out, quite without reference to the imagination or to the forms of nature.

Passing from ordinary space to its polar counterpart, we interchange the roles of point and plane. As noted previously, in Euclidean space the Absolute is a plane, but in this familiar space of the physical-material world, points and point-like entities predominate. The Absolute is infinitely distant and unattainable, and yet all the relations show that for this very reason the space determined by it will be predominantly “pointwise.” Points, or at least point-centered volumes, will be the spatial entities “inhabiting” such a space. In negative or counter-Euclidean space, on the other hand, depending as it does on a functional infinitude in a point, the exact opposite will be true. The constituent entities are planes—planes which are all of infinite extent and have, not a point-centered but rather a peripheral, enveloping quality. In the physical world, materials (even the living materials in plants) cannot carry out in full the planar formations characteristic of counter-Euclidean space; but in the enveloping gesture so peculiar to the living forms of the higher plants, for instance—a gesture shown often by a single leaf or by many leaves together—Nature reveals before our very eyes the kind of space in which the plane, not the point, is primary. Such a space will be found to be endowed with definite orientation, form and measure, for there will be somewhere an innermost point acting as the “infinitude within” (the point at infinity), just as the outermost plane gives form and measure to the space of Euclid.