Plant Growth and the Forms of Space by George Adams and Olive Whicher

Plant Growth and the Forms of Space

by George Adams and Olive Whicher

An article from the 1950 Edition of the Golden Blade

An Anthroposophical Journal

https://www.waldorflibrary.org/images/stories/Journal_Articles/GoldenBlade_1950.pdf

 

Many of Nature*s secrets are “open secrets.” The truth is manifest for all to see; yet generations of mankind, even epochs and civilisations can go by, and it is not seen. For man beholds reality not with the outer bodily senses only, but with the spiritual eye of the mind. The outer eye may be focussed howsoever sharply, conveying to the consciousness ever so accurate, a photographic picture of the external object; yet it will stare blindly and without understanding until one day the mind becomes imbued with the ideal concept belonging to the phenomenon in question and in some way revealed in it. Then and then only do we begin really to see.

Look at the growing plants in spring or early summer. See, in infinite variety of form, the characteristic gesture at the growing point of the vegetative shoot. The young and tender leaves appear to open out as if between them they would harbour something of great value in the hollow space immediately above the growing point of the stem. Plants such as woodruff bring forth a whorl of leaves at every node. The young leaves reaching upward form a hollow cone above the growing-point, slender and deep to begin with and opening out as the leaves grow older. So long as the vegetative phase of the plant’s life continues, the hollow space above the growing-point is maintained, node upon node bringing forth the leaf-buds. They first appear with this enveloping gesture and thence open and spread out towards the horizontal plane, which often tends to be the ultimate position of the mature leaf.

In the great majority of those plants which form but a single laterally placed leaf at every node, this tendency to form a hollow cup or cone at the growing-point, with the enveloping gesture of the young leaf-buds around it, is still in evidence. Here the leaves follow one another in some form of spiral sequence. Generally the growing-point the successive nodes have not yet opened out the young leaf-buds of several nodes form more or less of a spiral whorl, enclosing once again a hollow space within. Even a single leaf will often shew the enveloping and protecting gesture very eloquently.

Some hiterto underknown reality underlies this rather childlike impression which the mind receives from the gesture of the growing and unfolding plant. What is the essence of this hollow space enveloped by the young leaves above the growing point of the stem? Where shall we find the concepts which will help us really to see this process with the inner eye of understanding?

Physical science, since Faraday and Maxwell, is not unfamiliar with the notion that in a seemingly empty space there may reside the potential essence of those forces which the behaviour of visible objects will under given conditions make manifest. We speak of “fields of force.” We think the unseen nature of the field of force between two charged conductors accounts for the reactions of the latter, or of other objects introduced, when these potential forces are released. We find the explanation not in the tangible objects, but in the space between them. Thus it is not inconceivable that in the seemingly empty space lives a reality, unseen at first, determining the behaviour of what we see.

But we must ask: If this be so for the growing-point, what will its nature be? The image of an electrical field will not help us here. If in the hollow space between the unfolding leaves there is an unseen sphere of force, we might be tempted to think of some pole of force at the centre, radiating outward like an electric or magnetic pole or the centre of gravity of a material body. But the phenomena do not indicate anything of this kind. In pure geometry and mathematics, however

In pure geometry and mathematics, however, it is possible to describe not only points of the kind, which occur in physics as electric or magnetic poles or centres of gravity. “Singular points” of many kinds occur in geometry and in the theory of functions. Now, from elementary projective geometry we learn that every plastic form in space—every curve or surface—has a dual aspect. The surface of a sphere, for example, is not only the sum-total of all points at a fixed distance from a given centre. To describe it thus is only one aspect; in a qualitative sense it represents only half the truth. For at each point the sphere has also a tangent plane.

It is enveloped by its tangent planes, and we need only assign the right law of movement to a moving plane, to form the sphere as an infinite
assemblage of planes no less accurately and completely than by the usual definition, where we are really forming it as an infinite assemblage of points. Modem geometry has learnt to think of all forms in space, even of the very structure of space itself, in the planar as well as the pointwise aspect.

Known since a century or more ago – its depths and beauty recognised and much admired by the pure mathematicians—this form of thinking has found very little direct application in mathematical physics, for the simple reason that the fundamental entities of physics are pointlike, which is another way of saying, atomistic. We shall discover, however, that it has its application in the sphere of life—in the morphology and physiology of living things Nature reveals in her phenomena the planar and not only the pointwise aspect of space. For the planar aspect of the sphere, the elementary entities will be the tangent planes that touch and mould it from without. If we choose a finite number from among them and in a regular sequence or pattern, they will reveal a gently enveloping and enclosing gesture, like the petals of a half-opened flower.

In visible Nature, the ideal entities of pure thought are but approximately realized. Every particle of matter is pointlike, yet it is not an ideal mathematical point, however finely we grind the matter down. That this is so, makes the ideal conception of the point no less significant for physics. Where it applies, it applies exactly, not as an approximation but as an ideal key to the under lying law. The centre of gravity of a chair is in most instances not even a point of the material chair; it hovers in mid-air. Yet the dynamic behaviour of the chair can be described exactly by reference to it and in no other way.

Thus the notion of the point and of the pointwise properties of space (such as the relations of distance between points) is essential to an understanding of the realm of matter, yet this does not depend on the ideal point being realized in any material particle. And that the particles of matter, e.g., the drops of an emulsion, approximate to ideal points, is like a hint, an indication, of the kind of ideal truth
prevailing in the inorganic laws of Nature. To use an old form of expression, it is in the nature of a “signature”— a ‘signatura rerum‘.

So too, if it be true, as here suggested, that the plane prevails in the ideal space-formation underlying the characteristic phenomena of life, we shall not expect any exact reproduction of the plane in the phenomena. We shall have even less cause to expect it than with the point and the particles of matter. For to our outer senses only the material is visible. The “more than matter” in a living thing can reveal itself to our senses only by the character which it imparts to the material living body. And if this “more than matter” is in polar contrast to the realm of matter, as in pure geometry the plane is polar to the point, it stands to reason that the material entity will be able only to reach out, as it were, to receive it; will be able to manifest it only partially and transiently. It is an apparent truism that the whole of a point is there wherever the point may be. The geometrical plane, on the other hand, is of infinite extent; in the material world, therefore, only an infinitesimal part of any plane can be represented. The planar element in the visible form may nevertheless become for us a key to the understanding of just
that aspect of the living world which is not fully contained in or explained in terms of physical substance alone.

Profound and beautiful as the discoveries of the last century or two in pure geometry have been, it is at first not easy to see how they apply in practice, nor is it easy so to adapt our mental habits that the new insight thus gained becomes second nature. Our own self-consciousness is very much concerned with the point. A man must in any given moment be at a particular point in space; what he achieves in the outer world must at each moment radiate outward from the point where his body is. Unconsciously we project this spatial sense of our own entity into the material or quasimaterial entity whose movements and reactions we are conceiving, be it a visible body, animate or inanimate, or an imagined one—molecule or atom.

Now let us try to conceive a world in which the fundamental entitles are planar—in which, if real beings did exist, they would be plane-like instead of pointlike. They would, if gathering into themselves with the feeling “I am here”, be doing spatially the very opposite of concentrating. They would expand with the greatest possible intensity throughout their plane. We have no word to describe it: expansion with the quality of concentration, focussing not in a point but in a plane, finding one’s centre in the very opposite of centre. To enter such a realm not only imaginatively but with exact thinking—giving the several forms of thought the value that properly belongs to them in this domain—we must be able “to think the extensive intensively and the intensive extensively.” The words are Rudolf Steiner’s.

If then a realm of this planar character really exists, finding expression in the phenomena of living Nature, the material body of a living creature will be able only to reach out into it. Matter will have to use its properties in such a” way as to deny itself, in a manner of speaking, bringing to manifestation what is the very opposite of matter. Among the fundamental properties of matter are concentration about a point and movement of that point along a line. Matter will thus have to rise or in some way detach itself from massive Earth, its particles moving along lines which branch or ramify in such a way as to expand into a surface—to simulate the unbroken unity of a plane or of a plastic surface in its planar aspect.

This is precisely what we behold in the plant, for so the leaf comes into being as a material form. The watery sap rises through the veins, which branch and ramify as if into a pre-determined planar surface, and by repeated anastomosis sustain and supply the intervening cells, making the delicate and continuous expanded leaf-forms. The limitations of matter will enable it to do this only up to a certain boundary. Physically speaking, only a tiny portion of each plane will be made visible. Yet in the gesture and in the play of forms and forces it will be revealing. To the unbiassed eye it will tell its story. For such is Nature; in the most delicate ways, to the aesthetic sense she reveals the truth. Her phenomenon, as Goethe said, is theory, if only we can find it.

We have suggested that there is real significance in the hollow space enveloped by the unfolding leaf-buds at the growing-point of a vegetative shoot, or again by the petals of a half-opened flower. We have applied the geometrical picture of a sphere enveloped by its tangent planes. But we have now to look deeper. What will be the essence of such a sphere? What kind of forces will be prevailing in and about it? What kind of function shall we attribute to the centre or ideal focus there within it? Here once again, modern geometry will lead us forward, if with courage and imagination we take its thought-forms with us in our perception of surrounding Nature. For this geometry not only sees the surface of the sphere formed of its points or of its tangent planes. The very presence of the sphere gives rise to an all-prevailing relationship of points and planes throughout the space in which the sphere is hovering. This relationship is in the nature of a polarity—in the mathematical and also in the Goethean sense of the term.

To every point within a sphere there will be found a corresponding “polar plane” outside it, and to every point outside, a plane that traverses the inner space. The “polar plane” of a point on the sphere surface will be the tangent plane at that point—a picture we have just been describing. To points farther in towards the centre belong planes farther out towards the periphery of space, and vice versa. The points and planes as they move inward and outward follow the simple numerical law of inverse proportions. This leads to the conception that the infinitely distant periphery of space is indeed a plane, containing all infinitely distant points. Even as there is one innermost centre where all internal points will come together, so there is one infinitely distant plane into which all distant planes, as they go out into the infinite in all directions, will ultimately merge.

It appears, therefore, that a given sphere not only has the dual aspect of its outward and visible form, in that it can be formed of points or of planes alternatively. It has a far more mobile and organic function extending throughout the whole of space. It generates an all-pervading polarity of point and plane, wherein the innermost corresponds to the outermost, the most contracted to the most expanded. According to this polarity, the radial arid outward movement of points or pointlike entities from within the sphere will suggest the hovering inward of planes or planar entities from the infinitudes of space.

When we contemplate a sphere from this point of view, we shall no longer say so categorically that the space within is of finite volume, while that without is infinite and measureless. Rather shall we be ready to alight in one or the other direction according to the problem in mind. For physical matter it is quite true that the space within the sphere is finite; a given sphere will contain only a finite quantity of matter, according to its volume. The space without, on the other hand, containing the “plane-at-infinity”, is infinite. This is because physical matter is at home in the space of Euclid, of which the plane-at-infinity is a determining factor. In the space of modem projective geometry however—or in different kinds of space to which it can give rise—it is quite logical to conceive a sphere that is filled, as it were with “planar substance ,
from without inward, a sphere into which “planar forces may pour from the periphery of space. To such a sphere we have not to apply the metrics of Euclid. It is only for the latter that the outer space is categorically infinite, while the interior is of finite volume. Even the very concept of a measured volume may not apply.

A geometrical space is a pure form of thought beheld in the eye of the mind. The spatial concept which enables us to penetrate with understanding any of the phenomena of Nature evidently bears relation to the ideal realm to which these phenomena belong—the realm which they make manifest to our senses. Euclidean geometry has been our guide in penetrating to the ideal reality of the phenomena
of inorganic Nature. Another kind of geometry will provide an essential key to the phenomena of life, where inorganic matter often seems to rise beyond itself, and where the characteristic spatial forms and gestures are so very different.

To return to the question from which we took our start: What is the nature of that intangible sphere or hollow space of seeming emptiness above the growing-point of the green shoot, or in the hollow chalice of the flower?

We are suggesting that even as the phenomena of inorganic matter belong to the space of Euclid, – the infinitude of which is an infinitely expanded sphere or “plane-at-infinity”, so the phenomena of life reveal, intermingled with the other, a polar opposite type ‘of space: a space for which the infinite is innermost instead of outermost—a single point at the very heart of the living entity instead of an infinitely distant plane. Into such a space the “planar forces” of the Universe will spend themselves. In the region of the “infinitude within”, where there is very little matter or even none at all, that which is “more than matter”, the life which indwells the form, will be great. Here the created form will arise as though coming into being from an infinitude of life as yet unborn, in physical volume minute at first, yet vast in terms of ethereal vitality. Vivid and tender in its young life, as the leaf grows older and less vital it attains to greater physical perfection, and in its form bears witness to the planar world from which it emanates.

We have now to form some conception of the “planar forces” which will be prevailing in this other form of space. This will be the type of force of which Rudolf Steiner often indicated that before long it should become known to science. It may be called ‘negative gravity”, or “active buoyancy”, or simply “levity.”

With this, our thought will come to meet the mysterious phenomenon of the upward growth of plants, which for the Earth-planet as a whole is the primary manifestation of all life. It is to this day a vexed problem: How is the sap able to rise in the trees in spring time, even beyond barometric level? We need not assail the various theories which have been put forward to account for it—osmotic pressure, capillarity, molecular cohesion. It is far more important that the quality of the thought-picture to which we shall be led in this “ethereal” realm is directly akin to the phenomena we see, just as the quality of pure thought in mechanics is akin to the visible form—beautiful for this very reason—of an engineering structure, such, for example, as a suspension bridge.

We shall indeed describe these forces very naturally as “ethereal forces”. Their formative activity will obviously be related to their dynamic quality. The individual form in which a living body grows will be very largely determined by their specific distribution. We referred to them just now as “planar forces”; this is explained as follows.

In the physical aspect of matter, not only is each material body point-centred, but the mutual forces between bodies act from point to point; so we have various centres of force—centres of gravity, electric and magnetic poles, and so on. In a word, we have not only geometrical but dynamic centres. The typical forces of the physM-material world act along lines from point to point. Joining two points, geometrically speaking, there is always a single straight line; if there is a mutual force between them, it acts along this line. Likewise in the ethereal or planar spaces: any two poles have a life in common. If they are parallel, or if one of them is the infinitely distant plane, their common line is infinitely far, like the celestial horizon in which the Earth’s tangent plane meets the sky. The mutual force between two planar entities of the ethereal spaces will act about their common line, tending to draw them together, which means away from the infinitely distant point of the ethereal space to which they belong. The outcome will appear in physical space as an outward soaring, an expansion. In quality, however,
the expansion will not be like the outward-thrusting force, say, of the fragments of an exploding shell; it will be a drawing outward—a with-drawing, we might literally say—with the character of suction.

We will suppose now that the Earth-planer as a whole is ethereal as well as physical. The planet not only has its own field of gravity; it has its own field of levity also. It is not merely made of inorganic matter. Earth as a whole is a living entity; the single plants that grow upon it are like the organs of a larger, more differentiated organism. Or, to describe it in the more anthroposophical language, the Earth has not only its physical body; it has its “etheric body” also.

We gain a perfectly clear picture of what the levitational field of the planet will be like if we imagine that the “point-at-infinity” of the ethereal space is at or near the centre of the Earth, and that the archetypal and most potent “plane of levity” is in the infinite sphere of the Heavens. We make this dual co-ordination: at the very place where the main centre of gravity of the physical forces is located, there is the infinitude—as it were, the ideal void—of the ethereal; while far away in the Heavens, in what for physical space seems like the infinite void, there is the primal source of the ethereal and planar forces which draw all other planar entities upward and outward, away from the centre of the Earth. To put the two, mutually  interwoven thoughts side by side:—