# Golden ratio

The golden ratio is a special ratio with approximate value 1.61803..., an irrational number. The golden ratio is typically denoted with the Greek letter, phi (φ), and has been studied by mathematicians throughout history, including Euclid (~300 BC).

In definition, two values exhibit the golden ratio if the ratio of the sum of the two values to that of the larger quantity is the same as the ratio of the two quantities. Mathematically, given two quantities a and b, where a is the larger quantity:

This definition leads to φ^{2} - φ -1 = 0 and its positive solution is:

The following diagram shows what the golden ratio looks like visually:

### Uses of the golden ratio

From geometry, to art, architecture, and even nature, the golden ratio can be found in various aspects of our everyday lives.

In geometry, the golden ratio is often seen in figures that have pentagonal symmetry, since the length of a regular pentagon is φ multiplied by its side. It's also seen in the golden rectangle, a rectangle whose sides exhibit the golden ratio.

The golden ratio is often used in architecture in forms such as the golden rectangle because a number of people find it aesthetically pleasing. The golden ratio is also often used in art, like paintings, in the form of shapes; the canvas may be a golden rectangle, or the painting may include dodecahedrons with edges that exhibit the golden ratio.

In nature, the golden ratio may be found in patterns such as the spiral arrangement of leaves as well as other plants.

The golden ratio is also very closely related to the Fibonacci sequence, a special sequence that has also been widely studied by mathematicians. The limit of the ratios of the successive terms in the Fibonacci sequence is the golden ratio. So, the larger the successive terms used, the closer the approximation of the golden ratio:

Fibonacci sequence: 0, 1, 1, 2, 3, 5, 8, 13, 21, 34...