Golden ratio

Summary: The golden ratio of roughly 1.618 is found throughout nature and art.

It was Euclid of Alexandria, a well-known Greek mathematician, who in his book The Elements (300 b.c.e.) first wrote about the golden ratio. The golden ratio is denoted by the Greek letter φ (phi) and known also as the “golden section,” the “golden mean,” and the “divine proportion.”

This last name was given to φ because of the frequency with which the ratio exists in the natural world—leading many to hold it up as a mystical number. The golden ratio is, as all ratios are, a comparison. In his description, Euclid describes the golden ratio through the division of a line segment. A line segment whose length is A is divided into two smaller pieces, one of length B and the other of length C, such that the ratio of the original segment to the larger piece is equal to the ratio of the larger piece to the smaller piece. Mathematically, this ratio would be represented as the following:

A perfect rectangle is a rectangle in which the ratio of the length of the longer sides to the length of the shorter sides yields φ. Alternatively, the ratio may be expressed as follows:

and it is approximately equal to 1.16180339877.… As this is an irrational number, there is no end to its digits and no pattern among them.

The golden ratio may be used to create a golden spiral. Golden spirals are common in nature and can be found on shells, the caverns of the inner ear, the horns of various animals, and even some flowering plants. A golden spiral is a spiral that gets wider by a factor of φ for every quarter turn it takes as it opens outward from the point of origin (see Figures 1–2). If one considers the origin to be the eye of a hurricane, the spiraling out can be seen in the shape of the hurricane (the circling of winds that opens outward from the eye), and this provides yet another example of the golden ratio’s appearance in nature.

The golden ratio appears in many other areas as well, including science, art, and nature. For example, the work of Herodotus (fifth century b.c.e.), considered the first historian, indicates the use of the golden ratio in the construction of the pyramids (see Figure 3). Phiddias (490–430 b.c.e.), a sculptor, is said to have used the golden ratio in the creation of sculptures that were later found in the Parthenon. The Parthenon itself consists of many uses of the golden ratio, a simple example being the length and width of the building. Similarly, the golden ratio appears in modern architecture, such as the United Nations Building in New York City. Here the ratio of the height of every 10 floors as compared to the width of every 10 floors also yields the golden ratio.

The work of Leonardo da Vinci is also said to incorporate the golden ratio, including in the definition of the proportions in the Mona Lisa (see Figure 4).

The use of the golden ratio in art and architecture is common, especially when one considers that the ratio is pleasing to the eye. Gustav Fechner (1801–1887) performed many experiments with respect to this ratio. He found that rectangles, books, buildings, and other objects were more pleasing to individuals when they contained the golden ratio.

Music is another place where the golden ratio plays a vital role. Mozart’s piano sonatas use the golden ratio in the arrangement of sections of measures that make up individual pieces. Mozart’s piano sonatas are made up of two sections called the “exposition” and the “recapitulation.” In one 100-measure composition, Mozart divided the pieces into two sections between the 38th and the 62nd measures. The measures in the pieces, when compared, yield the closest approximation to the golden ratio that can be made when dividing a 100-measure composition into two sections. However, the pieces do not always make use of the golden ratio throughout. That is, subsections do not always include the golden ratio, leading some to question whether Mozart was conscious of his use of it. In addition, in many of the most successful musical pieces, the climax of the piece occurs in accordance with φ. That is, the ratio between the length of the piece prior to the climax compared to that after the climax yields, once more, the golden ratio.

Further, the golden ratio is apparent in proportions in the human body. If the distance from the navel to a person’s foot is considered to be “1,” then the height of the person is approximately φ. The ratio of the distance from the navel to the top of the head to the length of the head also approximates φ. In the idealized human face (that which is said to be most beautiful in terms of proportions, φ comes up when comparing the length of the face to the width; the length of the mouth and the width of the nose, and many other comparisons.

The golden ratio is also related to the Fibonacci sequence—a numeric sequence in which each successive term (except for the first two) is obtained by adding the two prior terms. This yields 1, 1, 2, 3, 5, 8, 13, 21,… . When the ratios between successive terms in the sequence are found, they approach the golden ratio.

Some question whether the golden mean is a number that is preferred or significant in nature or whether the number is so prevalent because the mathematical meaning of the number influences or biases perceptions of the applicability. The diversity of systems in which it appears, including multiple developmental markers of human growth, suggests that it may be broadly advantageous. Analysis shows that the ratio’s logarithmic spiral is a system that could theoretically self-replicate indefinitely. It also minimizes wasted space and gives new growth maximum exposure to necessary resources, such as sunlight. This makes a golden spiral an optimal and efficient design for growth in biological systems.

Bibliography

Dunlap, R. The Golden Ratio and Fibonacci Numbers. London: World Scientific Publishing, 1998.

Hemenway, P. Divine Proportion: Phi in Art, Nature and Science. Salt Lake City, UT: Sterling Press, 2005.

Livio, M. The Golden Ratio: The Story of Phi, The World’s Most Astonishing Number. New York: Broadway Books, 2002.