Golden spiral


The logo of EQUIDAS is an exponential spiral with a growth factor of phi (φ = 1.618), otherwise known as the golden ratio. This means that for every quarter turn that it makes, the spiral gets wider by a factor of phi. This exponential spiral is called a golden spiral.

The easiest way to draw the golden spiral, is by using a series of golden rectangles. A golden rectangle has a long side of φ times the length of the short side.

One starts by defining the golden ratio then proceeds to create the golden rectangle and finally draws the golden spiral.

The golden ratio

Given a line AC we want to find point B so that:    

     (Eq. 1)

If AB=a and BC=b then the equation above can be written as:

     (Eq. 2)

which reads: a+b is to a as a is to b. This results in: 

     (Eq. 3)

Replacing    a=b φ   then:

     (Eq. 4)

Using the quadratic formula, two solutions are obtained:    

           (Eq. 5)

Choosing the positive solution one finds that:  

It is easy to remember that φ is between 5/3 and 8/5 which brings the issue of the relation between the golden ratio and the Fibonacci series: the ratio of successive Fibonacci numbers converges on φ. It is also worth noting that from Eq.4 one derives:

              (Eq. 6)

a unique property of the golden ratio among positive numbers.

The golden rectangle

In order to create the golden rectangle, a square of side a is first created. Then the midpoint G of one side is found. From the right-angle triangle GBC, the lenght of the hypotenuse GB is calculated :

Using GB as the radius one draws a circle to locate point F. Then:

i.e. the long side is φ times the length of the short side. Hence we have a golden rectangle.

The golden spiral

Given a golden rectangle one can partition it into a square and a smaller golden rectangle a process that can be continued endlessly. It is no coincidence that the figure below reminds of the works of Mondrian, he has made extensive use of golden rectangles.

In each square an arc can be inscribed and thus the EQUIDAS logo is created, an approximation of the golden spiral.