The Golden Ratio definition, or **Golden Mean** or **Golden Section**, is a ratio expressed by the decimal value 1.61803... It is an irrational number, like {eq}\pi {/eq} or *e*, meaning that it never terminates and never repeats. Like pi, it is expressed with a Greek letter **phi** - {eq}\phi {/eq}. The Golden Ratio is a proportional concept that can be used to describe the relative lengths of two line segments.

In this figure, we can see line segment AB and line segment BC. The ratio of the length of AB to AC is {eq}\phi {/eq}, the Golden Ratio. However, the ratio of segment BC to the whole segment AC is also {eq}\phi {/eq}.

Another interesting geometric construction involving the Golden Ratio is the Golden Rectangle.

Begin with a rectangle whose sides obey the Golden Ratio. This is called a **Golden Rectangle**. Then, add a line segment separating the rectangle into a perfect square and another rectangle, and that new rectangle is another Golden Rectangle!

This process can repeat, and a curve drawn through the successive squares is called a **Golden Spiral**, or a spiral whose logarithmic growth factor is (you guessed it) {eq}\phi {/eq}.

Consider now another geometric object called a regular pentagon. If we construct a regular pentagon with side length *a*, and draw diagonals within that pentagon, then the length of any diagonal will be {eq}a\cdot \phi {/eq}. If that's not amazing enough, notice that if we draw every possible diagonal, we create a smaller regular pentagon inside the big one. The lengths of these sides are {eq}\frac{a}{\phi^2} {/eq}. However, another of the more interesting things about the Golden Ratio is that {eq}\phi^2 = \phi + 1 {/eq}, so {eq}\phi^2 = 2.61833... {/eq}, making this problem a little more easily solved. Finally, notice the triangle shaded in yellow within the diagonals. This is called a **Golden Triangle**, and it is a special isosceles triangle that has the unique property that if one of its base angles is bisected, the resulting smaller triangle is similar to the larger one.

Another fascinating place where the Golden Ratio appears is the **Fibonacci Sequence**. This is a mathematical sequence that begins with 1, 1, ... and then, each subsequent entry in the sequence is the sum of the previous two entries. So, the first two entries are 1, 1, the next is 2, followed by 2+1 = 3, then 3+2 = 5, and so on. The Fibonacci Sequence looks like this:

1, 1, 2, 3, 5, 8, 13, 21, 34, 55, 89, 144...

So, what does this have to do with the Golden Ratio? Well, if we take ratios of adjacent numbers in the Fibonacci Sequence, as the sequence gets larger, those ratios approach {eq}\phi {/eq}!

Let's talk about how to calculate the Golden Ratio. There are several methods by which we can see how to find the Golden Ratio, and we will look at a few examples here.

### Golden Ratio Formula

There are several ways to state the **Golden Ratio formula**. Recall the line segment example from above. In this example, the Golden Ratio can be found by solving the proportion {eq}\frac{AB}{BC} = \frac{BC}{AC} {/eq}. The Golden Ratio may also be found directly by evaluating {eq}\frac{1}{2}(1+\sqrt{5}) {/eq}.

Perhaps the most interesting way to calculate the Golden Ratio is by repeated division.

Begin by dividing 1 by 2. Then, add 1. This gives a result *q*. Then, proceed by dividing 1 by *q* and add 1 to that result. Repeat this process. Let's take a look:

New value | Divide by new value | Add 1 |

2 | 1/2 = 0.5 | 0.5+1 = 1.5 |

1.5 | 1/1.5 = 0.666... | 0.666+1 = 1.666... |

1.666... | 1/1.666... = 0.6 | 0.6+1 = 1.6 |

1.6 | 1/1.6 = 0.625 | 0.625+1 = 1.625 |

1.625 | 1/1.625 = 0.6153... | 0.6153+1 = 1.6153... |

1.6153... | 1/1.6153... = 0.619... | 0.619+1 = 1.619... |

So, repeating this process many times, each result gets closer and closer to {eq}\phi {/eq}! This leads to a self-contained formula for {eq}\phi {/eq}:

$$\phi = 1 + \frac{1}{\phi} $$

That the Golden Ratio can be defined in terms of itself is another unique property of this fascinating number.

### Golden Ratio Number

As mentioned, the Golden Ratio number is irrational and is equal to about 1.61803..., but much like we use {eq}\pi {/eq} to denote the number 3.14159... we use the Greek letter {eq}\phi {/eq}, or phi, to denote the Golden Ratio. It is an infinite, non-repeating decimal that cannot be expressed as a fraction.

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