The customary classroom and related practice, is to explain the construction of the Golden Section as necessary for the construction of the regular pentagon. This seemingly innocent practice has contributed to the circulation of much nonsense, nonsense which is avoided if the Golden Section is situated directly within a proper reading of the simple construction-proof of the uniqueness of the five Platonic solids.
Lyndon LaRouche, On the Subject of Metaphor
In "On the Six-Cornered Snowflake (1611)," Kepler draws significant attention to the characteristic difference between living and non-living processes. The question of why snowflakes always form six tiny, iced feathers, no more and no less, gave rise to the question of why most flowers of plants and trees form five petals or a derivative of fivefold symmetry in the seeds of certain fruit and other living processes. However, at first glance, the geometry of these structures, whether living or non-living, is not enough to explain the causality of why they take their form. We humans, like some other living creatures, have five digits on our hands and feet. Why not six or seven? How can we know, as Kepler would put it, the relationship between the material necessity in the symmetry of either living or non-living processes and the principle guiding the material to take the shape that it does? The one thing we can be sure of is that we are the only living creatures on this planet that have the ability to even begin to answer these questions.
The purpose here is to begin to make clear how Kepler's idea of a life-principle, or soul, generally reveals itself to the senses in such a way as to be expressed in the best possible manner. For if creation was expressed in the worst possible manner, then perhaps the "material necessity" of both processes wouldn't really be "necessary" but arbitrary. Then, in that case, what does it matter if humans were created with seven or eight tentacles instead of arms and legs, or if cats were created with legs as long as a giraffe's?
One of the biggest problems in investigating the Golden Section is approaching it with the assumption that what Kepler calls "the proportion of mean and extreme ratio" is already known in advance. However, how does one discover such a proportion? Through trial and error? Accidentally? In terms of Kepler's regular figures, when one has discovered means of knowability constructing certain polygons, then naturally one may begin to investigate the magnitudes and their relationships found in the division of a circle, as the ancient Greeks did and as was elaborated in Book I. Could this perhaps lead to an understanding of the Golden Section? Though you may perhaps stumble upon such a treasure, its significance is not fully understood competently unless one purges the algebraic formalisms and numerological hoaxes from their inquiry. Let's begin with the nature of proportion on a more general level.
The basic idea of a proportion is the way in which one thing relates to another thing by the mediation of something that is neither of those two things. For example, a pond is to a lake as a lake is to an ocean. Now although the proportion may not hold true in terms of quantity, like buckets of water or square feet for a measure of the bodies of water, but the proportion holds in terms of their magnitude. In other words, to someone who has no concept of number, the concept of proportion could be communicated through their relationship in magnitude and of what they are, that is bodies of water. If you replaced one of the terms with another object like a mountain or a creature, it would change the relationship of the proportion and perhaps make the entire relationship disproportional. For how would a pond be to a little rabbit as a little rabbit is to an ocean? Such a relationship would be absurd.
So now we begin to see that a proportion exists because of relationships, not necessarily because of the individual objects themselves. This is so with numerical proportion as well. The proportions that are expressed in number and geometrical relationships will be elaborated on in Book V in dealing with the ellipse and also see Kepler's "Political Digression on the Three Means" in Book III. A brief account of geometrical proportion will be dealt with here.
Nicolas of Cusa, in his De Docta Ignorantia, provides a distinct qualitative difference between the polygon and the circumscribing or inscribing circle. No matter how many sides you add to the polygon, it will not attain an identity with the circle, for the difference between the curved and the straight is unbridgeable. Therefore the circular arc and the straight line have a relationship to one another which is absolutely distinct from any other relationship among straight lines. This relationship would later be called "transcendental". One could say that the characteristic curvature of the circle is primary to any linear or geometrical relationships it is bounding, transcendentally. How would you begin to create a proportion between something which is curved and something which is straight? Archimedes did after all find an approximate relationship to the circumference of a circle to its diameter. Does this bring us closer to knowing the ontological nature of the circle to its diameter, or does it just make certain calculations easier, for example measuring the sine of an angle to the circular arc it is subtended by? These questions are more ready to be answered in Books II, III, and V, where certain curvatures, like the circle, torus and cylinder, and the ellipse, have certain proportions about them, or rather potential for certain proportions.
To get a glimpse of this transcendental idea of curvature, let's look at the following animation which was created out of the investigation of the snub cube (see animation).
Start with a black square, a purple square, and four red diagonals. The four diagonals move to their respective corner, all the while shrinking in size due to the shape of the square. The purple square's corners divide each of the red diagonals in the same proportion as the red diagonals divide the side of the black square. This happens on all diagonals but in the animation, attention is only brought to one. So for example, length DP is to PC as length BC is to CA, always. As the diagonals move away, the purple square has to shift and change shape to maintain the proportional division and thus has its corner, point P, trace out the curve in green. What is this curve? Is this curve perhaps creating the proportion among the diagonals and squares, or is it the other way around, the proportion creating the curve? As you'll see in the characteristic of the catenary chain in past and future pedagogicals, the physical universe adheres to specific types of curvature, but ontologically, the curvature itself is being governed by a principle which is unsensed and primary to any proportional or numerical relationships that are expressed about the curvature. The traces of the Golden Section are begun to be revealed when approached from this ontological perspective.