The Golden Mean (PHI)

A golden section is a line segment that has been divided into two parts in such a way that the ratio of the longer part (a) to the shorter part (b) is equal to the ratio of the entire segment (a+b) to the longer part (a). This can be indicated symbolically as a/b = (a+b)/a = φ (phi ), and this ratio, phi, is called the golden ratio, or the golden mean.

The concept of a golden section is of historical importance in aesthetics, art, and architecture. It has often been thought that a form, including the human form, is most pleasing when its parts divide it in golden sections.

A related concept is the golden rectangle, which is a rectangle that has adjacent sides with lengths in the golden ratio. The ancient Greeks felt that the golden rectangle had proportions that were the most esthetically pleasing of all rectangles. The most famous ancient Greek building, the Parthenon, is based on the Golden Mean.

The shape appears in many works from antiquity to the present. It is especially prevalent in Renaissance art and architecture. Leonardo da Vinci drew a human figure based on the Golden Mean, called "The Vetruvian Man."

A golden rectangle has the property such that if a square with side equal to the rectangle's short side is marked off, the remaining figure will be another golden rectangle; this process can be repeated indefinitely.

The golden ratio arises in Fibonacci Sequences and in the construction of some regular polygons. A Fibonacci sequence is a sequence in which each term is the sum of the two terms immediately preceding it. It is named for its discoverer, Leonardo Fibonacci. The Fibonacci sequence that has 1 as its first term is 1, 1, 2, 3, 5, 8, 13, 21, 34, 55, ... The numbers may also be referred to as Fibonacci numbers. Fibonacci sequences have proved useful in number theory, geometry, the theory of continued fractions, and genetics. They also arise in many seemingly unrelated phenomena, for example, the Golden Section, a shape valued in art and architecture because of its pleasing proportions, and the spiral arrangement of petals and branches on certain types of flowers and trees, such as in the seed pattern of the sunflower and the arrangement of points on a pine cone. The pattern is also found in the septa of the nautilus.

Short Answer Questions

1. What is a golden ratio?
2. What is the property of a golden rectangle mentioned in the passage?
3. Why is the golden section valued in aesthetics, art and architecture?
4. What is a Fibonacci sequence?
5. What do a sunflower, a pine cone and a nautilus have in common?

(Keys.)