Art / did you know? / Maths

Between reality and mystery: the Golden ratio

Each of us is a walking tribute to the Divine Proportion


Hi guys!
Doctors have discovered that the perfect ratio between systolic and diastolic pressure is about 1,618.

Mmm, do you know this number? Perhaps you’ve heard or read about it, as for example the “Da Vinci Code” talks about it.

I’m going to present this particular number: if you cut a stick into two parts, one measuring 38.2 cm and the other 61.8 cm, the ratio between the larger quantity and the smaller one is equal to the ratio between the whole stick and the larger quantity.

This ratio is called golden ratio, divine proportion, or also golden mean. Adjectives like golden or divine are used because of the several mathematical and geometrical properties and use of this ratio in different cultural contexts. This divine proportion has fascinated men so much that it’s considered a canon of beauty.

Who were “its inventors”? The ancient Greek mathematicians obviously, because of its frequent appearance in geometry. In particular, this discovery was attributed to Pythagoras and his followers, because the regular pentagram was the Pythagoreans’ symbol; indeed the golden ratio is demonstrated inscribing a pentagram in a pentagon.

The triangles ADC and DCB are similar so we can make a proportion: AB:DB=DB:CB. After several calculations, we get the equation φ2-φ-1=0 where φ represents the golden number and it’s the positive solution φ=(1+√5)/2=1,61803398…

The negative root of this equation is -0,61803398… and is called golden ratio conjugate.

The golden ratio is intimately connected with the Fibonacci sequence. The Fibonacci sequence is: 0, 1, 1, 2, 3, 5, 8, 13, 21, 34, 55, 89 and so on. If one of these numbers is divided by its “predecessor” in the sequence, the quotient approximates φ (e.g., 55/34≈1,61764…).

As I wrote, the golden ratio has been applied in many fields, like architecture, painting, music, literature. For example, the Parthenon façade and other elements are circumscribed by golden rectangles (whose side lengths are in the golden ratio!); the Cheops pyramid, Constantine arch, the Pantheon in Rome, the aqueduct in Segovia and in Gard have been built according to golden ratios; an artist who used the golden ratio in his works is the Swiss architect Le Corbusier: he thought that if the golden section reflects the proportions in the human body, then it had to be the basis on which the world around should be structured. We can find this “divine proportion” in many of Leonardo da Vinci’s paintings, too. They’re the Virgin of the Rocks, St. Jerome, the Head of an old man and the famous Mona Lisa: if we circumscribe the head of the mysterious lady in a rectangle, we get a golden rectangle.

Music has many connections with Maths, and especially with the golden ratio. Some people think that the beautiful sound of the violin comes from the building of the sound box according to particular geometries: the centre of the curvature of the arc is in “golden position” towards the total length of the violin. Instead, the keyboard of a piano follows the Fibonacci sequence:  the 13 keys are divided into 8 (white) and 5 (black), which in turn are divided into groups of 2 and 3… the succession 2,3,5,8,13!!! Is it a sheer coincidence? Although it may seem strange, we can find the golden ratio even in some poems. The mathematician Paul Bruckman wrote a poem called “Constant Mean”, in which he extols the algebraic properties of this number. Perhaps even the latin poet Virgilio structured his Aeneid in major and minor parts, according to the golden ratio. But there’s more! If we divide our height by the distance of our navel from ground, the result is the golden number (as Leonardo demonstrated in his Vitruvius), or there’s a golden ratio between the distance from shoulder to fingertips and the distance from elbow to fingertips, too!

A small number, indicating a mysterious ratio, is probably the secret of the harmony that exists in reality, in the infinitely large and infinitely small things. This is really a mystery and some people even think it’s a trace of God.

And you? What do you think about it?

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