![]() ![]() It may be overthinking that has led us to identify its seemingly overwhelming presence in nature. ![]() Much to our surprise, this might just be our wishful thinking. (Leaves at consecutive vertical levels are pseudo-colored differently from red to purple.) The frequent appearance of the golden ratio in artwork begs the question: Is the golden ratio really a solution to our pursuit of beauty?įigure 2 Spiral phyllotaxis. It is 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 . 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 (footnote 1 figure 2), is also prevalent in plants. For instance, phyllotaxis (arrangement of leaves around the stem) of certain plants was discovered to be related to the golden ratio. Often linked to the “beauty of proportion”, φ appears in various areas of nature and was said to have inspired artists and architects for centuries. In other words, the higher the Fibonacci numbers, the closer the ratio is to φ. The limit of the ratio between each number and its predecessor is, as you can probably tell, the golden ratio, φ. Each number in this sequence is the sum of its two predecessors: 0, 1, 1, 2, 3, 5 and so on. Another famous mathematical concept you may have heard of, the Fibonacci numbers ( F n) forming the Fibonacci sequence, is also closely related to this ratio. To find the value of such a “divine proportion” x, we can create a quadratic equation from the relationship above:īy solving the equation and rejecting the negative solution, we can get x equals to, approximately 1.618 – this value is the golden ratio. 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? The length of the longer part is normalized to one and that of the remaining part becomes x – 1, as illustrated in Figure 1.įigure 1 Division of a line segment of length x into two parts. ![]() The line segment is then divided into two parts, one longer than the other. In order to understand it, let’s assume a variable x, which represents the length of a line segment. You may be wondering exactly what the golden ratio is and what makes it so special. One such example is the golden ratio.ĭenoted 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. The most significant way we do this is by seeking patterns. Making sense of our place in the universe has always been a comfort to humanity in the face of the unknown. The world we live in is 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. ![]()
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