The Golden Section

In most objects that are aesthetically pleasing, you will often find that the proportions that make them up are in the golden section, also known as the golden ratio, golden rectangle, golden proportion, golden cut, golden number.

The ancient Pythagoreans discovered the golden ratio which is approx. 1:1,618033988749895a��a��a��. This ratio is used extensively in art, painting, architecture, music, design, book design, the human body Golden sec. 9

and can even be found in nature. This use of the Golden Ratio results in various objects being aesthetically pleasing to the eye.

In mathematics, two quantities are in the golden ratio if their ratio is the same as the ratio of their sum to the larger of the two quantities. The figure on the right illustrates the geometric relationship. Expressed algebraically, for quantities a and b with aA�>A�bA�>A�0,

 

Golde sec. 1A�A�A�A�A�A�A�A�A�A�A�Golden sec. 2.

where the Greek letter phi (I�) represents the golden ratio. Its value is:

Golden sec. 3

 

 

Some of the greatest mathematical minds of all ages, from Pythagoras and Euclid in ancient Greece, through the medieval Italian mathematician Leonardo of Pisa and the Renaissance astronomer Johannes Kepler, to present-day scientific figures such as Oxford physicist Roger Penrose, have spent endless hours over this simple ratio and its properties. But the fascination with the Golden Ratio is not confined just to mathematicians. Biologists, artists, musicians, historians, architects, psychologists, and even mystics have pondered and debated the basis of its ubiquity and appeal. In fact, it is probably fair to say that the Golden Ratio has inspired thinkers of all disciplines like no other number in the history of mathematics.

Architecture

The Parthenon’s faA�ade as well as elements of its faA�ade and elsewhere are said by some to be circumscribed by golden rectangles.Golden sec. 6 A 2004 geometrical analysis of earlier research into the Great Mosque of Kairouan reveals a consistent application of the golden ratio throughout the design, according to Boussora and Mazouz. They found ratios close to the golden ratio in the overall proportion of the plan and in the dimensioning of the prayer space, the court, and the minaret. The Swiss architect Le Corbusier, famous for his contributions to the modern international style, centered his design philosophy on systems of harmony and proportion. Le Corbusier’s faith in the mathematical order of the universe was closely bound to the golden ratio and the Fibonacci series, which he described as “rhythms apparent to the eye and clear in their relations with one another. And these rhythms are at the very root of human activities.

Music

ErnA� LendvaA? analyzes BA�la BartA?k’s works as being based on two opposing systems, that of the golden ratio and the acoustic scale, though other music scholars reject that analysis. French composer Erik Satie used the golden ratio in several of his pieces, including Sonneries de la Rose+Croix. The golden ratio is also apparent in the organization of the sections in the music of Debussy’s Reflets dans l’eau (Reflections in Water), from Images (1st series, 1905), in which “the sequence of keys is marked out by the intervals 34, 21, 13 and 8, and the main climax sits at the phi position.”

The musicologist Roy Howat has observed that the formal boundaries of La Mer correspond exactly to the golden section. Trezise finds the intrinsic evidence “remarkable,” but cautions that no written or reported evidence suggests that Debussy consciously sought such proportions.

Pearl Drums positions the air vents on its Masters Premium models based on the golden ratio. The company claims that this arrangement improves bass response and has applied for a patenton this innovation.

In musical instrumentsA�A�this is also prevalent and in Guitar making and Violin making there are plenty of examples and if examined closely Golden sec. 4you will find these ratios everywhere. These are adapted and combined with functionality, aesthetics and tonal quality to produce the musical instrument, which over time, the shape has not varied that much, but tweaked here and there to facilitate small improvements toward making the perfect instrument.

Golden sec. 5

Fibonacci

In a future blog I will discuss the FibonacciA�Series andA�its presenceA�everywhere around us…………….

 

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