The golden section

By Jørgen Mortensen

The golden section
The golden section in art
The golden section in the natural sciences
The Fibonacci sequence
The Fibonacci sequence in nature
The golden section in Per Nørgård's oeuvre



The divine proportion

The golden section is a particular ratio that has been known since antiquity and has occupied mathematicians, scientists, artists and composers up to the present day. It is surrounded by a certain aura of reverence, being regarded as a natural relationship of singular harmony and beauty, and has therefore sometimes been called the divina proporzione, the 'divine proportion'.

Euclid
The golden section can be traced back at least to Pythagoras (c 580 - c 500 BC). We find a precise description of it in Euclid (c 365-300 BC). In The Elements - his 13 volumes of geometry - he writes in Volume 6 about how a line may be divided on the following principle:

    The smallest segment shall have the same ratio to the largest as the largest to the whole line:

Based on the figure, this will mean that:

CB

AC

=

AC

  AB
0.618034
 If the whole line AB is given the length, 1, the point of division will be at 0.618034, or:

5 - 1

2



0.618034É

If the segment AC is given the length, 1, the whole line AB will be 1.618034. This number can be written as:

5 + 1

2


Phi
In specialist literature, the number 0.618034 is called phi (the Greek letter), whereas 1.618034 is called Phi.

It therefore holds that phi2 + phi = 1

phi is the only number which when added to its own square root gives 1.