The golden section and its relatives

Around 1600 Johannes Kepler – known for the Kepler Laws of the movements of planets – discovered the relationship between the Fibonacci numbers and the golden section. He observed that the relationship between a number in the Fibonacci sequence and the previous number more and more closely approaches the irrational number Φ ((phi)) the longer the sequence is continued. And Φ describes nothing other than the golden section.

It was the Greek mathematician Euclid who produced the first precise description of the golden section around 300 BC: two lengths are in the proportions of the golden section if the larger one is to the smaller ones as the sum of both is to the larger one. Later, in the 15^th century, the Italian mathematician and Franciscan monk Luca Pacioli took an interest in Euclid’s works and devoted a whole volume to this division of lines, which he called “divine partition.”

The golden section defines a proportion which is felt by the human system of perception to be especially harmonious. It plays an important role in the fine arts. For example, Leonardo da Vinci constructed the proportions of the human body on the basis of the golden section.

A rectangle in which the proportions of its sides correspond to the golden section is called a golden rectangle. Similarly, isosceles triangles in which two sides are in this proportion are termed golden triangles.

An important part is also played by the so-called golden angle Ψ ((psi)), which divides the angle of 360 degrees in the proportions of the golden section. It is – approximately – 137.5 degrees.