The Fourth Dimension
GA 324a
Questions and Answers VIII
13 February 1913, Berlin
QUESTION: Is the Golden Section based on occult laws?
Because [Note 26] it is founded upon the effect of what exists in space, the Golden Section is indeed based on an occult law. Goethe said of this law that what is most hidden is most revealed and vice versa—namely, the law that is intimately related to our human constitution, the law of repetition and varied repetition. [Note 27] [Note 28] If you look at the Buddhas talks, for example, you find that the same content is always repeated with slight variations that must not be omitted, because the content is not the only important factor. [Note 29]
The golden section is not simply a matter of repetition. We repeatedly discover the same proportion, since there are actually only three components. [Note 30] The self-contained character of a repetition, which, however, is not self-formed, is what makes the golden section so appealing to us.
Notes of a question-and-answer session after a public lecture in Berlin in the House of Architects on "Lionardos geistige Grosse am Wendepunkt zur neueren Zeit" (GA 62).
Goethe's Das Märchen. See Note 3, Lecture 3.
For further discussion of the general occult law of repetition and varied repetition, see Rudolf Steiner's Outline of Esoteric Science (GA 13), chapter 4, "Cosmic Evolution and the Human Being." On the law of repetition as an elementary principle of the etheric realm, see, for example, Rudolf Steiner's lecture of October 21, 1908 (GA 107), where he illustrates this principle using the example of plant growth and points out the varied repetition in the ongoing process of leaf formation.
The significance of repetitions in the Buddha's talks is also mentioned in lectures Rudolf Steiner gave on September 18, 1912 (GA 139), and on the afternoon of September 27, 1921 (included in GA 343).
Fra Luca Pacioli (ca. 1445-1517), who was influenced by Piero della Francesca (1410–1492) and Leonardo da Vinci (1452—1519), wrote the paper Divina proportion (Venice, 1509) using drawings copied from his friend Leonardo. This paper was the first thorough study to focus on the mathematical and aesthetic characteristics of the Golden Section.
The golden section (sectio aurea), also called "constant division," results from dividing a line segment into two parts in such a way that the ratio of the smaller portion to the larger is the same as that of the larger to the whole. When we continue to divide a line segment according to the golden section, the result is a sequence of line segments such that the proportion between any two adjacent sections is the golden section. This explains the term constant division.
A further indication of the principle of repetition and varied repetition in the context of the golden section is the appearance of the proportion of the golden section in continued fractions. Furthermore, the approximation fractions of these fraction sequences are the quotients of successive members of the Fibonacci series 1, 1, 2, 3, 5, 8..., which play a major role in the arrangement of leaves in plants (phyllotaxis) (see Coxeter [1981], chapter 11).