Out of all of the infinite numbers in the world, there are precious few that are given their own letter from the all-too-finite Greek alphabet. The golden ratio, also known by the letter φ, or phi (usually pronounced “fie” in English), is one of those few. An irrational number that begins 1.618…, it describes an important kind of geometrical proportion—specifically, an elegant way to divide a line segment. Imagine we divide a segment (a) into a longer part (b) and a shorter part (c). If the ratio of a to b is the same as b to c, then that single ratio is golden. A rectangle whose sides are lengths a and b is called a golden rectangle, and it’s found in the geometry of a regular pentagon and the Platonic solids, five fundamental 3-D shapes, including the cube. The golden ratio is also tightly connected with the mathematically important Fibonacci sequence: The ratios of successive numbers in the Fibonacci sequence converge to the golden ratio.
So, like fractals, the golden ratio unites different areas of mathematics together. Interestingly, it is also found in many places in nature. For example, evidence of the golden ratio has been detected at the quantum level, where magnetic atoms linked together seem to vibrate at frequencies described by φ. On the macroscopic scale, the Fibonacci sequence and golden ratio describe the natural arrangements of seeds and leaves on many plants. If you examine the packing of seeds on the head of a sunflower, there are a series of clockwise and counterclockwise spirals, which generally appear in successive Fibonacci numbers. The plants are precise enough with their math that one especially robust sunflower reportedly had exactly 233 spirals of seeds one way and 144 the other. In addition, the golden ratio can be converted into an angular version (approximately 137.5 degrees), which specifies the rotational gap between successive sunflower seeds, and also between leaves growing from the stems of many other plants. In both these cases, it is thought that the golden angle allows for the most efficient packing scheme possible, as described by Mario Livio in his book The Golden Ratio. Phi is sometimes called “the most irrational number,” meaning that it is the hardest to approximate with a ratio of rational numbers. This means that by offsetting each leaf by the golden angle from the one below it, the leaves will not line up in stacks, with one on top of another and gaps in between—they instead come in the tightly packed spirals we see in nature.
Phi has also had an interesting role in aesthetics. For centuries, some people have argued that rectangles that have golden proportions are the most pleasing to the human eye. This has inspired artists like Salvador Dali and the architect Le Corbusier to use the golden ratio in their art. Dali’s Sacrament of the Last Supper, for instance, is painted on a large golden-rectangle canvas, and Jesus and his apostles are framed by a somewhat incongruous dodecahedron, one of the Platonic solids. Le Corbusier developed an entire system of measurement called Modulor based on the golden ratio and the human body, which he used in many of his architectural drawings.
For all of the appearances of the golden ratio, there many be even more erroneous sightings of it. The spiral of the nautilus’ shell is often said to fit precisely within a golden rectangle regardless of its size. But that is untrue. Each nautilus shell does maintain the same proportions throughout the animal’s life (that is, it’s a logarithmic spiral), but that proportion is generally not the golden ratio. Many have also claimed that the golden ratio is found in the proportions of various parts of the human body, the shape of the Gutenberg Bible, the Mona Lisa, and the Parthenon. None of these assertions have stood up to skeptical scrutiny, yet these myths stick with us. The mathematician Keith Devlin once gave a talk about the golden ratio, discussing numerous misunderstandings and debunking them, but when a radio station re-broadcast a portion of his lecture, it crucially omitted the fact that the examples were all false. Why does this myth persist? What makes it so resilient and appealing?
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