**The Significance of Counter-Euclidean Geometry For an Understanding of Living Nature**

At a time in history when contemporary social and economic thinking has been dominated by the idea of an expanding economy, scientific thinking has taken a similar direction—especially as regards theories of an expanding universe. It is important to note this trend, for the ways in which men have at varying times in history conceptualized the universe in which they live—the forms of thought in which their science or their religious beliefs are expressed—have a direct effect on their ways of living and upon their social forms. The prevailing “thought forms” *(Denkformen) *of a time are reflected in the social organism.

For example, when the space of Euclid, with its rigid and right-angled three dimensions emanating from a central point, was taken absolutely for granted, social forms were organized in accordance, strictly graded around a central source of power, whether religious or secular. A dominion or kingdom is a finite area; the means of communication radiate quite naturally to and from the hub or capitol, relating it to the outskirts or frontiers. The area under rule may be large or small—it may expand or contract according to the exigencies of war—but the essentials remain the same. The ideal thought-form behind this type of picture is Euclid’s circle: a center with radii relating it to a more or less distant periphery or circumference—an outer limit.

When one comes to think of it, the thought-picture of the Euclidean circle, or its three-dimensional counterpart, the sphere or globe, has been for centuries —and still is—a powerfully formative image or ideal, both in science and sociology.

In the 1820′s and 30′s, the non-Euclidean geometries were discovered. Spaces of more than three dimensions were thought out, thus shaking the absolute nature of classical conceptions of space, and of the uniform flow of time. The result was the conjoining of what had been thought of as two radically different entities into one: space-time. In the 19th century, the unceasing efforts of scientists to adapt then-current forms of thought to the new phenomena led still further away from the old, hitherto unquestioned, framework, into which researches concerning “real” objects in space and time could no longer be made to fit. The result, of course, was the theory of relativity and subsequent developments, which so radically changed our picture of the world. As a result of these new concepts of matter and energy, the old ground with its familiar landmarks, appeared to dissolve away. Yet in practical, day-to-day matters, the space of Euclid is still valid—even if theoretically it is no longer finite but.has, in the new understanding, acquired an “infinitely distant plane” or “absolute.”

During the second half of the 19th century, Arthur Cay-ley and Felix Klein discovered that the specialized and rigid forms of geometry in which some specific measure appears could all be based on the more general and mobile projective form. In this way the ordinary space of Euclid, answering to our everyday experience in the physical world, and also the different non-Euclidean spaces, could *all *be approached from their appropriate points of view, which are embraced in projective geometry. (In Cayley’s words, “Projective geometry is all geometry.”) As Adams wrote in 1949, “Projective geometry has discovered that the ideal structure of three-dimensional space does not proceed one-sidedly from the point alone, but from two opposite entities—point and plane—which play a fully equivalent part in the fundamental structure.”^{ }^{1}

^{1}* *The Golden Blade, *p. 66.*

Amid all the possibilities of variation and change in relationships between the fundamental geometric entities in projective space (points, lines and planes) according to the definitions of this geometry, it was only necessary to propound an unique entity, as it were cosmically or absolutely given, whereby the measures of a particular type of space could be determined. This entity is called “the Absolute” of any space.^{2} In projective geometry, the infinitely distant line of the plane functions just as any other line does, and it contains the infinitely distant point of every line in the plane. In the special case of Euclidean space, the Absolute is the plane at infinity and the measures become equal and symmetrical around a central point. The Absolute, in the full sense of the word, is the ruling factor in projective processes taking place between points, lines *and planes *in the full three dimensions of projective space. (However, for the sake of simplicity, the example given below is restricted to a two-dimensional plane figure.) When in the full three-dimensional space the Absolute plane is the infinitely distant plane, then the forms take on the characteristics of what may be called *positive or Euclidean space. *It is the type of space necessary for the right-angled symmetries and parallelisms of architectural creations and the expansive and contractive rigid measures of plus and minus in the material world.

^{2}* It is more complete and mathematically exact to say that the Absolute of Euclidean space is an infinitely distant plane with an “imaginary circle” inscribed therein, and that the Absolute of counter-Euclidean space is a point-at-infinity containing an “imaginary cone.” “The conception of elements lying at an infinite distance is due to Desargues, who in 1639 explicitly considered parallel straight lines as meeting in an infinitely distant point, and parallel planes as passing through the same straight lines at an infinite distance. . . . Poncelet *

*[1822] 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 *Urraum *to 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 two *correlative*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.^{1}But 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.

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.

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In our times, it is of paramount importance to transcend the idea of a *one-fold,** *point-centered, material world, in which man (one of many similar units) lives his life automatically accumulating substance and losing it again. The material world of Euclidean space is, after all, not the only aspect of the universe, though perhaps of necessity man has had to become so deeply immersed in this aspect that he has not been able to see clearly beyond its limitations. As we have seen here, in the perspective of projective geometry, the universe may more adequately be seen as *three-fold,** *material and spiritual, with life sustained between these polar realms. A truer picture of such a cosmos reveals polarities which interplay and interweave to create the diverse forms in nature.

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.

*Olive Whicher, who is a member of the faculty of Emerson College in Sussex, England, where she lectures and gives practical courses in Projective Geometry and Plant Morphology, worked for twenty-eight years with George Adams. She is a co-author with him of several books on plant morphology, and has written a book on Projective Geometry which has been published in German and English. Ms. Whicher offers the following Selected Bibliography for those who would like to pursue further the study of projective geometry introduced in her article:*

*Adams, George,*Physical and Ethereal Spaces,*London 1965;**Stuttgart 1964;**Dornach, 1973.**Adams and Whicher,*Die Pfianze in Raum und Gegen-raum,*Stuttgart 1960.**Whicher, Olive,*Projective Geometry,*London 1971;**Stuttgart 1970.**Locher-Ernst, Louis,*Projektive Geometric*Zurich 1940;**Dornach 1957.**Steiner, Rudolf. The references are to many books and lectures, but three fundamental works are: —*Philosophy of Freedom,*London 1964;**New York, 1968;**Spring Valley, N.Y., 1973.**Unger, Georg,*Grundbegriffe der modernen Physik*(Teil III), Stuttgart 1967.**Lehrs, Ernst,*Man and Matter,*London 1958;**Frankfurt 1966.*