Essay, Research Paper: The Golden Ratio

The Golden Ratio and Our World
Leonardo of Pisa, better known as Fibonacci, was born in Pisa, Italy, about 1175 AD. He was known as the greatest mathematician of the middle ages. Completed in 1202, Fibonacci wrote a book titled Liber abaci on how to do arithmetic in the decimal system. Although it was Fibonacci himself that discovered the sequence of numbers, it was French mathematician, Edouard Lucas who gave the actual name of "Fibonacci numbers" to the series of numbers that was first mentioned by Fibonacci in his book. Since this discovery, it has been shown that Fibonacci numbers can be seen in a variety of things today.
He began the sequence with 0,1,… and then calculated each successive number from the sum of the previous two. This sequence of numbers is called the Fibonacci Sequence. The Fibonacci numbers are interesting in that they occur throughout both nature and art. Especially of interest is what occurs when we look at the ratios of successive numbers. The Fibonacci numbers play a significant role in nature and in art and architecture. When you construct a set of rectangles using the sequence (1, 1, 2, 3, 5, 8, 13, 21,), a design found in nature is revealed:

Next, when you construct in each square an arc of a circle with a radius the size of the edge of each respective square (a quarter circle), the organic design, which can be found in a snail shell can be seen:


Throughout history the length to width ratio for rectangles was one to 1.61803 39887 49894 84820. This ratio has always been considered most pleasing to the eye. This ratio was named the golden ratio by the Greeks. In the world of mathematics, the numeric value is called "phi", named for the Greek sculptor Phidias. The space between the collumns form golden rectangles. There are golden rectangles throughout this structure which is found in Athens, Greece.

He sculpted many things including the bands of sculpture that run above the columns of the Parthenon. Phidias widely used the golden ratio in his works of sculpture. The exterior dimensions of the Parthenon in Athens, built in about 440BC, form a perfect golden rectangle.
Many artists who lived after Phidias have used this proportion. Piet Mondrian and Leonardo da Vinci both thought that art should manifest itself in continuous
movement and beauty. Therefore, they both expressed movement by incorporating the golden rectangle into their paintings. The golden ratio expresses movement because it keeps on spiraling to infinity. They showed beauty in their paintings by using the golden ratio because it is pleasing to the eye. To express the Fibonacci Sequence in art one must pay close attention to beauty, proportions, and continuous rhythm.
Leonardo Da Vinci dubbed this proportion the “divine proportion.” If you draw a rectangle around Mona Lisa’s face, you would find that the rectangle is in the golden proportion. He did an entire exploration of the human body and the ratios of the lengths of various body parts.
A modern day artist that used the golden ratio in a numerous amount of paintings was Mondrian. Piet Mondrian avoided any suggestion of reproducing the material world. Instead using horizontal and vertical black lines that outline blocks of pure white, red, blue or yellow, he expressed his conception of ultimate harmony and equilibrium. His style, and its underlying artistic principles, he called neoplasticism. Here is an example of one of his angular paintings which employ the proportion:

Composition with Gray and Light Brown
by Piet Mondrian
1918 (170 Kb);
Oil on canvas, 80.2 x 49.9 cm;
Museum of Fine Arts, Houston, Texas



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