by Pier Francesco Sciuto
Let’s go
Imagine we are in a public park. It is an ideal place to grasp the similarities between the chemical and physical laws that govern nature, the mathematical thinking that describes and mediates its functioning, and the human artifacts that are its transient reflection. First, our attention falls on the trees, flower beds, and grassy expanses. Then, it briefly rests on a leaf, the flight of a butterfly, or a drop of dew. Each of these details refers to a complex universe and a vital process in which the moment joins eternity. This can be understood through many languages, but mathematics is the only language capable of offering everyone the same wealth of information without qualitative or quantitative alterations. Though seemingly distant from the world around us, mathematics is the distillation of observations, points of view, and facts reflected in our daily experience. When we perceive regularity and recursion in our surroundings, we are beginning to reason in mathematical terms, albeit unconsciously.
One of the most easily observable phenomena in an environment such as a garden is undoubtedly phyllotaxis. As we shall see, its numerical expressions coincide with those studied by the Pisan mathematician Leonardo Bonacci, also known as Fibonacci (c. 1170–1242). Phyllotaxis, from the Greek words φύλλον (leaf) and τάξις (order, sequence), is the branch of botany that studies and determines the order in which leaves (and flower petals) are distributed on the stems that bear them. Recognition involves counting the number of leaves that are regularly superimposed on each node and measuring the angle of leaves on each node in relation to the leaves on the next node. By counting the number of leaves perpendicular to the first leaf taken as a reference for each 360° turn around the stem, we obtain the phyllotaxis quotient. For example, in the case of the elm, there are no more than two perpendicular leaves in 360°, so the quotient is 1/2. The poplar has eight perpendicular leaves in three 360° turns, so the quotient is 3/8.
All the numbers in the quotients of phyllotaxis are also found in the sequence known as the “Fibonacci sequence,” named after the great medieval mathematician mentioned above. It is also known as the golden sequence, and it is formed by all integers, each of which is the sum of the two preceding ones, as in 0, 1, 1, 2, 3, 5, 8, 13, 21, 34, 55, 89, 144, and so on.
These are the phyllotaxis quotients of some common plants: elm, lime tree, cedar, grasses 1/2; beech, hazel, grasses, bramble 1/3 ; oak, cherry, apple, holly, plum, apricot 2/5; poplar, rose, pear, willow 3/8; American willow, almond 5/13. Similarly, counting the petals of some species of flowers, we have these numbers: lily, iris 3; buttercup, rose, dog rose, delphinium, columbine 5; delphinium 8; jacobaea vulgaris, cineraria 13; aster, rudbeckia hirta, chicory 21; plantain, pyrethrum 34; september aster 55, asteraceae 89. Depending on the type, daisies can have 13, 21, or 34 petals. On the head of a sunflower, the number of spirals often follows this pattern: 89 spirals radiating more steeply clockwise; 55 counterclockwise, 34 less steep clockwise. The largest known sunflower has 144, 89, and 55 spirals. As can be seen, these are quantities and ratios that are always found in the Fibonacci sequence.
The existence of mathematical constants in the plant kingdom had already been seriously considered by Theophrastus (371-287 BC), Pliny the Elder (23-79 AD), Leonardo da Vinci (1452-1519), and Kepler (Johannes Kepler, 1571-1630). It was Kepler first, then Linnaeus (Carl Nilsson Linnaeus, 1707-1778), who intuited the existence of a relationship between phyllotaxis and Fibonacci numbers. Subsequently, botanists Karl Friedrich Schimper (1803-1867) and Alexander Braun (1805-1877) and crystallographer Auguste Bravais (1811-1863) succeeded in establishing a general rule. In turn, botanist Julius von Wiesner (1838-1916) hypothesized that phyllotaxis was due to the plant’s need to optimize light absorption, given that the spiral arrangement was the one that best facilitated the passage of light between leaves.
Fibonacci and the golden ratio
Fibonacci is known for introducing Arabic numerals and the number 0 into mathematical notation, thus replacing roman notation, which was graphically very elegant but inconvenient from a computational point of view. He designated the number 0 with zephirus, the name of the wind that blows from the west, the greco-roman version of the arabic sifr, which in turn was taken from the sanskrit term śūnya, meaning “empty.” Zephirus became zevero in venetian, which eventually became the italian zero. Fibonacci is also credited with the discovery of the sequence we have already mentioned, which we reproduce here in a more extended version: 0, 1, 1, 2, 3, 5, 8, 13, 21, 34, 55, 89, 144, 233, 377, 610, 987, 1597, 2584, 4181, 6765, 10946, 17711, 28657, 46368, 75025, 121393, 196418, 317811 […]. In this sequence, as mentioned, each number is the sum of the two preceding numbers: 1+1=2, 1+2=3, 2+3=5, 3+5=8, etc.
Another interesting property lies in the relationships between the numbers that make up the sequence, where each number is first the dividend, then the divisor. Here they are:
2:1 = 2
3:2 = 1.5
5:3 = 1.666
8:5 = 1.6
13:8 = 1.625
21:13 = 1.615
34:21 = 1.619
55:34 = 1.617
[…]
Division after division, the fluctuations decrease until the ratio becomes constant and settles at 1.618, an irrational number of which we report only the first three decimal places here. In mathematical terms, we say that the limit of this ratio tends to 1.618. This number expresses the golden ratio (or golden section) investigated since ancient times by mathematicians and artists. Today it is commonly designated by the greek letter φ (phi), the initial of the name of the great greek artist Phidias. The use of φ to indicate 1.618 was introduced at the beginning of the 20th century by the american mathematician Mark Barr (1871-1950), based on the widespread, though not always accepted, idea that the athenian sculptor and architect who lived in the 5th century BC had deliberately applied the golden ratio to the design of his works.
What are the characteristics that make φ such a special number in the history of science and the arts, almost a cult object attributed with properties known only to those initiated into the universal laws of harmony? It is the only non-natural number whose reciprocal (i.e., the result obtained by dividing the number 1 by the number in question) and whose square (the multiplication of the number in question by itself) remain unchanged in their decimal part. Given that (for greater approximation, we will arrive here at nine decimal places)
φ = 1.618033989
then
1: φ = 0.618033989
while
φ x φ = 2.618033989
With these words, in book VI of his Elements, the greek mathematician Euclid (4th-3rd century BC) defines the golden ratio: “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 less”. Therefore, the golden ratio φ (1.618) is precisely the ratio between the length of the total segment (1.000) and its golden section (0.618).
The mathematical and geometric principles of the golden ratio were already well known to Pythagoras (c. 580-570 BC – c. 495 BC) and his followers. In their rich geometric symbolism, the Pythagoreans focused on the golden ratio in relation to the properties of the pentagon, which was the emblem and distinguishing mark of their community par excellence. In fact, there is a golden ratio between the diagonal and the side of the regular pentagon, as well as between the side of the regular pentagon and the side of the five-pointed star inscribed in the pentagon itself.
The tip of the iceberg
The golden rectangle is a geometric figure in which the ratio between the length of the longer and shorter sides corresponds to φ. If a square with sides equal to the shorter side is subtracted from its surface area, the remaining portion is still a golden rectangle, obviously smaller. Of course, the operation can also be imagined in reverse, starting with two equal squares side by side and adding a square with sides equal to the longer side of the construction to build a larger rectangle, gradually adding squares and rectangles. A golden (or logarithmic) spiral fits perfectly into a golden rectangle and its additions or subtractions, as we have just described.
The best-known example of the golden spiral in nature is the shell of the nautilus (Nautilus pompilius), a perfect logarithmic spiral. Inside the shell, the mollusk that inhabits it grows in size and builds increasingly spacious chambers, sealing off the old ones that have become too small. Because it is a logarithmic spiral, the shape of the shell does not change and the animal can grow while maintaining the same proportions. The golden spiral also occurs in other instances, such as the horns of the mouflon and the tail of the seahorse.
The proportions of the golden rectangle are particularly pleasing to the eye. This is not always for rational reasons, but rather because of natural empathy and an immediate correspondence with our field of vision. Many everyday objects refer to the golden ratio, including credit cards, identity documents, company logos, web page layouts, posters, and photographs. Relationships with Fibonacci numbers and the golden ratio can also be found in music, beginning with the shapes and sizes of instruments. Here, it’s clear that aesthetics and functionality must go hand in hand, as the wavelengths of sound in the atmosphere also respond to golden ratios.
Stele of King Djet, an Egyptian bas-relief preserved in the Louvre Museum, is one of the oldest works of art in which the golden ratio can be observed. It dates back to around 3000 BC. From top to bottom, it depicts a falcon, a symbol of the god Horus; a snake, a symbol of the pharaoh; and a city profile. The rectangle containing the snake and city, on which the falcon stands, has a golden ratio with the rectangle enclosing the snake and the square enclosing the city.
The proportions of the Parthenon, built on the Acropolis of Athens between 447 and 432 BC, broadly follow the golden rectangle pattern. Similar criteria can be found in hellenic statuary from the classical and post-classical periods, from the two Riace Bronzes, dating from 460 and 430 BC respectively, to Praxiteles’ Aphrodite of Knidos (circa 360 BC). In greek sculpture of the 5th and 4th centuries, the golden ratio became an essential rule. Each part of the body is in golden ratio with the others. This is particularly evident in Polykleitos’ Doryphoros (c. 450 BC). It is no coincidence, then, that Polykleitos theorized the division of the body into eight parts: five correspond to the height from the ground to the navel, three to the upper part. The numbers 3, 5, and 8 are present in the Fibonacci series and, as we know, are in a golden ratio to each other. The golden ratio is also applied in the architecture of Magna Graecia: for example, in the construction of the Temple of Athena in Paestum (circa 500 BC). In turn, the proportions of roman architecture do not differ from those already established in Greece. Temples, basilicas, and triumphal arches bear witness to this.
The Renaissance revival of the golden ratio is evident in the work of artist-scientists like Piero della Francesca (c. 1412–1492) and mathematicians like Luca Pacioli (c. 1445–1517), the author of the treatise De divina proportione (1497). These investigations, which ranged from optics to anatomy to mechanical physics, gave rise to some of the most famous graphic schematizations that the Renaissance left as a legacy to subsequent eras. One of the most notable examples is Leonardo da Vinci’s Vitruvian Man (c. 1490), in which the human figure is inscribed in a square and a circle. In the square, the man’s height is equal to the distance between the tips of his outstretched hands. A horizontal line passing through the navel divides the vertical sides of the square into two segments that are in a golden ratio to each other. The navel, the center of the human body’s gravity, is also the center of the circle in which the figure is inscribed with open arms and legs. In The Last Supper (1494–1498), Jesus’s figure is enclosed in a golden rectangle. References to golden measurements can also be seen in the Mona Lisa (1503–1506). However, Sandro Botticelli in The Birth of Venus (1485) and Domenico Ghirlandaio in The Birth of the Virgin (1486–1490) had already established spatial coordinates based on recognizable golden ratios.
The design of two- and three-dimensional space, using appropriate geometric and graphic constructs, is essential for understanding how complex concepts, which we imagine to be specific to pure mathematics, are found in the figurative arts and architecture, and for understanding how they occupy not only intellectual space but also, in the broadest sense of the term, cultural space. Geometry, understood as the tracing of flat and solid figures on a surface, encompasses scientific and mathematical thinking but, like an iceberg floating in the sea, only a small part of it, the simplest and most intuitive, is visible. It is that empiricism that artists and craftsmen resort to in order to master the tools of their trade, which are indispensable for shaping objects, from useful items for everyday life to the most exclusive works of art.
The rest of the iceberg remains submerged: not that it is unimportant (on the contrary, without it the buoyancy would be compromised), but simply that it acts by force of gravity, forming a whole with the part that is above water. It is important to emphasize this because, while it is true that every era has had cultured artists and artisans close to elites with complex knowledge, they were also required to have inventive and executive skills forged through experience and repetition. Mathematics was embodied knowledge and ready-to-use technology in their field, just like Pythagoras’ theorem – not yet actually “Pythagoras’ ” – in the minds and hands of the surveyors of the Fertile Crescent.
BIBLIOGRAPHY • I. Adler, Solving the Riddle of Phyllotaxis: Why the Fibonacci Numbers and the Golden Ratio Occur in Plants, World Scientific Publishing, Singapore 2012. • Ch. Boileau, The Painter's Secret Geometry: a Study of Composition in Art, Thames & Hudson, London 1963. • M. Livio, The Golden Ratio. The Story of Phi, the World's Most Astonishing Number, Broadway Books, New York 2002. • M. Muccillo, Fibonacci, Leonardo (Leonardo Pisano), in Dizionario biografico degli italiani, Roma, Istituto dell'Enciclopedia Italiana, Roma 1997, vol XLVII. Homepage: Comparison between leaf and flower budding modes, drawing by Nadia Borgetti (N. Borgetti-D. Isocrono-DISAFA UniTo/Wikimedia). Below: Stele of King Djet, 3100-2900 BC, limestone, 143 x 65 x 25 cm (total), Paris, Louvre (photo credits Musée du Louvre/Christian Décamps).
