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5. July, 2010 / Idun

Fibonacci Numbers and the Golden Ratio

By definition, the first two Fibonacci numbers are 0 and 1, and each subsequent number is the sum of the previous two. This forms the sequence 0, 1, 1, 2, 3, 5, 8, 13, 21, 34, 55, etc.

The Fibonacci sequence is named after Leonardo of Pisa, who was known as Fibonacci (a contraction of filius Bonaccio, “son of Bonaccio”). Fibonacci’s 1202 book Liber Abaci introduced the sequence to Western European mathematics, although the sequence had been previously described in Indian mathematics.

It is important to mention the close relationship between the Fibonacci numbers and the Golden Ratio. Two quantities is in the golden ratio if the ratio of the sum of the two numbers to the larger quantity is equal to (=) the ratio of the larger quantity and the smaller quantity.

 \frac{a+b}{a} = \frac{a}{b} = \varphi\,.

This ratio is an irrational mathematical constant, approximately 1.618, and is often denoted by the Greek letter phi (φ). As we say in Norwegian; a loved child has many names. This also applies to the golden ratio. Other names frequently used are the golden section (Latin: sectio aurea) and golden mean. Other terms encountered include extreme and mean ratio, medial section, divine proportion, divine section (Latin: sectio divina), golden proportion, golden cut, golden number, and mean of Phidias.

When dividing any Fibonacci number with its preceding number in the sequence, you obtain very similar ratio numbers. These numbers become more and more similar the higher the Fibonacci numbers are, and after the 13th sequence number this ratio is fixed to 1.618, the approximate value of the golden ratio.

Particularly since the Renaissance, many artists and architects have proportioned their works close to the golden ratio. This has especially been in the form of the golden rectangle, in which the ratio of the longer side to the shorter is the golden ratio, believing this proportion to be aesthetically pleasing. I remember when learning about the golden ratio back in school we were asked to divide a stick into two parts, and that this divide should be what we found more aesthetically pleasing to the eye. Everyone would divide it so that the ratio would be close to the golden ratio. I always found that immensely interesting.

The golden ratio has been used (and is still used) by artist and architects. Some works and structures that are proportioned according to the golden ratio include The Great Pyramid of Giza, the Acropolis in Athens, including Parthenon, and many sculptures by the Greek sculptor Phidias (who gave name to the golden ratio, phi). Many works by Leonardo Da Vinci feature the golden ratio, including the Mona Lisa, and the classic violin design by the Masters of Cremona have proportions that relates to the the golden ratio.

In more recent time, the golden ratio has been used in Cubism by Juan Gris, and by Piet Mondrian in his neoplastic, geometrical paintings. The Swiss architect Le Corbusier centred his design philosophy on systems of harmony and proportion, extensively using the golden ratio and Fibonacci numbers.  Also in the Eden Project in the South-west of England they have used the golden ratio. The education centre The Core has been designed using Fibonacci numbers and plant spirals to reflect the nature of the site, which contains plants from all over the world. The logo of the centre shows the Fibonacci pattern that is the roof panels.


Fibonacci Sunflower

Fibonacci sequences often appear in the natural world, in two consecutive Fibonacci numbers, such as branching in trees, the flowering of artichoke, the uncurling of a fern, the arrangement of a pine cone, and the seeds in a sunflower. It is often said that sunflowers and similar arrangements have 55 spirals in one direction and 89 in the other (or some other pair of adjacent Fibonacci numbers), but this is true only of one range of radii, typically the outermost and thus most conspicuous.

What many find the most interesting is how the golden ratio appear in our own bodies, or atleast in the ideal human body. For example, the ratio between the whole height of the body and the height of the navel is approximately 1.618, the golden ratio. The same applies to several other distances all over the body, and in the internal structure of the body itself. Please have a look at the videos below for a more detailed explanation about the Fibonacci numbers, and the golden ratio, and how these appear in the human body and the world in general.

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