Phi number story

Golden Section (Phi)

The Golden Section = 1.61803398….is an irrational number. That magic number has many names such as Golden Section, Golden Number, Golden Ratio, Phi, the Divine Proportion.. In the early 20th century Mark Barr gave this number Phi Greek name in honor of the Greek sculptor Phidias, one of the greatest of Greek artists at the same time a painter, architect and goldsmith (5th century B.C.). He built the Parthenon in Athens using principles of harmony and beauty of the Golden Section.


Golden Section (Phi) definition

My favorite definition of the Golden Section (Phi) comes from Vitruvius (Roman architect, 1st century B.C.):

“Three points in line, determining two segments, form a Golden Section if there is between the small part and the longer one the same ratio as there is between the longer one and the whole” (A is the long part, B is the smaller one and A+B is the whole. Thus A/B=(A+B)/A).



I believe the pentagram is the maximum geometric expression of the Golden Section and that’s why you can find this perfect star everywhere on this Web site.

Something interesting regarding Golden Section (Phi)

There is something so interesting regarding this special number:
adding 1 to Phi one obtains its square root and subtracting 1 from Phi one obtains the reverse (see below):
  • Phi = 1.61803398…
  • Phi + 1 = Phi2 = 2.61803398..
  • Phi - 1 = 1/Phi = 0.61803398..
If you look at the part after the point of those three numbers you can notice it is the same .61803398.. . Is not it incredible? It is an irrational number…

When I noticed this in 2008 I said to myself “Wow.. this is really amazing, it’s really interesting, let me see better..’ and I begun to study this special number.

Over time, this magic number has been attracting me more and more and I decided to draw and paint a geometry based on this amazing number.

Fibonacci numbers and Geometric sequence

Let’s look at two famous sequences of terms: Fibonacci numbers and Geometric sequence:
  1. Fibonacci numbers, the infinite sequence 1 1 2 3 5 8 13 21 34 55 89 …where each terms is defined as the sum of its two predecessors and the ratio between two consecutive terms is close to Phi

    (8/5=1,6), (21/13=1.615), (89/55=1.6181), (144/89=1.6179)…

    The Fibonacci numbers depend on the first two terms.
  2. Geometric sequence is a series of numbers in which the term is the product of the previous number by a set number that is called the common ratio of the sequence.

    For example: 1 2 4 8 16 32 64 128 256 … in which the common ratio is 2.

    The geometrical sequence depends on its first term and the common ratio.
The Geometric Sequence can be Fibonacci on one condition only: its common ratio should be

             Phi (1,6180..): 1 Phi   Phi2   Phi3   Phi4


             1/Phi (0,6180….): …1/Phi4   1/Phi3   1/Phi2   1/Phi   1

Geometry of my paintings is based on two Geometric Sequences that are Fibonacci at the same time with the common ratio Phi and 1/Phi.

I match these two sequences and get the following sequence

              ….1/phi4 1/phi3 1/phi2 1/phi 1 phi phi2 phi3 phi4 ….

and use this as only one sequence when I draw my paintings.

It’s like going deep and at the same time going out of painting…

It seems like this Noble number is the Beginning and the End.

Just follow … it.

Golden Section can be found everywhere

The Golden Number can be found everywhere especially in nature:
  • Shells such as Nautilus pompilius and Argonauta
  • Flowers like Iris, Sunflower, Orchid..
  • If you cut an apple into halves transversally you’ll see the presence of a star-shaped pentagon and of a decagon.
You can find the Golden Number in art and architecture:
  • Many of the painter based their paintings on the Golden Number such as Leonardo Da Vinci, Michelangelo Buonarroti, Raffaelo (1483-1520). You can find the Golden Number in some painting of Georges Seurat (1859-1891) and Salvador Dali (1904-1989), Ibrahim Mirza, Moyen Age, and many others.
  • As for the architecture you can find the Golden Number in many buildings all around the Earth:
    • Everywhere in Egypt
    • In many Romanesque and Gothic buildings like Notre-Dame in Chartres, Paris…
    • Some Greek architecture, sculpture
    • Some of the Roman architecture, especially Vitruvius (1st century B.C.),
    • Nowadays, the pyramid of the Louvre, the Geode (Paris) and...
Many people in the past studied this Magic Number and wrote a lot about it like Pythagoras (580-500 B.C.), Plato (427-347 B.C.), Euclid(325-265 B.C.), Vitruvius (Roman architect 1st century B.C.), Leonardo Fibonacci (1170-1250), Luca Pacioli (1445-1517 a Franciscan friar), Heinrich Cornelius Agrippa (1486-1535), Gustav Theodor Fechner (1801-1887)...

If you want to know more about this amazing number you can read a following books from nowadays like:
  • “Geometry of the Golden Section” Robert Vincent, Chalagam Publishing
  • “Divine Proportion. Phi in Art, Nature, and Science”, Priya Hemenway, SPRINGWOOD SA, Lugano, Switzerland
  • “The Ancient Secret of the Flower of Life I and II”, Drunvalo Melchizedek, Light Technology Publishing
  • “LE NOMBRE D’OR”, Matila C. Ghyka, Editions Gallimard, Paris 1931
  • “Le Temple de l’Homme 1, 2” , R. A. Schwaller de Lubicz, Copyright 1957, 1993 by E’ditions Dervy, Paris, France