FIBONACCI: FROM A SIMPLE SEQUENCE TO THE GOLDEN RATIO

Evelina Petitto

“How many pairs rabbitsof rabbits will be produced in a year, beginning with a single pair, if in every month each pair bears a new pair which becomes productive from the second month on?”

This is the problem that mathematician Fibonacci (Leonardo of Pisa) proposed in 1202, in his treatise “Liber Abaci”. The problem assumes the following conditions:

  • The first couple of rabbits have just been born
  • Rabbits reach sexual maturity after one month
  • The gestation period is also one month
  • After reaching sexual maturity, the female rabbit give birth every month
  • The female rabbit always gives birth to one male and one female
  • The rabbits never die

Following these indications, it is easy to see that, for the first two months, we will have only 1 pair of rabbits; in the third month, we will have one new pair, bringing the total number of pairs to 2. In the fourth month, there will be 3 pairs, in the fifth month, 5 pairs and so on, until one year passes and there will be 233 pairs of rabbits.

What we get is the following series of numbers:

1,1,2,3,5,8,13,21,34,55,89,144,233,….

This is an example of a recursive sequence, where to find out what the next term is, one simply adds up the preceding two numbers:

F(1) = 1

F(2) = 1

F(n) = F(n-1) + F(n-2)

This simple recursive sequence has fascinated mathematicians for centuries; it also appears in nature, from sunflowers to galaxies. For example, consider this diagram:

golden-rectangle

Each square has sides whose lengths correspond to the terms of the Fibonacci sequence – here, they are arranged in an “outwardly spiralling” pattern. A Fibonacci spiral is a series of connected quarter-circles drawn inside an array of squares with Fibonacci numbers for dimensions. Because of the nature of the sequence, where the next number is equal to the sum of the two before it, the squares fit perfectly together. Furthermore, any two successive Fibonacci numbers have a ratio very close to the Golden Ratio, which is approximately 1.618034. This number was known to the ancient Greeks, who assigned it the letter ϕ (phi). They believed that the proportion ϕ : 1 was aesthetically perfect, and the majority of their artwork, sculptures and architecture was built using this proportion as a foundation. A rectangle whose sides have this proportion is called a golden rectangle.

As we mentioned before, the Fibonacci sequence and the consequent Golden Ratio appear in nature, in many instances. Here are some examples:

1. Flower petals

The number of petals on a fbuttercaplower consistently follows the Fibonacci sequence. Examples include the lily, which has three petals, buttercups, which have five petals (see picture), the chicory’s 21 petals, the daisy’s 34 petals, and so on. Phi appears in petals on account of the ideal packing arrangement, as dictated by Darwinian selection; each petal is placed at 0.618034 per turn (of a 360° circle) allowing for the best possible exposure to sunlight and other factors.

2. Tree branchestree

The Fibonacci sequence can also be seen in the way tree branches form or split. A main trunk will grow until it produces a branch, which creates two growth points. Then, one of the new stems branches into two, while the other one lies dormant. This pattern of branching is repeated for each of the new stems.

3. Seeds and pinecones

seeds

Also, the heads of flowers are subjected to Fibonaccian processes. Typically, seeds are produced at the centre, and then migrate towards the outside to fill all of the available space. Sunflowers provide a great example of these spiralling patterns: the numbers of spirals in each direction are invariably two consecutive Fibonacci numbers. Similarly, the seed pods on a pinecone are arranged in a spiral pattern. Each cone consists of a pair of spirals, with each one spiralling upwards in opposing directions. The number of steps will almost always match a pair of consecutive Fibonacci numbers.

4. GalaxiesGolden-Ration_Galaxy

Spiral galaxies also follow the familiar Fibonacci pattern. The Milky Way has several spiral arms, each of them a logarithmic spiral of about 12 degrees.

5. Facesface

Faces, both human and non-human, are abound with examples of the Golden Ratio. For example, the mouth and nose are each positioned at golden sections of the distance between the eyes and the bottom of the chin.

6. DNAdna

Fibonacci also affects the microscopic domain. DNA molecules measure 34 angstroms long by 21 angstroms wide for each full cycle of their double helices. These numbers, 34 and 21, are numbers in the Fibonacci series, and their ratio 1.6190476 closely approximates Phi, 1.6180339.

References:

http://platonicrealms.com/encyclopedia/Fibonacci-sequence

http://www.livescience.com/37470-fibonacci-sequence.html

http://www.maths.surrey.ac.uk/hosted-sites/R.Knott/Fibonacci/fibnat.html

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