arrived at the conclusion that the points in space which lie at an infinite distance must be regarded as all lying in the same plane.” (L. Cremona, Elements of Projective Geometry, N. Y. Dover, 1960, p. ix.)
An elementary or imaginative pictorial approach to projective geometry is sufficient to introduce us to the vast range of possibilities of formative relationships between the simple elements of point, line and plane which come about by introducing movement into a configuration. This geometry used to be called the Geometry of Position (analysis situs), which may be interpreted as saying that no amount of movement in the relative positions of the geometrical elements would destroy the underlying law of a configuration.
Let us now take a simple example. Define a triangle as follows, dispensing entirely with any attribute concerning measurement: A triangle is determined either by any three points in a plane, but not in a line; or by any three lines in a plane, but not in a point. In other words, it matters not where any of the members (points or lines) of a triangle are positioned in the plane in which it lies, whether in the finite or infinitely distant. Dispense with measurement, but include the concept of the infinite, and a triangle is a triangle wherever its parts are located in the plane, even if one of its points or one of its lines is in the infinite and it does not therefore look like a conventional triangle; in thought it nevertheless is one. We are led to realize that points or lines at infinity function just as any other points do, and in fact are indispensable to the whole. Old distinctions resting on measurement cease to have any great significance; right-angled, isosceles or equilateral triangles are all included in the archetypal idea of a triangle.
Projective geometry rests on beautiful and harmonious truths of coincidence and continuity, and provides us with a realm of unshakable mathematical validity in which to move about freely in thought. The creative process can be continuous and sustained, passing through the infinite point or plane and returning again. The very fact that the creation of forms in this geometry is not dependent in the first place on any kind of measurement, but is concerned primarily with relationships between geometrical entities, gives access to processes of creative thought entirely different from those used, for example, in building a house. In the latter case, a beginning is made from, say, a cornerstone, some predetermined measure is laid out from corner to corner and so the building grows. In projective geometry, on the other hand, the actual measures of the once completed form may be quite fortuitous and appear last, not first; yet the form, however it may arise—however it may, so to speak, crystallize out in the process of construction, and however bizarre its appearance—will always remain true to the archetypal idea underlying it.
Let us take as an instance the theorem of Pascal (1623-1662) concerning the hexagon inscribed in a conic: The common points of opposite pairs of sides of a hexagon inscribed in a conic are collinear (i.e., lie on the same line).3
3 A conic is any curve which is the locus of a point which moves so that the ratio of its distance from a fixed point to its distance from a fixed line is constant. (Mathematics Dictionary, G. & R. C. James, eds.)
The theorem deals with six points in a plane and the remarkable fact that, provided all six points are points on a conic (any conic will suffice), then the “opposite” pairs of sides of the hexagon determined by these six points— three pairs of them—will each have a common point, and all three of these points will always be in line! Of course, the term “opposite” is a spatial one and correctly refers to the regular, Euclidean hexagon in the circle. This turns out to be a special case of the far more general, projective form, and the three points common to its pairs of opposite sides are all infinitely distant. The inclusion of the concept of points at infinity provides the clear thought that, in this special case, the three points are points of the infinitely distant line of the plane in which hexagon and circle lie. The variety of possibilities in the positioning of the points on the conic and also the choice of the cylic order in which to join them, added to the fact that the conditions still hold for a hexagon inscribed into any of the projective forms of a circle, fills the mind with wonder at the mobility of the spatial interpretation of the all-relating concept as expressed in the statement of the theorems. (See figures 1-4.) In this example, the line of three points or Pascal line, as it is called, may be regarded as the Absolute of the configuration, in the sense in which this term has been used above.
Because this is a realm which contains the archetypes and also the actual laws of forms as yet uncreated, it is important to understand and to become creative in the development of projective geometrical relationships. It is a realm of clear mathematical thought with its own remarkable laws, among which the laws of Euclidean geometry are one of the particular cases. The British scientist George Adams called the domain of protective transformations “Archetypal Space”; Louis Locher-Ernst used the German word Urraumto describe what we might conceive to be an ever-moving, fluid continuum of projective possibilities, containing the seeds or potentialities of all types of form.
In addition to the all-pervading element of movement in projective geometry, there is a fundamental principle of great significance—the principle of Duality or Polarity, the ideal and mutual equivalence of point and plane. Polar to the geometry of points and lines in the plane there is a geometry of lines and planes in the point;the two mutually complement one another. As Cremona wrote, “There are therefore always twocorrelative or reciprocal methods by which figures may be generated and their properties deduced, and it is in this that geometric Duality consists.”1 Laws at work in the “extensive” two-dimensional field of the plane are found again in polar opposite form in the “intensive” field of the point. So, too, the three-dimensional constructions of positive Euclidean space have their polar counterpart in a world of forms held in a point. These forms, and the laws according to which projective transformations take place among them, are of significance for a field of research which seeks a deeper approach to the living kingdoms of nature and man.
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-infinityon 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.
There can be no question, in a short, elementary statement such as this, of doing justice to all the work which has been done on the basis of this new mathematical development. Published work has been primarily in plant morphology and metamorphosis, related also to the work of Goethe. There is abundant evidence to suggest that the form-giving life of Nature is determined not only in the Euclidean universal space, in which matter qua matter (as it is usually understood) is at home, but also by the polar opposite type of space formation. In respect of time-duration, and in respect of “one” and “many,” this other kind of space plays quite a different role. It is a type of space-formation, not a single universal space given once and for all. Spaces of this kind come into being and pass away again with the life-cycles of living creatures or of their several organs. Wherever, in effect, there is a living seed, a germinating point, a special focus of life or growth —whether within the watery substance of a living body or hovering just outside it as at the growing point of the higher plant—there we may look for the “infinitude within” of such a “time-space.” We find evidence of the planar formative activity around the point in question, or in the gesture of the leaf-like organs that envelop it and thence unfold. The higher plant creates its living, counter-Euclidean spaces as it develops, and grows from thence outward into three-dimensional Earth space.
The concept of the two types of space also requires another approach to the concept of forces. Here, too, the primary polarity of space has been the guide in George Adams’ work on the planar counter-Euclidean aspect of force-fields. The idea of force in classical Newtonian mechanics (no longer, indeed, the only idea of what a force can be) is, if we reflect, in harmony with the pure geometry and kinematics of the space of Euclid. The primary characteristic of this kind of force is that in its spatial activity it is directed along a line, from point to point. We may describe the typical forces of the inorganic world as “centric forces”—forces working from center to center, that is, from point to point along the line that joins them. The archetypal instance of such a force is gravity; allied thereto are all the characteristic forces of pressure and contraction.
What kind of “force,” then, will be at work in the negative-Euclidean realm? The clear conclusion is that the primary force of such a space will be levitational, suctional, planar. The balanced duality of spatial theory will express itself also in the organic balance of a living form.
When man in his conscious activity of thinking has taken a more profound step in the understanding of polarity, as distinct from mere contrast, he will come to a creative and fundamental use of the imagination in many fields. Contrasts such as expansion and contraction in physical space take on quite another aspect in the realm of the living. To the extent to which we can learn to understand the laws of the interweaving polarities and how to put them into practice, we shall perhaps be enabled to sort out the complicated tangle of modern life.
Current methods of investigation into substances and the forces to which they are subject do not yet gain access to the whole of the living process; biochemistry and biophysics lean heavily on concepts which rest on quantitative mathematics and apply in the physical-mechanical realm. The impetus given by a quantitative mathematics has led to the development of a quantitatively minded world; it is an essential task for the future to develop the qualitative aspect of mathematics, so that the generations to come may in time achieve a true science of the living, conscious aspect of the world.