Numbers in Nature and the Golden Ratio with Fibonacci Sequence

What is the Golden Ratio?

The golden ratio is a number which is obtained by dividing a line into two parts so that the longer part divided by the smaller part is also equal to the whole length divided by the longer part. It is represented by Greek letter phi, Φ. The golden ratio is also known as the golden mean or golden section. Other names include divine proportion, divine section, golden proportion, golden cut, nature golden ratio fibonacci, and golden number. The golden ratio in nature can be seen if observed carefully.

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Golden Ratio in Nature with Explanation

According to the definition of the golden ratio,

\[\frac{a+b}{a}\] = \[\frac{a}{b}\] = Φ

Then,  \[\frac{a}{a}\] +  \[\frac{b}{a}\] = Φ

⇒ 1 + \[\frac{1}{\phi }\] = Φ

⇒ Φ\[^{2}\] - Φ - 1 = 0

On solving, we have 

Φ = \[\frac{1+\sqrt{5}}{2}\] and Φ = \[\frac{1-\sqrt{5}}{2}\] 

Since we have defined Φ as the ratio of two positive numbers, therefore 

Φ = \[\frac{1+\sqrt{5}}{2}\] ≈ 1.618

Fibonacci sequence and its relation with fibonacci golden ratio in nature.

The golden ratio is closely related with the Fibonacci sequence too. Mathematician Leonardo Fibonacci discovered the unique properties of Fibonacci sequence.

The Fibonacci sequence starts from 0, 1, 1, 2, 3, 5, 8, 13, 21 and so on. It follows the rule

F\[_{n}\] = F\[_{n-1}\] + F\[_{n-2}\] (for n>1) , where F\[_{n}\] is n\[^{th}\] Fibonacci number

This magical sequence ties directly with the Golden Ratio because if we take any two successive Fibonacci numbers, their ratio is very close to the Golden Ratio. As the numbers get higher, the ratio becomes closer to 1.625.

Fibonacci Spirals in Nature

The golden spiral in nature or divine ratio in nature can be observed in flower petals, shells, tree branches and in several other objects. Some of the examples golden section in nature are -

Shells:

The rectangle in which the ratio of the sides a/b is equal to the golden mean (phi) is known as a golden rectangle. The unique properties of the Golden Rectangle can be observed in shells. This shape can result in a nesting process that can be repeated into infinity — and which takes on the form of a spiral. It is called the logarithmic spiral, and it is also present in nature. The cochlea of the inner ear also follows this logarithmic spiral. It can also be seen in the horns of certain goats, and in the pattern of the spider's webs.

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Flower Petals:

The number of petals present in different flowers is another example of the Fibonacci sequence present in nature. Famous examples include the lily, which has three petals, buttercups, which have five, the chicory's 21, the daisy's 34, and so on. The explanation for this phenomena is the ideal packing arrangement of petals as selected by Darwinian processes. Each petal which is placed at 0.618034 per turn (out of a 360° circle) are the best possible exposure to sunlight and other factors.

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Tree Branches:

The Fibonacci sequence can also be seen in tree branches form or split. The main trunk will grow until it produces a branch, which further 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 new stem. A good example is sneezewort. Root systems and even algae exhibit this pattern.

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Family Tree of a Rabbit:

Suppose a newborn pair of rabbits, one male and one female, is put in the wild. The rabbits’ mate at the age of one month. By the end of its second month, a female can produce another pair of rabbits. Suppose that the rabbits never die and also each female always produces one new pair, with one male and one female, every month from the second month on. How many pairs will form in one year?

The answer is found in a series of numbers now known as the Fibonacci series. Consider pair A of rabbits giving birth to pairs B, C, D and E. Each of these new pairs gives birth to other pairs B1, B2, B3, C1, and C2, who in turn give birth to B11, etc.  At the end of each month, the total population of rabbits forms the Fibonacci series.

Seed Heads:

Fibonacci numbers can also be seen in the arrangement of seeds on flower heads. Seeds are produced at the centre and then migrate towards the outside to fill all the remaining space. Sunflowers is a great example of these spiralling patterns. The reason behind this arrangement forms an optimal packing of the seeds so that, no matter how large the seed head, they are uniformly packed at any stage. All the seeds being the same size, no crowding in the centre and not too sparse at the edges.

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