The golden ratio, approximately 1.618, is found commonly in nature. It is the ratio of two numbers such that the sum of both divided by the largest number is equal to the largest number divided by the smallest.
A Beautiful Ratio
The golden ratio is used often in art and design, with most picture frames having side lengths calculated with the golden ratio. It is also found in architecture, with buildings such as the Taj Mahal, CN building and the Parthenon all with dimensions which use the golden ratio.
It is believed that the golden ratio produces shapes which are attractive to look at and are perceived as more beautiful than shapes made without this ratio.
If you take any two consecutive Fibonacci numbers, the ratio between them will be close to the golden ratio. The higher you go the closer they get to the golden ratio.
For example: 5/3 = 1.666 but 377/233 =1.618
The golden spiral uses squares with side lengths as Fibonacci numbers, this spiral is seen in shells, hurricanes and also matches the shape of the spiral arms of spiral galaxies.
Calculating the golden ratio
Given that the larger rectangle and shaded rectangle are similar (meaning the ratio between the two sides are the same), it can be written as:
Multiplying both sides by Φ-1:
Solving this by completing the square:
This is also equal to:
1.61803398875… which is the golden ratio.