Examples of the Golden Ratio and the Fibonacci Sequence:

The Fibonacci Sequence and The Golden Ratio occur frequently in both art and science as well as in nature, often together, due to their close relationship.

There is evidence to suggest that the ancient Greeks and Egyptians used these two concepts. The Greeks believed that the golden rectangle (a rectangle whose sides are in the golden ratio) was the most aesthetically pleasing shape and so they incorporated it into their art and architecture. A famous example of this is the Parthenon, whose base was a golden rectangle and its height is in the golden ratio with its width. The ancient Egyptians also are believed to have known about the golden ratio, calling it the 'sacred' ratio. They used it in all temples and places of the dead, believing it to be both pleasing to the Gods and to the eye. The Great Pyramid is an excellent example, having its slant edge in golden ratio with half its base. The sides of the pyramid are also golden triangles (triangles with similar properties to the golden rectangle). Their hieroglyphs are also believed to use the golden ratio.

During the Renaissance, this concept was again brought to the forefront with artists like Leonardo da Vinci using it in their artwork.

Recently this concept has been applied to cosmetic surgery on the basis that facial parts in the golden ratio are the most visually pleasing.

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However, the occurrence of the Golden Ratio and Fibonacci numbers are not restricted to human creations, they are found abundantly in nature as well. The human body is full of Fibonacci numbers; 2 hands, 5 fingers, 3 parts to a finger and 2 knuckles. As well as that, each segment in the finger is in the golden ratio with the one next to it.

The Fibonacci sequence is also found in the genealogy of the bee. A male bee has only a female parent, whereas the female bee has both a male and a female parent. This results in a family tree of the bee being the Fibonacci sequence.

The numbers of petals on flowers are Fibonacci numbers. The number of spirals in pinecones and sunflower seed heads are Fibonacci numbers. This is because this causes optimal packing of seeds that are equidistant from each other. The arrangements of leaves on the stems of plants are Fibonacci numbers. The number of times one goes around the stem from leaf to leaf is a Fibonacci number. The number of leaves one encounters until they find a leaf directly above the original is a Fibonacci number. This is because this gives the leaf the best possible exposure to sunlight and insects for pollination. Why is another question?

The division of a golden rectangle into smaller and smaller rectangles forms the golden spiral or logarithmic spiral. This is found to be very similar in shape to the nautilus shell.
There are many other examples of the Fibonacci sequence and the golden ratio occurring in real life. From this it is clear that they are very useful and fascinating numbers.

Neil Kim
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