The organization, or better the self-organization in biological systems, is usually considered under the perspective of natural laws, including thermodynamic and kinetic factors. However, there is a form of regularity in nature that appears to challenge or even be above such natural laws. It is the so-called “golden ratio” (or “golden section”), a ratio which corresponds to an irrational number usually denoted, since the 20th century, by the Greek letter Φ (phi) [1], in honor of the sculptor Phidias.

What is the golden section, and why is so important and so peculiar? For its definition, let us go back to Euclid, who, already in the 3rd century BC, had to say: « A straight line is said to have been cut in extreme and mean ratio when, as the whole line is to the greater segment, so is the greater to the lesser» (see Figure 1). Expressed algebraically, this proportion reads as in the following equation: (a + b)/a = a/b = Φ

And what there is so noteworthy in this relation? This is its ubiquity, namely the fact that this ratio appears to be present in many geometrical figures, as well as in ancient architecture constructions, in paintings, in living structures, in self-organized natural phenomena, and to leave the word to Mario Livio: «The fascination with the golden ratio is not confined just to mathematicians. Biologists, artists, musicians, historians, architects, psychologists, and even mystics have pondered and debated the basis of its ubiquity and appeal. » (Livio, 2002).

The geometrical proportions in nature have attracted thinkers since antiquity, much before Euclid. Actually, the first inquiries and speculations about what today we call golden ratio [2] go back to the 6th Century BC, or even before. A concrete start can be found in the studies about the pentagon and pentagram by Pythagoras (c. 570 – 495 BC) and his school, down to his disciple Hippasus of Metapontum (5th century BC), sometimes credited with the discovery of irrational numbers [3] (see Livio, 2002).

Hippasus was – most likely – the first to recognize that the ratio between the diagonal and the side of the regular pentagon is an irrational number (see note 3), as well as some other ratios showed by the pentagram (see Figure 5). Plato (c. 428/427 – c. 348) related how the arithmetic approach to this study was initially (so, in 5th century BC) blocked by the Pythagorean prejudice that any number was expected to be rational. For them, the study of geometry, even simple figures – like pentagon, square, circle… and their mathematical relations – were sacred and expected to be in exact ratios. He found the first proofs of irrationality in some diagonals of regular polygons, citing his preceptor, Theodore of Cyrene (465 – 398 BC), a Pythagorean who proof the irrationality of √5.

In the Elements of Euclid, the relation of the “extreme and mean ratio” with the pentagon, the icosahedron and the regular dodecahedron is highlighted. This ratio, in fact, plays a crucial role in the construction of two Platonic solids, the icosahedron (with twenty triangular faces) and the dodecahedron (with twelve pentagonal faces). In both cases, their properties are based on the remarkable symmetry of the regular pentagon and pentagram, as shown in Figure 5. Furthermore, Euclid considered the rectangle with sides showing the same proportion (now called the golden rectangle).

Figure 3 shows that it has the unique property that the smaller rectangle, generated by cutting off a square from the original rectangle, is again a golden rectangle. If this procedure is continued, the points dividing the sides of the rectangles in golden ratios are connected by a spiral (often called “God’s eye). It can easily be constructed by inscribing quarter circles into the “whirling squares”. This is a good approximation of a particular logarithmic spiral, known as the “golden spiral” (see last panel of Figure 3 and later on).

In view of its remarkable properties, it is no wonder that the extreme and mean ratio has deeply captivated mathematicians and philosophers also for all the following centuries. Here we have no space to mention the studies, and related works. So, let us make a jump, going to the Renaissance period.

Luca Pacioli (c. 1447 – 1517) wrote his most famous book, De Divina Proportione (see Note 2). After three manuscript copies, it was printed in 1509. Its main subject was the mathematical proportions (the title refers to the golden ratio and its particular, fascinating character) and their applications to geometry, art and architecture. The book was illustrated (see Figure 4) also by Leonardo da Vinci (1452 – 1519) while living in Milan and taking mathematics lessons from Pacioli.

The clarity of the written material and Leonardo’s excellent drawings and diagrams helped the book to achieve an impact beyond mathematical circles, popularizing contemporary geometric concepts and images. Renaissance artists and architects started to consider that the “divine proportion” was pleasing to the eye, in particular the golden rectangle. It is however still debated whether, and to what extent, in renowned paintings, carvings and buildings, the artists have intentionally incorporated approximate golden ratios into their works (see Livio, 2002). Figure 6 shows some stimulating examples. Moreover, things become even more intriguing if one considers the Fibonacci’s sequence.

The Fibonacci sequence and Nature’s spirals

In the 13th century, Leonardo of Pisa, better known as Fibonacci (ca. 1175 – ca. 1250), one of the greatest mathematicians of the Middle Ages, formulated a number sequence in an attempt to model the growth of an ideal rabbit population. It is a sequence in which each number is given by the sum of the two previous ones: 0, 1, 1, 2, 3, 5, 8, 13, 21, 34, 55 and so on.

The relation with the golden ratio, discovered later by Kepler (1571 – 1630), is indeed remarkable: the ratio between two consecutive numbers in the Fibonacci series tends towards the closest rational approximation to Φ, so that greater is the chosen pair, better is the approximation [4]. It has been observed that, in some plants, the geometrical growth patterns, and often the consequent phyllotaxis, follow this sequence (see Huntley, 1970; Runion, 1990).

There are also natural patterns related to Φ, as the striking growth pattern of sunflower seeds [5] that creates an optical effect of multiple crisscrossing spirals. As the helical pattern, the sunflower pattern also may lead us to another fascinating pattern, the logarithmic spirals. They have the unique property known as self-similarity (they do not alter their shape as their size increase). They are ubiquitous in nature, as shown in Figure 2. Although they are not related to the golden ratio, the golden spiral [6] exist and it is a logarithmic spiral. As the golden ratio, also spiral and helical patterns can be recognized in some art and architecture examples. With his frequent use of “spiral body configurations” (see Figure 6, panel d), Leonardo da Vinci created several forms of classical elegance (Capra, 2013; Capra and Luisi, 2014).

Thinking to the golden ratio, to the Fibonacci series, and to helical and spiral patterns one can think that there would be a “rational” explanation of this, maybe some laws in natural shapes. But no any clear relation between them is given in the literature – and this is indeed a point to clarify!

The sketchy notes given in this short contribution highlight the important question, whether and to what extent the golden ratio should be considered a kind of natural law; and whether the corresponding geometrical figures have to do with our sense of beauty; and, last but not least, whether there is something very special in the relation between the plant geometry and the ideal geometrical figures of Pythagorean and Platonic memory. We will try to consider these points separately later on.

Text by Pier Luigi Luisi and Angelo Merante

Notes:
[1] We use here the symbol, Φ, as it was chosen, among other recent Authors, by Mario Livio in his well-known book, The Golden Ratio (2002). Actually, some Authors use φ, instead of the capital letter Φ.
[2] The term golden section was introduced – as far as we know – in 1835, by Martin Ohm (1792 – 1872) and, a few later, golden ratio. About three centuries before, the fascinating definition, Divina Proportione (Divine Proportion), was introduced by Luca Pacioli and became the title of his famous book on mathematics, geometry and architecture, and wonderfully illustrated by Leonardo da Vinci.
[3] In mathematics, the irrational numbers are all the real numbers which are not rational numbers, the latter being the numbers constructed from ratios (or fractions) of integers. When the ratio of lengths of two line segments is an irrational number, the line segments are also described as being incommensurable, meaning that they share no “measure” in common, that is, there is no length (no matter how short) that could be used to express the lengths of both of the two given segments as integer multiples of the length itself.
[4] The reader may consider, as example, some pairs in the sequence, as 8/5 and 13/8, and greater ones, as 610/377 and 987/610. So, being Φ (= 1.618034...), the results given by greater Fibonacci pairs, as 610/377 = 1,618037... or 987/610 = 1,618032..., are better than smaller pairs, as 13/8 = 1.625 or 8/5 = 1.6.
[5] Studies focused on the particular spiral pattern of sunflowers (and some other plants), based on the so-called golden angle (1/Φ2 ~ 137.5°) as generative principle of the spiral, indicate this one the pattern which permits the closest packing of seeds (Vogel, 1979).
[6] The golden spiral (a good approximation of this spiral was shown in Figure 3) is a particular logarithmic spiral that grows by a factor of Φ for every quarter turn. Although it is a logarithmic spiral and has the self-similarity property, it is very rare in natural shapes. In fact, other logarithmic spirals are frequent in nature.

References and cited books
Capra, F. (2013). Learning from Leonardo. San Francisco, CA: Berrett-Koehler.
Capra, F., and Luisi, P. L. (2014). The Systems View of Life. A Unified Vision. Cambridge University Press. Euclid, Elements (3rd century BC).
Huntley, H.E. (1970). The Divine Proportion. New York: Dover.
Livio, M. (2002). The Golden Ratio. The Story of Phi, the World's Most Astonishing Number. New York: Broadway Books. Luisi, P. L. (2016). The Emergence of Life: From Chemical Origins to Synthetic Biology. 2nd Edn., Cambridge University Press.
Pacioli, L. (1509). De Divina Proportione, (Antonio Capella), Venice: Paganinus de Paganinis de Brescia. Runion, G.E. (1990). The Golden Section. Palo Alto, CA: Dale Seymour Publications. Stewart, I. (2011). The Mathematics of Life. New York: Basic Books. Vogel, H. (1979). A better way to construct the sunflower head, in Math. Biosc. 44: (3-4), p. 179–189.