Artists, designers, scientists, and most importantly mathematicians around the world have always been interested or rather captivated by the concept of the Fibonacci series also known as the Golden Ratio.

What fascinates them the most is the fact that this ubiquitous presence of the ratio can be seen in the natural world around us.

So what does this suggest?

This only suggests that this ratio is not only an important but also a fundamental characteristic of the entire Universe, and it is something that students of math, irrespective of whether they avail online classes from sites likes Buy Online Class or not, finds it fascinating as well.

But what exactly is the Fibonacci Series or the Golden Ratio anyways?

This is how a Fibonacci series progresses: 0, 1, 1, 2, 3, 5, 8, 13, 21, 34, 55, 89, and so on. Therefore, you can see clearly that each number is nothing but the sum of the 2 numbers that precede it.

Now, although it might look relatively easy, but the pattern appears to be present in each and every aspect of our cosmos.

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Given below is a list of some fascinating examples of the presence of this ratio in our surrounding nature:

It was Leonardo Fibonacci, the famous mathematician from the Republic of Pisa in Italy, who came up with the sequence while calculating the growth of the population of rabbits over a period of 1 year.

And today this ratio known as the Golden Ratio (the ratio between 2 successive numbers in the Fibonacci Sequence) can be observed in almost every biological and inanimate objects.

So, here the examples:

  • Flower petals

If you look closely enough, then you will be able to notice that the number of petals in flowers strictly follows the Fibonacci sequence.

Some prominent examples include the following:

  • Lily – has 3 petals
  • Buttercups  – has 5 petals
  • Chicory – has 21 petals
  • Daisy  – has 34 petals; and the list continues.

Moreover, each petal is placed in such a way so that it can get the maximum exposure to sunlight.

  • Branches of trees

The presence of the Fibonacci sequence can also be seen clearly in the way tree branches are formed.

Typically, the main trunk grows first from which a branch grows, which, in turn, creates 2 growth points. After this, 1 of these new stems branches into 2 and the other lies dormant. And the pattern is followed for each and every new stem.

This progression can even be seen in the root systems of plants and also in algae as well.

  • Shells

The Golden Rectangle where the length and breadth of the rectangle are in a ratio that matches the Golden Ratio is also a relevant example. This ratio can easily result in a nesting process that can even continue to infinity, and which also takes on the shape of a spiral.

This spiral is known as a logarithmic spiral, and it can be found in abundance in the nature.

Probably the starkest example of this is the nautilus shells and snail shells as well, as both of them follow the logarithmic spiral.

It can also be seen in the cochlea which is the inner part of a human ear and also in the horns of certain goats and in certain specific spider webs.

  • Faces

The Golden Ratio can be seen even in the faces of both humans and non-humans. For example, the nose and mouth are both positioned at the Golden Section of the distance between that of the chin and the eyes.

Similar proportions can be seen between the eyes and ear when viewed from the side.

It also needs to be said that although it is worth mentioning that everybody is different, the closer the proportion gets to the Golden Ratio, the more attractive a person will look.

  • Fingers

The Golden Ratio can even be seen in our fingers. For example, if you closely observe from the tip of a finger to the wrist, then you will see that each section is bigger than the one preceding it and this ratio is approximately equal to the value of phi.

  • Animal bodies

Many will be surprised to know that even our human bodies show clear signs where the proportions are completely consistent with the Fibonacci ratio.

For example, if you measure the distance between the navel to the floor and the distance between the top of your head and your navel matches exactly with the golden ratio.

  • Honeybees

We all know what comprises a honeybee colony. It consists of a queen, a lot of drones, and workers. The female bees which consist of queens and workers, each of them having two parents, a drone and a queen.

On the other hand, drones are borne out of unfertilized eggs. This clearly indicates that they have one parent.

Hence, a person who has the simplest knowledge of the Fibonacci sequence can figure out a family tree of a drone.

  • Pinecones, fruits, seed heads, and vegetables

If you look at the seeds that are situated at the centre of the sunflower, you can easily figure out the spiral patterns curving left and right.

However, the main thing that needs to be said is that if you count the number of these spirals, your total will coincide with a Fibonacci number.

Moreover, if you count the spirals pointing left and the ones pointing right you are bound to get 2 consecutive Fibonacci numbers.

Similarly, you can also find this sequence in pinecones, cauliflower, and pineapples.


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  • DNA Molecules

If you thought that the microscopic realm would be immune to the Fibonacci sequence, then you have been grossly misinformed.

And probably the starkest example of this phenomenon is the DNA molecule which measures 34 angstroms long and 21 angstroms wide for every complete cycle of a double helix spiral.

Now, we all know that these 2 numbers 34 and 21 are numbers that appear in the sequence and the ratio between the 2 is approximately very close to the value of the Golden Ratio.

Therefore, if you are a math student who choose online learning from websites like Buy Online Class and others or follow the traditional mode of learning, you should be well aware of this sequence, the reasons for which has been provided in the blog.