Regenbogengruppe Anhängsel

‡ Folgendes hat anthroposofische Einschlüße ‡

[Walther Buehler, M.D].

My first talks on the Golden Section [= golden ratio if the ratio of the sum of the quantities to the larger quantity is equal to the ratio of the larger quantity to the smaller one.

The golden ratio is an irrational mathematical constant, approximately. = golden section (= sectio aurea) and golden mean/= medial section/= divine proportion/= divine section

(Latin: sectio divina)/= golden proportion/= golden cut/= golden number/denoted by the Greek letter phi].

Walther Buehler Pentagramm

Goldenes Rechteck und Dreieck

Ein Rechteck, dessen Seitenverhältnis dem Goldenen Schnitt entspricht, bezeichnet man als Goldenes Rechteck. Ebenso nennt man ein gleichschenkliges Dreieck, bei dem 2

Seiten in diesem Verhältnis stehen.

Ein Goldenes Rechteck lässt sich in ein Quadrat und ein weiteres Goldenes Rechteck zerlegen. Durch wiederholte Teilung erhält man eine Figur, in die sich eine gewisse

logarithmische Spirale einzeichnen lässt, die Goldene Spirale, in der Architektur auch Volute genannt. Sie wird oft, wie in nebenstehender Abbildung, durch eine Folge von

Viertelkreisen approximiert. Ihr Radius ändert sich bei jeder 90°-Drehung um den Faktor Φ.

Goldener Winkel

Blattstand einer Pflanze mit einem Blattabstand nach dem Goldenen Winkel

Durch wiederholte Drehung um den Goldenen Winkel entstehen immer wieder neue Positionen, etwa für die Blattansätze in nebenstehendem Bild. Wie bei jeder irrationalen

Zahl werden dabei nie exakte Überdeckungen entstehen. Durch die Drehung um den Goldenen Winkel wird erreicht, dass die Überdeckung der Blätter die Photosynthese

behindernd, in der Summe minimiert wird.

Goldene Spirale

Die schneckenförmigen Kalkgehäuse einiger Tierarten haben eine ähnliche Steigung, wie beispielsweise das des Nautilus. Bei den meisten dieser Tierarten ist die Steigung

jedoch eher geringer.

Goldener Schnitt im Ikosaeder

Drei Goldene Rechtecke im Ikosaeder

Die zwölf Ecken des Ikosaeders bilden die Ecken von drei gleich großen, senkrecht aufeinander stehenden Rechtecken mit gemeinsamem Mittelpunkt und mit den Seitenverhältnissen des Goldenen Schnittes. Die Anordnung der drei Rechtecke heißt auch Goldener-Schnitt-Stuhl.

and the significance of the pentagram in the human organism.

Zeising found the golden ratio expressed in the arrangement of branches along the stems of plants and of veins in leaves/extended his research to the skeletons of animals and the branchings of their veins and nerves, to the proportions of chemical compounds and the geometry of crystals, even to the use of proportion in artistic endeavors. In these phenomena he saw the golden ratio operating as a universal law/this discovery is not given the recognition it deserves. He found that the Golden Section was the key to the human form sought by artists. Anatomically, this is a truly Goethean approach that enables us to see every part in the light of a greater, ideal whole. This is a classical example of taking the purely analytical method, in which things are taken apart, a method that also has its justification but takes us away from the spiritual and creative aspect, and being able and compelled to bring in a different, holistic approach. Then, the image of man that sustains us will not be wholly lost. On the other hand, realization that the human form is wholly organized and differentiated in the "divine proportion" can make us aware of this as a direct expression of the work of creative etheric powers. The life body as "architect of the physical body" (R.S.) has five main currents running through it that combine in a pentagram. "These five currents lie hidden in the human being; the healthy ether body presents in such a way that they are its skeletal structure, as it were“. The pentagram is the archetypal form of the Golden Section, an important bridge linking the study of physical objects with an approach in which the etheric aspects are considered at the level of images.

The question as to the nature of the Golden Section as it presents in the realms of nature, in the planetary system and in the human organism has occupied the writer's mind throughout life. With a certain inner inevitability, it ultimately led to the question of the measure of the rainbow. The answer will be pursued in what follows.

R.S. gave a meditation that encourages us to consider the threefold plant and trace the three fundamental processes known as Sal, Mercury and Sulfur. These create a link between medicinal plant and threefold human organism.

As we consider the mercurial processes that mediate the polarity between light-related flowering impulses and earth-related root powers, we discover a connection with the morning hours. anthro.

The meditator is asked to pay special attention to the dew drops that line the margins of leaves like bead necklaces, catching the light of the sun. The observer has to find exactly the right position relative to the incoming ray of the sun if the glitter is to change into the enchanting sparkle of all the colors of the rainbow, bringing to mind the pure colors of precious stones. Often, the whole color spectrum will shimmer in the carpet of tiny droplets covering the surface of the leaf.

Every drop is a reflection on the small scale of the macrocosmic world and its spherical bodies. In terms of physics, it reflects part of the light which is passing through it in the form of a circular cone of color, which is due to diffraction, refraction and reflection. This delicate "light flower" holds all the colors of the spectrum. Deviation from the central sun ray is always approx. 42°. The observer, his back to the sun, must be within this angle, looking at the droplet from the left, the right or from above if his pupil is to capture the play of colors.

Here, we see on a small scale a process that happens thousands of times as a rainbow is created. When the tremendous abundance of streaming raindrops crosses the stream of light from the sun, millions of those color cones combine to form a flowing sea of color in the atmosphere. The human eye takes from this the great arc of color that rests in itself in majesty. A kind of reversal occurs, creating a "macrocosmic" counter image of the small droplet flower which is its polar opposite. If this reversal of rays reaching the human eye were visible to us, a half cone of color would appear, its tip touching the eye. Every human being sees only his own rainbow, which belongs to him, and this shows the special relationship we have to this marvelous natural phenomenon.

The relationship to the colors that develop in tiny droplets can also be seen in the proportions of the rainbow on the macrocosmic scale. From the center of the arc, which is always directly opposite to the sun, every point on the outermost red margin is at an angle of 42° to the central line. If the sun is relatively high up in the sky, we do not, of course, see the whole semicircle but only part of it. The fact that the rainbow only appears in its full glory at sunrise and sunset reveals its special relationship to the hours of morning and evening. What follows must, therefore, have its beginning in these ideal moments.

The rainbow may be seen to play a mediating role in many natural processes. It appears between light and darkness, needing both the cloud which obscures and the brightening blue sky. Belonging in the 6 hours of morning and evening, it is essentially connected with the sun when it is in the East or the West.

The chapter on northern lights, lightning and the rainbow shows in detail how the harmonious sequence of colors and the whole mysterious phenomenon of the rainbow hold a middle position between the vehemence of lightning striking the earth and the evanescence (= dahinschwinden) of the northern lights. These arise in the hours of noon and midnight respectively and relate specifically to Summer and Winter. This takes us to the threefold light organism of the earth, where the flashing belt of tropical thunderstorms is connected with the northern light crowns in the polar regions through the two semicircular bands of color that span the globe at dawn and dusk. The day-side of their mercurial zones is the home of all the rainbows; these vanish as the sun rises higher towards noon.

The next step was to see if a relationship exists between the rainbow and the unique ratio of the Golden Section. We know that however much this is divided, every part retains its proportional relationship to the whole. In abstract mathematical terms and relating to a line segment, this means that on asymmetrical division, the smaller part (the minor) shows the same proportional relationship to the greater part (the major) as the latter does to the whole, undivided segment. The same relationship may be seen in areas and in angles. Works of art (buildings/sculptures/paintings) created in these proportions always look harmonious and well balanced.

In the natural world, we find the Golden Section in the arrangement of the leaves on the stems of many plants, i.e. in the mercurial region of the shoot. Zeising was the first to point this out. Even in Goethe's day, botanists had noted that the spiral sequence of leaves moving up the stem and their rhythm involves specific angles that are always greater than 90°. At the Sorbonne in Paris, the Bravais brothers measured the angles of divergence in hundreds of plants and found the average to be 137° 30° 48° This seemed a chance result, requiring no further discussion.

Braun and Schimper, german botanists studied the spirals and rhythmic arrangements of scales on the cones of coniferous trees. They found a geometric sequence of "leaf cycles" which governed all kinds of different plant species. If phyllotaxis was in the 3/8 cycle, for instance, the denominator of the fraction tells us that eight leaves make a cycle, with only the ninth leaf pointing in the same direction as the first. The numerator indicates that the leaf spiral goes 3x round the stem. The whole fraction shows that the next leaf on the stem will never appear after one eighth (45°) or two eighths (90°) but only after three eighths of 360° (= 135°) around the circle. This holds true for the Cruciferae.

The laws of phyllotaxis may thus also be put as follows:

1. The number of turns in a leaf cycle always relates to the number of leaves as minor to whole;

2. The same proportion is seen between the angle of divergence for two consecutive leaves and the whole stem circumference;

3. In their number of leaves per cycle, the different plant species present a regular series relative to each other, with every less complex one showing the minor:major relationship to the next higher in order of complexity, thus uniting with it in a proportionally-ordered whole, and being part of a continuous scale of proportional stages with all that precede and follow it.

Zeising put an end to a scientific dispute when he made his fundamental discovery. Until then, German botanists had accused those at the Sorbonne of having fallen into mathematical and intellectual abstraction. The scientists at the University of Paris, on the other hand, felt that the cyclic system used by the Germans was a hypothesis, rounding up to whole numbers, with romantic overtones, relating to angles that were within the range of the variation seen in the sphere of life and its sources of error. The Golden Section provided the bridge that united the two factions. With its inner mercurial quality it proved a peacemaker.

Zeising went on to explain:

To grasp and understand the truth and living reality of the natural world, we must above all find a law that offers the transition from one opposite to the other and allows us to see that the law itself is the source-spring of freedom, with deviation from it merely a consequence and further effect of a law that takes manifold forms. This requirement is, however, met in a better way by our law than by any other. The law reconciles the idea of unity and uniformity on one hand with that of variety and manifoldness on the other, taking us from the realm of the finite into that of infinity, where it bases itself on the firm measure of a whole that has definite limits and by an infinitely subtle division of the whole which is capable of being continued to infinity, shows that within this finite element, infinity, too, is to be found.

The discovery of the Golden Section function in the sphere of the plant shoot shone like a ray of light in the midst of the darkness of the 19th Century, a presage of the age of light that was to come.

To illustrate his approach, Zeising produced two drawings, among other things, in which the 360° of the circumference are divided to show the main angle of the leaf sequences.

In Figure 3, the 136.5078 degree angle ABC represents the minor relative to the circumference; it is the angle of divergence the plant seeks to achieve in its phyllotaxis. The larger residual angle of 222.4922 degree is the major. If the leaf spiral goes in the opposite direction around the stem (Figure 4), the leaf at D, which follows A, is, again, at an angle which is in the Golden Section relationship. It is in the nature of the Section, however, that as soon as the minor is positioned within the major, this is, in turn, divided in the same proportion, with no need for tedious construction. The new third 85 degree angle DCB is the minor of 222 degree.

Readers will have realized that the proportions that determine the rainbow are also appearing in the plant world. Zeising produced the key to the rainbow more than 130 years ago, though he was not aware of this. Think of a leaf growing towards the rising sun on its shoot. The next leaf, feeling its way around the horizon, as it were, seeks the point, in its angle of divergence, where the rainbow meets the earth. We might also say it grows into the world around it following the laws that are reflected a thousand times in the beads of dew which adorn the leaf.

The same proportions appear if we consider the circumference or horizon as a straight line AB and divide this so that AC and CD are in the Golden Section ratio. The smallest new minor section CD would mark the measure of the rainbow. This division occurs at the intersections of every diagonal in the pentagram. The intersections recreate the original pentagon in inverted form, the length of its sides corresponding to the diameter of the rainbow.

The rainbow developes when opposite extremes come together in the elements. (Ianus) in width and height, it is a mercurial, natural mediator; its proportions relative to the periphery are indissolubly bound up with the laws of the "divine proportion“.

Light is processed in every dew drop as it is in a raindrop. Quietly observing the play of colors, we unconsciously experience the same laws in the interplay of the elements that also govern etheric activity in the plant, letting leaf slowly follow leaf up the stem in rhythmic sequence.

This unique interaction between circumference and center, the fleeting play of light created by the sun and the organic forms created in life, stirs the human heart and mind at its very depths. It addresses the same laws in the meditator that out of the ether body organize him in terms of the pentagram. The sensual and moral feeling corresponding to the "leafy morning" loosens the body of creative etheric powers. It prepares the way for new faculties of imaginative thinking, taking hold of the secrets of nature. "We must find ourselves again and again in the rhythm of nature if we want to gain living rather than dead knowledge.... If you thus direct your constant attention to these wonderful secrets of nature, you enliven your medical knowledge in a practical way“.

A German proverb says that the morning hour has gold on its lips. Perhaps the Golden Section has been given that name for good reason. The sun is the cosmic source of the metallic gold process, in the 6th lecture of the Course for Young Doctors, R.S. referred to its mercurial harmonizing function. In the planetary system, where the Golden Section is one of the constituent principles, the sun creates the balance between spirit and matter by virtue of its position which goes beyond all physical phenomena. "Because of this, the sun is a cosmic body both maintaining order in the planetary system and establishing order among the forces that enter into our material system“. This cosmic heart function may be seen as reflecting the activity of the Spirits of Wisdom who endowed the developing human being with an ether body during the Old Sun stage of earth evolution, thus raising him to the plant level of existence. In the cosmic intelligence and the activity of the Kyriotetes, we get a sense of the original source of the process that relates the part to the whole, a process we can perceive wherever the "divine proportion" presents itself.

Vorwort/Suchen Zeichen/Abkürzungen Impressum