Plant Between Earth & Sun Chapter 3: The Polar Forms of Space by Olive Whicher & George Adams

PLANT BETWEEN EARTH & SUN

CHAPTER 3 : THE POLAR FORMS OF SPACE

Chapter 1: The Language of Plants

Chapter 2: Science of the Future

Chapter 4: Physical & Etheric Spaces

Chapter 5:  Ethereal Space of the Plant Shoot

Chapter 6: Staff of Mercury

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16 – Perspective Transformations

The fundamental notions of projective geometry are not really difficult of attainment; they are less difficult and certainly far less abstruse than many of the mathematical ideas applied, for example, in the physics of our time, in which large numbers of people take an intelligent interest. These fundamental notions are, however, comparatively little known. To enter into them requires only a certain effort in active and imaginative thinking, such as does not always come easily to people today.

We shall introduce them here descriptively, as simply as possible, with the help of illustrations, it being of paramount importance to activate one’s picture-thinking in a mobile and qualitative wat, thus evoking a healthy feeling of form. The relationships and metamorphoses are related to the living rhythms of our life, which penetrate not only our abstract thinking, but our whole being. The meaning and content of the geometrical truths will become apparent through description and illustration, but we shall leave aside proofs. Reference can be made by those requiring a systematic treatment of the geometry to the relevant publications to which reference is made in the Notes and References.¹⁶

Historically, projective geometry, the modern form of geometry, which transcends Euclid’s ancient form, arose of the the transformation of geometrical figures. Their transformation to begin with, as the name implies, is by perspective, where a plane figure, for example, is “projected” from one plane into another. “Transformation” means “metamorphosis”; it is not surprising that this kind of geometry is akin to the Goethean morphology.

Figs. 19, 20, & 21 show, for example, the perspective transformation of the circle in the oblique plane (pictured as an ellipse) into the curves in the horizontal plane below. In such a way, the parabolic or hyperbolic form of light is thrown on the surface of the road by the cone of light from a lamp with a circular aperture. According to the angle of the cone of light, the circle will appear in a different form on the the horizontal plane, appearing as one or other of the “conic sections” – we will call them “circle-curves” : ellipse, parabola, or hyperbola. ¹⁷

The parabola appears when one of the rays (a so-called generator) of the light-cone is parallel to the horizontal plane. Set the projecting eye-point a little higher up and the parabola changes to an ellipse, a little lower down and it changes to a hyperbola, the ray which pointed to the infinitely distant point of the parabola having inclined to the other side of the oblique plane. Whatever properties remain unaltered by the transformation are held to belong to the form, as it were, upon a deeper level; they are more fundamental than those that suffer alteration, when the original aspect of the form is changed. A circle, for example, changed by perspective into an ellipse but with the other, still more different curves into which it can be transformed, the parabola and the hyperbola. Fig 22 shows, in an oblique perspective, the transformation of these curves one into the other in the sequence resulting from the movement of the raying point.

Fundamental is the theorem discovered by Blaise Pascal around 1640, when he was only sixteen years of age, about the inscribed hexagon. Join any six points of any circle-curve to form a hexagon of any shape, and the three points common to “opposite” pairs of sides will be in a line (called the Pascal Lines). * [The word “line” in the geometrical part of this book always means “straight line”. (See Note 17).] It is a remarkable phenomenon, capable of being expressed in an infinitude of ways, in which – it should be noted – there is never a question of measure, neither linear nor angular (Figs 23 & 24). The constant element is simply the relative positions of the points and lines and their qualitative relationships brought about by the curve. When the curve with inscribed hexagon is projected, as in Fig 19, from one picture-plane into another, each line of the first plane together with the eye-point forms a plane, which meets the second picture-plane in yet another line. The linear quality of the relationships is preserved in projection, and though measure and form are radically changed, the idea of the inscribed hexagon always remains. ¹⁸

Modern Geometry leads to a conception akin to Goethe’s concept of an ideal Type. Each of the fundamental forms it recognizes (curves, surfaces, and so on) has many manifestations, outwardly often very different from one another. The ideal form as such cannot be identified with any one of its aspects; it hovers, as it were, over and among them all, recognizable to pure thought. It is made manifest in all of them, yet fully manifests only to a “metamorphic” thinking, which can pass freely and intelligently from the one to the other. It is in this sense like Goethe’s “archetypal leaf”.

 

17 – The Infinitely Distant Elements

The young dawn of the new geometry was at the very beginning of modern time – in the fifteenth to seventeenth centuries. It occurred in the first place in the form of a practical theory of perspective, in connection with the work of artists and architects. From this beginning there gradually developed in the new field of pure mathematical and geometrical thought. The actual sunrise was, however, at the beginning of the nineteenth century, still just in Goethe’s time, and it then became more and more clear to the mathematicians that this discipline of projective geometry included and illuminated all the other known forms of geometry, revealing much which was foreign to the classical geometry of the old Greeks and Arabs.

Not only did geometrical thought come to terms with the infinitely distant elements – point, line, and plane, but because of this very fact it became able for the first time to understand on a deeper level and to formulate the laws of polarity as expressed in point, line and plane and their mutual relationships. ¹⁹  This prepared the way for many possible conceptions of space to arise, beyond the classical Euclidian conception and including the idea of polar-Euclidian space or counterspace (Gegenraum), to which reference has already been made in the previous chapter.

For the geometry of Euclid, the point is minute, a centre of no dimensions, the line denotes direction and a definite distance in one dimension, while the plane is an extent of flat surface, an area stretching away two-dimensionally on all sides towards its boundary. In Euclidian geometry the primary element is the point; lines and planes and also volumes being built up from points by methods based on measure.

In modern geometry the three ideal entities interweave to create one another; they are in themselves formless, ie. without measure. A drawing can only reveal them partially. Each entity may be though of as existing in its own right, or as “manifold” – an organism whose parts or members are the other two elements.

For example, the plane is woven of points and lines. In Fig 25, the plane is shown consisting of a multitude of lines and points in which they interweave. Obviously, only part of the plane can be depicted and only a few lines, the pattern of lines and points being extensible in all directions in the plane. The ideal plane is of infinite extent.

The point is a manifold of planes and lines. Figs 26 and 27 show a point determined by interpenetrating planes and lines. The latter, though each of infinite extent, will all be contained in the one point. The ideal points is of infinite content.

The lines is formed of points or of planes. In Fig 28, both aspects of the line are illustrated; it may be formed by an infinitude of points or by an infinitude of planes.

In this way of thinking about and defining the three basic elements, the plane is just as primary as the point. Two planes are sufficient to create a line (as are two points); three planes, unless they are all in the same line, will always create a point. The lone point, shorn of its parts, which is the basis of Euclidian geometry, no longer has a place in the new geometry; it always appears embedded, as it were, in a matrix of lines and planes. Its members are the planes and lines which are in it, just as the plane is membered by the lines and points which lie in it.

It may be stated as follow:

As points are to planes, so are planes to points. The line is related equally to points and planes. Point, Line, and Plane thus form of a trinity, with point and plane representing the polar opposites, and line in the intermediate, balancing factor.

The completeness of this formulation rests on the acceptance of the idea of the infinitely distant elements. For example, consider the statement concerning three planes: Any three planes not all in the same line will create a point (just as any three points in a line will create a plane). Any two of the planes in Fig 28 will create the line, and it is easy to picture a third plane striking through this line and thereby creating a points, as in Fig 26. But now the third plane to be parallel to the common line of the first two; in case the common point of the three planes will be at infinity. It is an ideal point.

Fig 29 is an aid towards this conception: the two infinitely distant points in the two opposite directions of a line are identical. The lines of a fixed point ray out and relate in perspective each one with a point of a fixed line. Beginning with a point somewhere in the line and allowing the ray to turn in the direction of the arrow, the point on the fixed line will move away more and more quickly to the right. When the ray reaches the parallel position with respect to the fixed line, the point disappears momentarily into the infinite to the right, only to reappear in the next moment out of the infinite on the left, as the ray continues to turn.

To gain an impression of the nature of the idea line at infinity of a plane, it is not difficult to picture this figure as a section of a spatial form. The fixed line becomes a resting plane, while the fixed point becomes an axis around which as whole plane turns, the planes have a common line, which move parallel to itself out to the infinite (when the two planes are parallel) and back again.

Now give the revolving plane more freedom of movement, allowing it to pivot on a fixed point, inclining at all angles, without leaving this pivot-point. One can picture the common line of the two planes sweeping out to the infinite in a particular direction, only to merge into the one infinitely distant straight line of the fixed plane, each time the pivoting plane comes into the parallel position. It matter not in which direction of the plane the line is moving; it will merge in the last resort into the same line at infinity.

Such exercises of geometrical imagination help in the realization of what the mathematician means when he says: “The line-at-infinity of a plane is the locus of all the infinitely distant points of al the lines which lie in that plane. Furthermore: The plane-at-infinity of space is the locus of all the infinitely distant points of all the lines in space and of all the infinitely distant lines of all the planes in the space.

Thus, the ideal points, line and plane at infinity are assumptions in thought, understandable and entirely reliable concepts, the validity of which can be demonstrated and proved.

 

18 – Polarity: Point, Line, Plane – (“Principle of Duality)

It becomes obvious that in the membering of space there is a kind of symmetry in regard to point, line, and plane and the idea begins to dawn that maybe there is justification in considering space itself and all spatial forms from both aspects – not only as formulations of points, but also of planes.

Thus it was the two French mathematicians of the beginning of the nineteenth century (Ponceclet and Gergonne) discovered what they called the Principle of Duality which was then further developed by significant thinkers (Jacob Steiner, Chasles, v. Staudt, Plücker, Cayley, Grassman and other). They called the space in which this balanced interplay of pint, line, and plane rules “projective space”.  It is the space which results when in inner imagination and pure thinking one does not consider the infinitely distant elements as being any different from all the others, they all have the same value. The forms in this space are changeable and never rigid. In this book we call this space the “free, archetypal space”. ¹⁷ One might almost say, it exists as a potential in the pure light of thought. Forms become created within it through the projective process by means of the activity of thinking. It is like a matrix, a realm in which all possible forms exist, awaiting creation, and when once a form appears, it may take on all the possible aspects derived from the interplay of the archetypal entities, planes, lines, and points.

The basic relationships between these elements may be described quite simply; we have seen there is a balanced pairing of point and plane, with the line mediating between the two. They give rise to the axioms upon which the whole discipline of thought rests; these Axioms of Incidence may well be called the Axioms of Community of Point, Line and Plane in three-dimensional space:

Two lines either have both a point and a plane in common, or they have neither a point nor a plane in common. In the latter case we call them “skew”.

For three-dimensional projective space, this may be briefly formulated: “As point is to plane, so is plane to point; as point is to line, so is plane to line.” All the fundamental relationships of point, line, and plane, evident to the imagination, can be stated in pairs, showing that point and plane play corresponding parts, with line related equally to either. Figures of propositions, mutually related in this way, are called the dual of one-another; the mental process of deriving the one from the other is called dualizing.

The principle might better be described as one of Polarity, even thought the world has already been adopted in a more specialized sense (polarity with respect to a conic, quadric, etc).

This archetypal reciprocity between the points and the planes of space manifests in all forms; for any assemblage of points in space there will be a reciprocal, planar form. Thus every form has an answering form. Moreover by “form” is meant not necessarily something rigid and permanent, but something changing and transforming – in other words, it can mean the metamorphosis of an ideal Type.

There is a very significant qualitative difference between a perspective transformation (they are called collineations) which shows gradual change from one extreme manifestation of the archetypal form to the other, and the kind of change of form which comes about when a form is turned into its polar opposite. This kind of transformation is polar reciprocation (correlation), in which a radical change is revealed between the original form and its transformed state, with no intermediary steps. In Figs 19, 20, and 21, the sister forms of circle, ellipse, parabola, and hyperbola are shown in three different pictures. Fig 22 shows a whole family of sister, revealing the perspective (projective) transformation between one form and another, such as would occur during the movement of the projecting point. Ideally there are of course an infinite number of curves between the point or focus on the horizontal plane and the line in which the two planes interpenetrate (directrix). In this homology type of transformation, all possible intermediary steps are ideally present. The transformation is very reminiscent of the variations – leaf-forms which can be seen up and down the stem of a plant, or between different varieties of the same plant.

 

19 – Polarity called forth by the Sphere – Polar Transformation

The sphere – the purest and most ideal representation of the polar aspects of form as such – rules in space with a kind of active, balancing potentiality; it brings about the polar reciprocal type of transformation. This is understandable, when one considers that the sphere itself (and all its sister forms – or perspective transformations of itself) has two aspects, similar to the polar aspects of the curves in the plane (Figs 30 to 33). The sphere, which is a spatial form, can be though of as an assemblage of points, all equidistant from the central point; this is usually the only way we think of a sphere. In the new geometry it has, however, also to be understood as an assemblage of planes; ideally, to every point on the sphere’s surface there is a tangent plane. The concept of a sphere is therefore only fully present, if we are able to think also of the assemblage of tangent planes, all of infinite extent, which envelop it on all sides, shaping its hollow form from the periphery inward. This is an unusual and perhaps difficult thought; it is, however, essential to the understanding of the idea of metamorphosis. The quality of transformation exercised by the sphere is the change from centric to peripheral types of form, where the relationship from one to the other may be quite unrecognizable outwardly. These forms are called “polar conjugate”, an interesting use of terminology, meaning something quite different from a “family” of curves, such as concentric circles or their perspective transformations, as pictured in Fig 22 and in Plate IX.

The understanding of the difference between these two basic projective processes can be an important and reliable guide in the study of organic morphology and in coming to terms with the idea of metamorphosis.

In three-dimensional space, the five regular polyhedra are beautiful examples of the polarity of point and plane. The cube in Fig 34 is seen created in the interweaving of planes and lines, which bring about the points, and this is how we should think of the other, more complicated forms, with their many planes, lines, and points. Rather than calling these forms the “Platonic Solids”, as is usually done, we prefer to call them the “Platonic forms“, for we are not thinking only of a fixed form in its finite aspect, but of the form-types, considered also from the aspect of the plane. The forms consist of so and so many plane surfaces, which, interpenetrating one another, create the lines as edges and the points. The planes and lines are in reality of infinite extent, though the drawings only show the finite surfaces.

It is in fact the sphere which call forth for a given form its polar reciprocal form. In Fig 35, the sphere is set within the cube, so that the mid-point of each of the six faces of the cube touches the sphere’s surface. These six points give rise to a second form, now within the sphere, – the octahedron. The two forms reveal a complete polarity in regard to point and plane; cube and octahedron show themselves to be polar conjugate forms. They arise out of one another by the mutual exchange of point and plane called forth by the sphere. In Fig 36 the roles are interchanged; the sphere is now inside the octahedron, and in the eight triangular planes of the octahedron there arise the eight corners of the cube, in which three lines and three planes meet.

Fig 37 shows a similar pole and polar reciprocity with respect to the pentagon dodecahedron and the icosahedron, and Fig 38 the self-polar forms of the tetrahedron.

These five forms are the only possible quite regular spatial forms, apart from the sphere itself, which, as we have seen, is also self-polar, like the tetrahedron, for we must think of the sphere from within and the sphere from without, – the pointwise and the planewise sphere. Put together, these two aspects reveal the whole concept sphere. 

We have said that the sphere rules in space, balancing polarities. This is not only so when, as in the previous illustrations,  the sphere touches the one form from within and the other from without. Let the cube, as shown in Fig 36, diminish, towards the mid-point, while the sphere remains unchanged. Through the relationship of cube-planes, -lines and -points to the sphere, there still arises an octahedron, for the  sphere calls forth to every plane in space its point or “pole” and to every point its “polar-plane” – the so-called law of pole and polar with respect to the sphere. To the extent the cube leaves the surface of the sphere and diminishes towards the centre, the octahedron grows outward towards the periphery. Contraction and expansion keep pace with one another.

Figs 39 and 40 help in the understanding of this process. In Fig 39, plane and point are co-incident, as in Fig 35 in the case of the cube and octahedron. When, as in Fig 39, the plane is raised above the sphere, its pole is determined by the tangent cone, which determines a circle of contact, which in turn determines a plane passing through the pole. The important Law of Pole and Polar with respect to the sphere is thus demonstrated; a point within the sphere is related to a particular plane outside it and a plane outside to a particular point within. Following the process to its ultimate extreme, it will be seen that when the plane outside the sphere reaches the plane at infinity, its pole will be at the centre of the sphere, forming a functional infinitude within. If on the other hand, the plane moves right in, penetrating the sphere, its pole will move outward towards the infinite, which it will reach, when its polar plane actually reaches the innermost centre (at which moment the tangent cone becomes a cylinder; in projective terms, it is a cone with its apex in the infinite.)

Following the process taking place in Fig 36, as the octahedron grows, its points and lines get further and further away, until they merge into the infinitely distant plane of space, and all its eight planes disappear into this plane at infinity. There remains three points and three lines in the plane at infinity. At the moment when this happens, the cube points disappear into the centre of the sphere and there remain three planes and three lines in this central point. Thus the ultimate polarity is between the plane at infinity of space containing three points and three lines (a triangle) and the central point, or point at infinity within,  in the centre of the sphere, containing three planes and three lines (a trihedron).

It is important to try to visualize the polar conjugal form of the triangle – the trihedron – a form of infinite extent, opening out to the infinite in b oth directions, its three planes and lines all held in a single pint. The law of “Pole and Polar with respect to the Sphere” is absolutely fundamental to all considerations of morphology and of space itself. Stated simply : the sphere (or any of its sister forms) call forth, or accords, to each point in space a plane and to each plane a point.

Looking again at Fig 39 the tangent cone is (partially) depicted, which determines the plane, which is polar to a point outside the sphere; this polar plane is determined by the circle of contact which the cone has with the sphere, and in this drawing, the plane, which is of course in reality  of infinite extent, is only shown as an ellipse. Now move the point anywhere in the horizontal plane, and the cone will change its position, resulting in the movement of its circle of contact with the sphere and therefore of the polar plane. In the drawing, the point is marked upon which this plane would pivot, this point being the pole of the plane in which the apical point of the tangent cone is moving. So long as the moving point in the plane above remains in this plane, so long will the moving plane interpenetrating the sphere pivot upon and never leave the point within the sphere. It is from such considerations that the relationship of polar plane and pole is determined; it is an entirely mutual, reciprocal relationship, permeating the whole of space. Sphere, Polar Plane, and Pole belong together as do parts of an organism.

Now take the thought one stage further. We have given the point in the horizontal plane complete freedom of movement within the plane;  so its polar plane has complete freedom of movement within the point or pole.  Thus, to whatever form the moving point might draw in the horizontal plane, there will be a polar reciprocal form in the point or pole. This is a very valuable thought form, or ideal picture, which turns out to be of unique importance in the study of plant morphology. It is very worth while gaining a facility in this type of mobile, polar reciprocal transformation; it gives a far wider range of interpretation of any given phenomena in the living world than does the one-sided and indeed old-fashioned approach to morphology, which relies only on Euclidian concepts.

For example, proceed, to begin with, quite simply, by allowing the point to draw a circle in the horizontal plane. To begin with it is easiest to take a circle symmetrically poised above the sphere, with its centre at the point where a vertical axis (not shown in Fig 39 and 40) would strike  through the horizontal plane. It will be easily recognized in imagination that the plane which is polar to the point drawing the circle will move in such a way that it will describe or plasticize a cone of planes, all passing through or held in the point-pole. The aperture of this cone will depend  on the radius we have chosen for the circle in the horizontal plane, for this will determine the angle of inclination of the plane which is polar to the moving point (Figs 41, Plate IV).

 

20 – “Extensive” and “Intensive” Forms

Our emphasis and terminology departs from what is customary in the textbooks of the new geometry, in what we give the fullest possible scope to the Principle of Duality and Polarity, and indeed express ourselves in such a way as to help to make the application of this principle a matter of instinctive feeling, as well as of pure thought. Thus we feel free to say that a plane is in a point, just as a point is in a plane, so we shall frankly speak fo the geometry in a point, putting this over against the geometry in a plane. ¹⁷ We abandon the use of special terms – such as a manifold of planes and lines, but we introduce the idea of “extensive” forms in the extensive, two-dimensionality of the plane, and speak of the “intensive” forms in the intensive world of a point, which involves also the idea of the polar reciprocal aspects of two-dimensionality. ²⁰ To some extent this manner of speaking has of course been used already, more in some textbooks than others. The departure is logically justified and indeed necessitated by the fundamental hypothesis of this book , which is to attribute to the idea of Polarity a universal significance for the spatial structure of the world, not only in pure thought but in the real activities of Nature.

Our circle in the horizontal plane, which we have learned to think of as formed by all its points and all its lines, is transformed by the sphere into a cone in a point, which we must think of as being formed by all its planes and all its lines. This is no ordinary cone in the sense of our familiar three-dimensional world; it is just as two-dimensional as is the circle, but ina polar reciprocal sense. We must learn to picture this cone as being swept out, or moulded from without by a moving plane, the plane which is polar to the point which draws the circle; it is a surface-form in the “intensive” world of the point-pole, which sustains it (Fig 41).

No matter what form or figure may be described in the extensive realm of a plane, the sphere will call forth a polar reciprocal form – a conjugal form – in the intensive realm of a point. To comprehend such forms quite exactly, we must, as it were, allow thought to take wing and to reveal to the inner light of imagination forms which do not fully exist in the familiar earth-world of three dimensions. Indeed, all forms in this familiar world take on a new light, for ideally, according to this new way of thinking, to any form considered pointwise, there is a planewise aspect.

Furthermore, not only does the sphere or spheroid call forth, for any form in a plane, its reciprocal form in a point, but every continuous surface in space; formed as it is of points and of planes, will have its polar counterpart in another surface which it is possible to find accurately in “intensive” space. All forms are a synthesis of the two aspects, through the one or the other aspect might be dominant. Think of a surface as formed by a point that moves according to some determined law, assign, on the other hand, an appropriate law of movement to a plane ins space, and the plane in its determined movement will mould and plasticize, or as mathematicians generally say, it will “envelop” a surface. These two polar aspects of form are related and organically joined by the sphere, which is itself a synthesis of the two. Just as the sphere has its pointwise, filled aspect and its planewise, hollowed out aspect (p 55) so too, have all forms. The cone of planes in a point, for instane, is enveloped by planes from without, leaving the “inside” hollow. Thus the convexity of any surface will take on for us a deeper meaning in the new science of morphology, as will aslo the concept of “expansion and contraction”.

 

21 – Expansion and Contraction as Qualitative Forms

Let us go a step further, and picture for a moment that we have drawn not only one circle, but a whole family of concentric circles in the plane, growing outward from the central point in some kind of measure towards the infinitely distant line of the plane, into which at long last the largest circle will end. Then surely the polar form will be a family of cones, rather than just one cone; they will also be one inside the other, but in a quite different way from the family of circles. We see such a family of cones depicted is tangent to a sphere in the same way as in Fig 39, but in the plate the sphere is not drawn in. This plane is therefore polar to the central point of the concentric circles, a few of which are drawn in. Looking back at Fig 40a, it is not difficult to see that the points of a smaller circle in the horizontal plane will give rise to polar planes inclined more obliquely, and that smaller and smaller circles, finally sharpening into the central point, will relate, in the polar forms of the cones, to wider and even wider open cones, until in the last resort, when the circles will have shrunk to a point, the cones will have flattened into a plane (the plane which is polar to the central point of the circles). On the other hand, following the circles in the plane outward towards their infinitude, which is the infinitely distant line of the horizontal plane, we shall see that the cones close in towards a smaller and smaller aperture, towards what for them is an infinitely innermost line – the vertical axis of the picture. As the points of the circles merge into the infinitely distant line, the planes of the cones merge into the innermost axis.

This significant thought-form may be stated as follows:

Just as the circles have a central point from which they expand out into the infinitudes of the extensive realm of the plane in which they lie, so the cones, which are held in the point and open out (upward and downward), have a “central” plane from which they begin to close in towards their common axis, which is for them an innermost infinitude. We use here the word median plane, pairing it with the mid-point of the circles and thus differentiating between two kinds of “middle”. ¹⁷

It is a valuable and for our purposes indispensible exercise to familiarize oneself with this concept, to practice it in many possible variations and experience its polar qualities. In Fig 40 (Plate VI), for example, the spiral surface would arise in the cone-point, if in the horizontal plane a point were to draw a logarithmic spiral.

Expansion and contraction now appear in a new light, or at least, the whole question may be thrown open. Quite evidently, expanding and contracting circles or sphere picture the idea of expansion and contraction, but what of the polar reciprocal process? When circles, beginning in a point, expand to the infinite periphery, in the polar space of the cones, the process starts from the median plane and proceeds towards the inner infinitude of the vertical axis.  Is this to be thought of as expansion or contraction? And what is happening in the reverse process, when the circles shrink to a central point and the cones die back into their median plane?

The innermost point, which functions as an infinitude within, will play an important part in our further studies. The nineteenth-century mathematicians had a good intuition, when they called it a “star”. The sphere, as we have siad, has two aspects considered spatially, it has a central point and when it expands it tends towards the plane at infinity of space. Considered in is polar aspect, as an assemblage of planes, it has has its plane at infinity, and the other extreme, towards which it tends in the polar reciprocal process, is its “star” – the point at infinity, or functional infinitude, within it.

In the great advance of science from the time of Newton onward, it is an essential feature that ideal entities contain the key to understanding and mastery of phenomena in Nature in which they are only approximately realized. Indeed, they owe their power to their ideal purity and clarity – the very feature which makes them unrealizable to outer senses. For example, in mechanics, material particles can at most approximate to the character of a mathematical point, and yet this concept is essential to that of a centre of gravity, and without this and others like it we could not master the science. A scientific theory must not only allow for this fact; it must be founded on it. Truth is, the ideal entities are not approximate mental copies of sense-perceptible objects; they are the ideal aspects of a reality which presents itself to us only partially by sense-perception. Therefore, till joined to this ideal aspect, the sense-perceptible object remains dumb and inarticulate; united with it, it unfold in full reality and only now we “know” it.

Goethe’s Idea of “expansion and contraction”, so fundamental to his idea of the “archetypal leaf”, is in effect an expression of polarity between a most contracted and a most expanded entity within the space in question. To entertain this idea not only in the aspect of the finite, already created, physical forms in space, but also according to the form-creative process underlying it, brings us close to Goethe’s thought. The growing plant brings forth manifold shapes and geometrical patterns, on which the botanist bases his recognition of plant family and type. The plant reveals itself, now in expanded planes, now in hidden points. The living geometry and dynamics which Nature here unfolds belong to the more original and mobile projective and polar forms of space, wit its balanced and reciprocal polarity of plane and point – expansive and contractive tendencies. It recognizes the shapes and patterns on a deeper level than the merely external and spatially sense-perceptible level. Concepts aof size and quantity in outer measurements take second place, or at least refer better to end-results rather than beginning. So it is with leaf and bud or eye; the expanded leaf-form has grown out into space and is finished and visible; it is, however, only complete when taken together with its bud or eye, the “intensive” form which harbours the future.

Such are the questions concerning expansion and contraction, which will interest us in later chapter of this book. We shall need to know and understand the ideal entities point, line, and plane, each in their intensive as well as their extensive character of quality. This is a conception as yet unfamiliar to experimental science in genera; it has, however, long been close at hand in modern mathematics. The intense activity of pictorial but sense-free thinking here required leads beyond the boundaries determined by a study of morphology, which takes only into account the already created and sense perceptible phenomena.

 

22 – Continuous Transformation & Polarity – Metamorphosis

In all that we have been considering since entertaining the idea of the interplay of points, lines, and planes at the beginning of this chapter, we have gone far beyond the fixity and rigidity of Euclidian thought-forms. At first the new element to enter in has been that of forms of movement.  We have contemplated the transformation of forms one into the other by perspective, as for instance in the change from circle to ellipse and on into the parabola and hyperbola. These are all sister forms within one family, and to pass from one form to the next is to be involved in a continuous movement and variation of shape. (The same process can be applied in the realm of the cone-type forms in a point and also of a spherical-type forms in space.) This has only been possible through the acceptance of the idea of the infinitely distant elements, and in so doing, we have advanced from the idea of the discreet and separate forms of Euclidian geometry to the idea of continuity and synthesis, in which the whole process is primary and the parts are secondary.

The inherent polarity in the idea of point, line, and plane and their inter-relationships, has however, also led us to the far more radical, discontinuous and sudden transformations in the idea of the Polarity called forth by the Sphere. Movement and Polarity are as it were the basic ingredients of transformation and metamorphosis. To progress from fixed and rigid forms to forms in movement and then to polar transformations is a very useful geometrical exercise, undertaken by the mathematicians in recent centuries, which goes hand in hand with a study of morphology. It leads to a clearer and deeper understanding of metamorphosis than is at all possible with the limited and finite concepts of form and space, based on Euclidian geometry or even on the one-sidedly pointwise thought-forms of modern, “non-Euclidian” (analytical) mathematics.

We have been introduced to the possibility of a world of moving, changing forms, in great variety, in which the perception of an all-prevailing polarity of spatial structure is fundamental. Each form (curve of surface) has many manifestations; as we have said, we are not even limited in the polarizing process to the fixed form of the transforming sphere, for, like the circle, it has its sister forms. ²¹

Goethe’s idea of metamorphosis is indeed inspired out of the same realm of spiritual activity as is the idea of the polar transformation of forms.

In the fifteenth century, man had reached a stage in history in which he was developing an intense inner activity of thinking and at the same time was awakening in a new way to the realm of outer phenomena. The era of modern science was dawning. The overcoming of the limitation s of Euclidian mathematics, as we have seen, brought the light and mobility of perspective into men’s manner of observation in the sense-world. In many different ways, the universe was open for exploration. Science was armed with the analytical geometry of Descartes, which though freed from the fixed framework of the right-angle and the idea of rigid measurement, led, nevertheless, in the first place, to the one-sided “exact science” of our time, which has abandoned itself to the material, atomistic, pointwise realm.

The next stage, which has reached around the turn of the nineteenth century, has hardly yet been noticed; it is in the nature of things that processes set going in some direction tend to run on in the old lines as if by dint of inertia. This would seem to be a law in the process of evolution. Looking back into history and even into our present time, the origins of a particular development usually lie far back in the past, half hidden in the debris of opinions and theories, like half-buried seeds. This is as true of science as it is of Christianity.

The work of Goethe as a scientist is beginning to trickle into the science of today, notably in botany and theory of colour. Rudolf Steiner, introducing Goethe’s writings on organic morphology, call Goethe the Kepler and Copernicus of the organic world, in that he has laid the theoretical foundations and stablished the methods for the study of organic nature. ²² The great difference between the phenomena of inorganic and organic nature is that in the former, the sense-perceptible occurrences are determined by conditions which likewise belong to the sense world. Here the concept and the phenomena coincide. In the organic world, however, sensible qualities appear, the causes of which are not immediately clearly perceptible to the senses. In this realm it is not easy to grasp the unity of concept and percept – to perceive exactly the truth underlying the observed phenomena. It is in this realm that Goethe’s great contribution to science lies.

In the organic realm, full understanding of the whole phenomenon is not to be found in a study of the finished, material form alone, but only when the spiritual idea or concept which belongs to it is joined with the percept; in the inorganic realm the material form or physical process itself furnishes the required data. Compare this with what we have been considering in terms of the development of mathematics. In Euclidian terms, the comparison, for example, of similar triangles rests with their actual, observable measurements, just as an inorganic form is adequately described in terms of its observable measurement and material constituents. In the examples afforded by modern geometry, such as the cube and octahedron in their intimate relationships with one another through the law of polar reciprocal geometry, the statements are pure ideas, which are then revealed in manifold external ways, as soon as one puts them into practice in pictorial imagination, or reveals them in a drawing. The idea hovers among the many possible manifestations and their changes from one form to another, just as Goethe’s Idea of an archetypal plant hovers above all individual plants and plant species. ¹ Here we touch the nerve of what was being envisaged as a future direction of science by Goethe and many of the seekers of his time, and the great mathematicians did not lag behind the task, even though their efforts destined to be lost sight of for a time.

 

23 – Goethe & Modern Geometry

The morphological ideas derived from modern geometry give a basis for a systematic training in “metamorphic” thinking. Goethe did not reflect much on his own theory of knowledge, yet a true method and consistent theory of knowledge underlay his work. He believed that the Theory is contained in the Phenomenon. The phenomena our senses see are not unlike the ideal reality which brings them forth. As the Greek origin of the word implies, the Theory is the true seeing of the thing – the insight that should come with healthy sight. Yet man is so constituted that he does not really see, unless he meets what he sees with spiritual activity on his own part. It becomes the role of thought to interpret the language of phenomena. If what our thinking reveals is in the phenomena the senses see, so that, enlightened by thought, we feel that we are seeing the new-awakened eyes, then will our thinking have been true; it will have met and united with the archetypal Thought whereby the phenomenon was created. Scientific thought should not lead away from the phenomena into some remote construction from which the thing perceived derives by elaborate causation. It must gives us the faculty – latent but unawakened in the merely passive mind – to read in the phenomena, so that the chaos of detail becomes articulate language.

Goethe’s awakened faculties were expressed in his whole scientific outlook. It may at first cause surprise to claim there there is an affinity between the modern geometrical approach to form and Goethe’s work. Yet as some modern authors have recognized, there was far more of the mathematical in Goethe’s thinking than might first appear. He was far too universal a spirit to have allowed the new mathematics, which was coming into being in his time, to pass him by. But mathematics, too, has gone a long way since Goethe’s time, and we are nearer today to an appreciation of the qualitative and form-creative as against the merely quantitative aspect of this science. It is this qualitative aspect which reveals the air in which Goethe breathes.

The “holistic” strain in Goethe’s thinking allows him to perceive the organic relation between part and the whole, or between part and part. He begins by observing the whole, – the whole which contains the parts, and then he sees that in each part the virtue of the whole is contained. For Goethe, this became an immediate aesthetic insight, and so it is form may a biologist in gifted moments. The concept of and ideal Type, having many manifestation, outwardly often very different form one another, perhaps even recognizable, we have seen revealed in the few geometrical studies we have undertaken in this chapter. We have seen it, for example, in the family of curves in Fig 22 , where the law of the whole is to be found in each separate curve, however different they may be from one another externally; we have see it in the examples of polar reciprocation, where the ideal type reappears transformed beyond all external recognition.

In a still deeper sense the change of outlook in science is contained in the later work of Rudolf Steiner, who related mathematics and imagination rather in the spirit of Goethe’s essay on “Perceptive Judgement”. ²³  In selfless contemplation of Nature, says Rudolf Steiner in effect we can so train our imaginative faculties that they become an instrument of cognition no less conscious than mathematical reasoning. Our mathematical thought-forms – such as the three perpendicular axes of analytical geometry – spring from the form of the human body and from the upright stature we acquired when in infancy we learned to stand and walk. All our cognitional faculties in earthly life are in some way a sublimation of the powers of life and growth which placed us bodily into the world in embryo-life and early childhood. Many-sided is our common bond with Nature; only by virtue of it can we know here.

Euclidian thought is of such a nature that it supplies the concepts which fit the world of created forms, which may be the object of our attention in the external world of space at any one moment in time. trivially speaking, as a material body, the plant form or any organic form is there, of course, in three-dimensional space. The question is, whether that is the whole story, if one would understand what has taken place before that moment, and what will eventually be determined through the process of Becoming. The organism lives through the sequences of time, the changing forms of the plant arise even as colours arise in and not only the part; colours arise with their different qualities and shared between the polarities of light and darkness. In “Light & Darkness” Goethe divined an ideal polarity, fundamental to the structure and processes of the world; the visible polarity, fundamental to the structure and processes of the world; the visible phenomena were for him an outcome of the interplay of both components. ²⁴ In theses realms live the thought-forms of projective morphology.

 

24 – Concerning Conceptions of Space

Our considerations so far involve another outlook into the spatial universe as such. The cosmic outlook in the age of science hitherto has been instinctively and naturally pointwise: problems connected with the seeming infinitude of the universe, with the nexus of celestial phenomena and the place therein of the solar system and the earth, have always started from this premise. If now the basic polarity of space – that of point and plane – is to have meaning for the real universe, there will also be an aspect of cosmic and earthly nature as a whole, polar opposite to what is pointwise. we have to look for this aspect and find for it an appropriate designation.

Here, too, the history and development of modern geometry points the way. It was discovered by Arthur Cayley and Felix Klein, during the second half of the nineteenth century, that the more specialized and rigid metrical forms of geometry 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 of the physical world – and also the different non-Euclidian spaces could be brought together. It was only necessary to propound a unique entity – as it were, cosmically and absolutely given – whereby the definite measures of a particular space should be determined. They call this entity “the Absolute”. For Euclidian geometry it is an infinitely distant plane (with an “imaginary circle” therein inscribed). ²⁵

Mathematicians are used to many different conceptions of space, – Einstein’s space among them. Around the middle of the nineteenth century, when the discipline of projective geometry was already well-established, there lived in Germany Hermann Grassmann, ²⁶ a man of repute in two widely differing realms of culture; he was a scholar in old Indian manuscripts and in mathematics. His most significant work remained long unnoticed; according to A.N. whitehead, he was “a hundred years ahead of his time”.

Grassmann proved, following the direction of projective thinking further, that a polarity of contracted and expanded entities, analogous to point and plane, will hold good in spaces of any number of dimensions. He founded in this sense what he described as a “science of extension” (Ausdehnungslehre). We shall contend that in the relatively simple threefold space of our universe, the living plant is like a realization of such a polar “science of extension”. We shall attempt to recognize the plant as a being in Nature which lives and makes manifest the processes of its becoming. Unlike the material, finally created, dead world of the mineral, which  lies as a finished object in a fixed space, the plant in becoming, forms its own space, and it does so according to the same ideal laws which underlie the creation of universal space; laws – or ideas – which can light up in a the inner perception of man in the activity of thinking. In the three-dimensional space of our universe, the plant lives out, as it were, in the real world, a “science of extension.”

In this spirit we shall attempt to bring together with phenomena in the plant world, and in particular in regard to the leaf, the idea of a plane as an original, form-creating entity, equal in importance to the point. We will put to the test, without bias, what arises when the world of phenomena is approached with the following direction of thought. Just as in the condensed and rounded off material body – a stone, for example – the idea of the point (as centre of gravity and so on) applies, so in the leaf, revealer of all plant nature, it is the plane. If this be true, it will show how point, line and place have their counterpart in the plant, namely the live very obviously in the stem, the point in the seed or eye, and the ideal plane in the leaf.

This co-ordination, however, will become meaningful only if we deepen our conception of “point” and “plane”, as we have begun to do, seeing the plane not only extensively, as the sum of all its points, but also intensively, as a single whole; likewise the point, not lying dead as a speck of dust, but alive, for example as a moving organism of planes. On the accustomed level of Euclid the comparison would, of course, be trivial.

Above all, we shall need to understand how the leaf is Nature’s adumbration of the ideal plane. Then, too, we shall attach fresh significance to the universal association of leaf and ey – the organ that shows forth the archetypal form with the on e that only bears it for a potential future. Goethe perceived that the expanded, planar organ has its inseparable counterpart in the contracted. In a letter to Herder on May 17, 1787, he wrote: “It dawned upon me that in the organ we are wont to call a leaf the Proteus lies hidden, who can reveal or veil his presence in all the forms the plant brings forth. Forward and backward, the plant is ever leaf, inseparably united with the germ of the future plant, so that the one is unthinkable without the other.”

 

25 – Archetypal Space (Ur-raum)

The mathematicians, following Cayley, Klein and v. Staudt, call the “space” in which the balanced, creative interplay of polar forms takes place the (three-dimensional) “projective space”. It is the “space” which presents itself to the activity of thinking and inner imagination, when one moves freely in the realm of the exact relationships between Point, Line, and Plane, considered as manifolds or organisms, and once one recognizes that the ideal, inifintely distant points and lines, and even the infinitely distant plane function in just the same way as all the others. It’als a realm attainable in thought, in which one works with continually transformable form-types, and not with ready-made, rigid figures. We call this mathematical thought-realm the free Archetypal Space (Freie Ur-raum.) ¹⁷ 

To be active in this realm of thinking, one sets aside, to begin with, any idea of measure, and one takes the infinitely distant elements for granted. In the weaving interplay of planes, lines and points in Archetypal Space there are myriad possibilities for the creation of all manner of forms – not only forms with straight edges and flat surfaces, but curves and curved surfaces also. It is a realm of infinite potential. Once a process has been begun, the planes and lines have the tendency to weave together according to some superior law; they will not be in disarray, but will create sensible patterns. Out of these patterns and spatial forms (one might call them “achieved relationships”) measures then arise functionally; they are not predetermined. Thus, in contrast to the ancient geometry of Euclid, we do not begin with a measure, or measures, and then build up forms from these; it is just the other way round. If amid the moving, weaving, mutual interplay between point, line, and plane, we establish some “Absolute” – some invariable element – then a measure will become manifest in some definite type of form or in a metrical space of some kind or other.

The rhythm of measure is thus freed from any preconceived fixity. In the Archetypal Space of projective geometry there are no rigid measures; there are only functional relationships, which find expression in pure number or form.

With this concept of archetypal space, the mathematicians reach into a realm of thought for which the words “space” and “geometry” are too narrow – even misleading; for as yet no actual metrical space, nor measurable forms have been created, and even when they come about, it is an exception rather than the rule to find them expressing the rigid measure of geo-metry. It is a realm in which the archetypes of all manner of spaces and forms are to be found. Perhaps one is here permitted a glimpse into eternity – into the time when God created the world, and man in his own image.

Out of an infinitely free potential of rhythmic and polar interplay between three archetypal elements, the many metrical space-concepts and the geometries therein contained may be derived, by planting in the projective space an invariant Absolute, by relation to which the prevailing measure will arise.

 

26 – The Three Fundamental Measures

In the so-called “projective determination of measure”, there arise to begin with three basic types: hyperbolic, parabolic, and elliptic. We, using simpler language, will call them Growth-measure, Step-measure and Circling-measure. For the purposes of this book, we will approach the ideas concerning the different measures as simply as possible, and will refer the reader to the Notes and References and to other writings for a fuller and mores systematic approach. ²⁷

Step-measure. For Euclidian space – that is for the familiar space of Earth, which we experience in everyday life – the Absolute is an infinitely distant plane, bearing an “imaginary circle.” ²⁵ Euclidian space is a three-dimensional projective space, in which this plane and circle are kept fixed. The resulting measures are parabolic, that is to say, of equal steps (arithmetical progression).  In physical space, we take a ruler, and then repeat the process again and again. Ideally, we could continue this repetitive process ad infinitum. The laws of equidistance, of parallelism and of translational movement are due to the infinitely distant plane, and the basic forms of the right-angle and of rotational, circular movement are due to the invariant imaginary circle. ²⁵

As we have learned, projective geometry treats as equivalent forms which are capable of mutual transformation by perspective, or by a sequence of perspectives, and does not regard the planes or lines at infinity as being different from any others.

In constructions of perspective, we put “vanishing points” on “vanishing lines” bringing into the actual picture what is otherwise ideal, namely, the infinitely distant line of the plane; this line represents the horizon from any assumed point on the plane. The same applies in the construction of plastic (three-dimensional) perspectives – as in the theatre, or in a bas-relief – where there will then be a “vanishing Plane”, representing the infinitely distant plane as such.

In the projective plane in Fig 43, the chosen Absolute is the horizontal line, on which there are four points from which the lines ray forth to create a network of quadrangles side by side. Once one quadrangle has been freely drawn, as for example in Fig 44, all the others arise as though of their own accord. Indeed, one must realize that once the weaving process has been set going, the whole net is already there, throughout the whole plane – invisible, until it has been drawn! The lines weave throughout the whole plane, above the horizontal line as well as below, tending towards it, as in a perspective picture, no one quadrangle having the same measurements as any of the others. It is called a harmonic net; there is evidently a hidden law at work, which maintains order throughout the plane. ²⁸

Now bring movement into picture: let a point move, say the one to the left on the horizontal line. Then all the others on the whole network – will move accordingly, changing to outer appearance, but always remaining true to the original idea. It will be interesting to try out what will happen, if one of the points move to the infinite point of the horizontal line. Surprisingly enough, it will then be found that the net becomes more regular; a step-measure will appear on all the lines raying in from the point at infinity! Rigidity has begun to set in.

Fig 45 shows what happens when the whole (horizontal) line has moved to the infinite. The Absolute now is the invisible, infinitely distant line of the plane, and the figure reveals the rigid step-measure, which results when the whole line, with its four points functioning as the Absolute in the projective picture, moves to the infinite! At this moment, the Absolute is the invisible line-at-infinity of the plane. From this it can be seen that the vanishing scale of measure in Fig 43 is a picture of step-measure in perspective. 

Fig 46 shows the similar process in three-dimensions; here a plane containing an archetypal triangle functions as the Absolute for a three-dimensional form. From the points and lines of the archetypal triangle, the lines and planes ray out, crating the three-dimensional form in three-dimensional projective space. It is in all respects, apart from measure, a cubical form (parralelopipedon).

Allow the archetypal plane to recede to infinity and, geometrically speaking, the elementary crystal lattice is such a harmonic network, formed from an invisible, archetypal, harmonic pattern in the plane-at-infinity of Euclidian space. The projective picture allows us to see what is really happening ideally, even in the actual parallel-sided form, such as a crystal. ²⁹

See the cubic forms one beside the other, as was done in the two-dimensional picture of quadrangles, and the ideal picture revealing the quality of Euclidian space become evident to the mind. It is a space capable of being entirely filled with forms side by side, stretching away to the infinite distances, with no particular centre – one might also say, where each single form can be regarded as the centre. This space has many centres and only one infinitude, the archetypal plane. For the cubical form there will be other patterns in this archetypal plane.

Here is revealed the true nature of Earth-space, with its repetitive step-measure and its finite forms, here is the old Euclidian idea, but with the addition in thought of the infinitely distant elements, the existence of which Euclid denied. It is as though Euclidian space itself were a texture of woven light, or were shot through with countless textures of this kind, formed from the plane-at-infinity, and every crystal a partial embodiment of such a texture, rhythmicized in its own individual measures. In effect, the measure that prevail throughout the mineral kingdom and that dominate the crystal form with its internal lattice, derive in the purely geometrical sense from the infinitely distant plane of our familiar space. More than that, the crystal form itself is a projective construction, with its archetype in yonder plane.

This is the inner reason why crystallographers portray the crystal types by projection on to a surrounding sphere. The latter has no particular radius; it is in fact a finite image of the infinitely distant. Upon this finite sphere we draw an image of the infinitely distant archetype. To form the crystal, geometrically speaking, it as though the archetype sent its rays inward. Potentially, it could fill the whole of space with its crystal lattice. Where there is mother-liquor to receive this in-forming, the potential form is materialized and made visible to outer sight.

The significance of these discoveries – essentially, they date from about a hundred years ago – can surely not escape unbiassed thinking about Nature. But it has been the fashion to suppose that all scientifically real thought-forms must be derived by abstraction from the properties of tangible, sense-perceptible objects. Hence, if a valuable thought-form, such as these infinitely distant or even imaginary elements, is obviously not of this kind, the tendency has been to regard it as a mere arbitrary definition or convention, – a mere word we find convenient to use in a fictitious sense.

This way of thinking was no doubt natural to the materialistic realism of the nineteenth century. Today there is every reason to go beyond it.

Growth-measure. To make the transition in a simple manner from step- or parabolic measure to growth- or hyperbolic measure, all we need do is to bring in the idea of forms one inside the other (Figs 47-49), instead of forms side by side. In so doing, the circle and the sphere are close at hand.

Take a square and inscribe within it, for example, a circle; or take a cube, as in Fig 35 and inscribe in it a sphere. In the projective picture, freed from the rigidity of the right-angle and the element at infinity, the harmonic net will reveal forms one inside the other, as in Fig 47 (in two dimensions) or in Fig 49 (in three dimensions). Here the octahedron arises in relation to the cube.

The measure arising in this type of net will be of steps growing – or diminishing – in such a way that two adjacent steps will always shows the same proportion (geometrical progression). We call this “Growth-Measure”. It will be seen that there are here two infinitudes, for the sequence progresses outward towards the infinitely distant line – or plane – and inwards towards a “star point” – a point functioning as an inner infinitude. In Fig 47 the forms close in towards such a point within and open out, to flatten in the last resort from both sides into the horizontal line.

Once again, what wee see here is a perspective picture of what in ordinary space would be a logarithmic process – a growth-measure taking place between an innermost point functioning as an infinitude, and the line or plane at infinity of space. In Fig. 49 the perspective picture does the same for a sequence of three-dimensional forms; cubes and octahedra would alternate one within the other in polar fashion, between the innermost infinitude and the plane at infinity of space.

It is important to appreciate the difference between step-measure and growth-measure, in that in growth-measure we have to do, not only with one, but with two infinitudes; here the measure runs between the functional infinitude within and the actual infinitude without, the latter is same  as the one distant infinitude upon which step-measure depends.

Circling-Measure.  The third basic measure is circling-measure, and this plays a part in the realms of both the other measures. Consider a family of concentric circles and their radii, as for example, in the top left-hand drawing in Plate X. Here there are two possibilities : either we may follow the growth-measure of the circles inward or outward, or we may go round and round in the measure given by the radii; this latter is a circling-measure of lines in a point (the point-at-infinity without). In Fig 50 we see the same circling-measure projectively portrayed along a visible line; were the whole family of curves to be drawn in, we should see a projective transformation of the concentric circles, which would look something like the curves in Fig 22.

PLATE X

In circling-measure, the archetypal principle of the rounding off of spatial forms comes to expression. In its physical, spatial manifestation, it appears in its purest form in the plane as a circle, in in the point as a circular cone and in space as a sphere. It is related to everything which circles or makes vortices in the universe, and also the the archetype of everything which is closed in upon itself and has come to rest.

Every living being tends in some way towards a rounded off and finished form. It is perhaps natural that the most primitive living forms tend in their visible spatial body towards the sphere.

This principle, too, arises to begin with out of the free Archetypal Space and not in the rigidity of Euclidian measure. Between the extremes of expansion and contraction, together with the circling principle, all forms come into being. Unite the growth-measure with the circling-measure, and those forms arise which are most ideally balanced between the two principles, namely the spirals. On the basis of the growth-measure  in Fig 48 the spiral in Fig 51a arise, where the radial as well as the circling component is brought into play. There arise spirals, which when drawn out in their continuous form are to be seen in Fig 51b. According as to whether the radial or the circular component dominates, the spirals will be less or more extremely curved. These spirals run between the two infinitudes: the innermost, which they never reach, and the outermost line-at-infinity which is also physically unattainable. Based on growth-measure, these spirals are called logarithmic, or as theory are often called in a more metaphysical sense the “Spirals of Life”. ³⁰

Spirals do arise also in the interplay of step-measure with circling-measure. take, for example, a family of concentric circles, getting larger in step-measure from a central point. Here the circles, together with their radii give rise to spirals which have no inner infinitude. These are the so-called Archimedian spirals, which run from a finite point out to the infinite and are qualitatively very different from the logarithmic spirals.

The grow-measure spiral, or spiral of life, has long been associated with living forms, especially the plant world; it come to expression also in water, upon which all life depends. It represents the rhythm of the nodes up and down the stem of the plant, in the spiral phyllotaxis. We see it pictured again and again in nature, most often in the growing-point of a plant, as in Fig 52, where it is always to be found, as though telling of the origin of the tiny leaf primordia. We see it frozen into the forms of shells, left behind like a record of its life by the watery organism.  (Fig 53).

The difference between the Spiral of Archimedes and the Spiral of Life is fundamental: the former with its one infinitude and its repetitive step-measure (addition), the latter, as in Fig 51, spanned between two infinitudes, with its proportional measure (multiplication). Circling measure belongs to both realms; both the Archimedian spiral and the logarithmic spiral circle round and round, inward and outward. In the former, however, there is a “dead end” in the centre, while in the Spiral of Life the curve continues on and on inward, just as it goes on and on spiralling outward. it spirals in towards a point, which has the same quality as the infinite periphery towards which it spirals out; this point plays the part of an innermost infinitude. We have called such points “functional infinitudes” because , although in ordinary space, they function as though at infinity.

Unlike the inorganic forms, all living forms have an inner realm as well as the outer one, and they live their life according to rhythms, which play between the two. We touch here upon the difference between “beat” and “rhythm”. Beat  is the mere repetition of sound at equal intervals, while rhythm is spanned between infinitudes.

The logarithmic spiral is beautifully portrayed in the distribution of florets in a composite flower, such as the Sunflower in Fig 54. It is pictured in a more spatial form in the Italian Cauliflower in Fig 55. It is to be seen in pine-cones, in the horns of animals, in bones, in muscular organs, such as the heart, and in the forms made by water, air and warmth. Indeed, this curve and other akin to it are dear to Nature’s hart (see §46 and Note 64 – [both to be published in succeeding chapters on AF soon]).

 

Gallery with All Plate Images from Beginning of the Book

Gallery with all images from Chapter III

Chapter 1: The Language of Plants

Chapter 3: The Polar Forms of Space

Chapter 4: Physical & Etheric Spaces

Chapter 5:  Ethereal Space of the Plant Shoot

Chapter 6: Staff of Mercury

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