Science Focus (issue 022)

The wor ld we l ive in i s a big place – a vast expanse of land and sea, in an even bigger cosmic web known as the universe, much of which remains a mystery to us. Making sense of our place in the universe has always been a comfort to humanity in the face of the unknown. The most significant way we do this is by seeking patterns. One such example is the golden ratio. Denoted by the Greek letter phi (φ), the golden ratio is an irrational number – an unending number with infinite digits that cannot be expressed as a ratio of two integers, just like π – that has caught the attention of mathematicians, biologists, artists, and architects across the world throughout history [1]. You may be wondering exactly what the golden ratio is and what makes it so special. In order to understand it, let’s assume a variable x, which represents the length of a line segment. The line segment is then divided into two parts, one longer than the other. The length of the longer part is normalized to one and that of the remaining part becomes x – 1, as illustrated in Figure 1. So, what would the ratio between x and the longer part be, such that it is equal to that between the longer and shorter parts? To find the value of such a “divine proportion” x, we can create a quadratic equation from the relationship above: By solving the equat ion and reject ing the negative solution, we can get x equals to , approximately 1.618 – this value is the golden ratio. Another famous mathematical concept you may have heard of, the Fibonacci numbers (Fn) forming the Fibonacci sequence, is also closely related to this ratio. Each number in this sequence is the sum of its two predecessors: 0, 1, 1, 2, 3, 5 and so on. The limit of the ratio between each number and its predecessor is, as you can probably tell, the golden ratio, φ. In other words, the higher the Fibonacci numbers, the closer the ratio is to φ. Often linked to the “beauty of proportion” [2], φ appears in various areas of nature and was said to have inspired artists and architects for centuries. For instance, phyl lotaxis (arrangement of leaves around the stem) of certain plants was discovered to be related to the golden ratio. While the angle between successive leaves or leaf pairs can be 90 degrees (decussate pattern) or 180 degrees (distichous pattern), spiral phyllotaxis with an angle close to the golden angle, approximately 137.5 degrees ( foot note 1; f i g u r e 2 ) , i s also prevalent in plants [3]. It i s not hard to imagine that such recurrent encounters with this ratio in nature have probably intrigued and awed ancient Greeks and us, as we can identify the incorporation of the ratio into the Parthenon and da Vinci’s The Last Supper [1]. The frequent appearance of the golden ratio in artwork begs the question: Is the golden Figure 1 Division of a line segment of length x into two parts. By Aastha Shreeharsh MYTHBUSTERS : The Golden Figure 2 Spiral phyllotaxis. (Leaves at consecutive vertical levels are pseudo-colored differently from red to purple.)