Fibonacci and the Golden Ratio: Divine Geometry?

“God geometrizes continually”, Plato (427-347 B.C.).

Phi (Φ,φ) –the golden number or Fibonacci’s number– is a very familiar concept, and one that has been studied by mathematicians of all ages. Nor is it unknown to lovers of art, biology, architecture, music, botany and finance, for example. You’re very likely to have come across it in any of these disciplines. Does this mean it’s possible to find a numerical translation for everything we see, hear or build around us? Perhaps the closest answer to this question is the phrase by Plato that opens this article.

But let’s take a closer look at a mathematical phenomenon that has attracted the attention of thinkers from all disciplines and periods since it was discovered: the golden ratio or the divine proportion. Before we start, we must go all the way back to the story of the mathematician Leornardo Bigollo (Leonardo Pisano or “the one from Pisa”)Fibonacci.

Fibonacci’s spiral

Phi (Φ,φ) is called Phi after the famous Greek sculptor Phidias (5th century B.C.), the creator of towering architectural landmarks like the Parthenon in Athens. According to Mario Livio in his book The Golden Ratio: The Story of Phi, the World’s Most Astonishing Number”, certain historians maintain that Phidias successfully used the golden ratio in his works. This was why the American mathematician Mark Barr decided to honor him by giving the symbol Φ his initial in Greek (Phi). So Phi was neither discovered by Fibonacci (it had already been defined and studied by Euclid), nor owes its name to the Italian. That said, however, we need to turn to the Italian’s discovery if we want to learn more about the potential harmonious capacity of Phi and its derivatives. The Fibonacci succession and the golden number are two sides of the same coin.

The series discovered by the mathematician from Pisa (0,1,1,2,3,5,8,13…) falls within the field of arithmetic (it studies the numbers and the elementary operations that can be done with them). The golden number, represented by the Greek letter Phi (Φ,φ), is derived from this succession, and expresses the relationship between two segments of a straight line. That is, Phi is a geometric construction (concerning the properties of figures) that occurs as follows:


Phi shown as a line divided in two segments, a and b, so that the whole line (a+b) is to the longer segment a the same as a is to the shorter segment b φ = (a+b)/ a = a/ b / Image: Wikimedia commons

If we turn to algebra to obtain the numerical value of Φ, we use an equation whereby Φ= a/b. We then apply this to the graphic representation of the previous segment, and when the total length of the segment (a+b) is divided by the longer part (a) we obtain the same result as when the longer part (a) is divided by the shorter part (b). The result of this operation is 1.6180339887… which is the same as the golden number defined by Euclid, “an infinite and unrepeatable number” (Mario Livio).

Curiously, this figure is very similar to the result of dividing any of the numbers in the Fibonacci series by its predecessor (example: 5/3= 1.666; 13/8=1.625). Combining these two aspects –that is, using geometry to represent the arithmetic concept– produces a key image which is why you find this article fascinating although you may not be a mathematician or have even understood the numerical basis underlying Leonardo el Pisano’s discovery: Fibonnaci’s spiral.


Reproduction of the process forming Fibonacci’s spiral in relation to the numbers forming the series (length of the rectangles whose union results in the form of the spiral itself) Image:

Ubiquity, science or coincidence?

The properties of the number Phi are genuinely surprising, and its discovery as a ratio or proportion led to a thorough analysis of different forms, objects, graphic representations and even patterns of movement that occur in our world and which theoretically are more or less directly related with this measurement –the golden ratio or divine proportion. This article describes only the golden rectangle or Fibonacci’s spiral, but it is also possible to identify golden triangles and pentagons. All these forms are defined by having a common property –they respect the golden proportion.

Video: Cristóbal Vila

So is it so easy to find these “golden” or “divine” forms in the environment around us? That is, outside disciplines like architecture or design, which clearly use forms and geometry intentionally? What about nature, or even the cosmos? The golden proportion can be found in the Pyramids in Egypt, the Google logo, in rose petals and even in the shapes of galaxies. In La Gioconda by Leonardo da Vinci, the microscopic structure of certain crystals, and Debussy‘s musical scores. Could this possibly be the most astounding number in the world? Or alternatively, are we distorting reality in our desire to see mathematics where none exist? Without doubt these facts force us to acknowledge that mathematics has a curious tendency to contribute something to the knowledge of even subjects to which it is –or at least appears to be– totally unrelated.

If you want to learn more about the ever-present golden proportion and find out the amazing variety of objects, natural elements and even parts of the human body in which you can find this measure, don’t miss this article where we analyze the  nine things that are surprisingly “conditioned” by mathematics.

Dory Gascueña for OpenMind