The Golden Ratio

The golden ratio (symbol is the Greek letter "phi" shown at left) is a special number approximately equal to 1.618

It appears many times in geometry, art, architecture and other areas.

 

The Idea Behind It

If you divide a line into two parts so that:

the longer part divided by the smaller part
is also equal to
the whole length divided by the longer part

then you will have the golden ratio.

The Golden Ratio is sometimes called the Golden Mean.

Leonardo Fibonacci discovered the series which converges on phi or the Golden Mean.

In the 12th century, Leonardo Fibonacci discovered a simple numerical series that is the foundation for an incredible mathematical relationship behind phi.

Starting with 0 and 1, each new number in the series is simply the sum of the two before it.

0, 1, 1, 2, 3, 5, 8, 13, 21, 34, 55, 89, 144, . . .

The ratio of each successive pair of numbers in the series approximates phi (1.618. . .) , as 5 divided by 3 is 1.666…, and 8 divided by 5 is 1.60.

The table below shows how the ratios of the successive numbers in the Fibonacci series quickly converge on Phi. After the 40th number in the series, the ratio is accurate to 15 decimal places.

1.618033988749895 . . .

The pictures below utilize the Golden Mean to form a spiral that begins with 2 squares with side length 1 and successive squares drawn along one side. We can continue adding squares around the picture, each new square having a side which is as long as the sum of the latest two square's sides. This set of rectangles whose sides are two successive Fibonacci numbers in length and which are composed of squares with sides which are Fibonacci numbers, we will call the Fibonacci Rectangles.

This is a famous drawing by Leonardo da Vinci who was somewhat obsessed with the Golden Ratio.

 

The Golden Ratio was often used in architectural design.

 

Golden ratio at work in webpage design!

The Golden Ratio comes into play in solid geometry, too.

There are five Platonic solids. They are the tetrahedron, hexahedron, octahedron, dodecahedron, and icosahedron. They are said to form the foundation of everything in the physical world. One way they are related to the Golden Ratio is shown by the Golden Rectangles inside the dodecahedron below:

 

Most people thought that idea was ridiculous until in the 1980s, Professor Robert Moon at the University Of Chicago suggested that the entire Periodic Table of Elements is based off the Platonic Solids.

 

Plato suggested that God used the dodecahedron as a model for the twelvefold division of the Zodiac. Some people say the shape of our universe is in the shape of a dodecahedron!

 

Role-playing games often use the 12-sided dice.

 

It's often used in art, like M.C. Escher's "Reptiles."

 

Mysterious Roman dodecahedra have been found in various Roman ruins in Europe. Nobody really knows their purpose.

 

So then people make conspiracy theories about the dodecahedron!

Back to Mathsongs