It was sunset time Abir was sitting on the cemented stairs of the large pond attached to his ancestral home at Rajballabhpur a quiet village near Diamond Harbour in West Bengal His eyes were focused on the still water that got disturbed only occasionally by the fishes jumping out playfully His ears were fully concentrated on the sound of the birds flying back to their nests and of the motion of trees due to the gentle breeze blowing around Compared to his fourteen years of age Abir is a little too romantic Nature allures him even when he is in his hometown that is always crowded and noisy Once in a year when he comes to this village he makes the most out of the pristine environment surrounding their centuries old home with the smells and with the sounds or the lack of it of this rural area

Arohi his sister who is only three years elder suddenly appeared from behind the bush Whats cooking on your head today my dear poet she asked her brother in a mocking tone of course She has a deep and unwavering love for her brother although they have quite different personalities and interests Both are exceptionally good at their studies but Abir loves to gets absorbed in literature and history while Arohi is a genius in mathematics and science subjects

Nothing that I can make you understand Here I am enjoying the nature and it is not your subject Father told me you are driven by your left brain while I mostly use my right one So please go study your boring science books Abir replied angrily

Why Do you think nature is not about science

Abir gave a little thought He realized that many topics from his physics books are actually about explaining natural phenomena through reasons

At least not your math books nature has nothing to do with math Abir played safe and restricted himself to the subject he hates most even after securing attractive marks in every exam

Arohi giggled Annoyed Abir asked Tell me a thing in nature that is designed based on mathematical rules that we study in our school And dont blabber about the golden ratio

Oh so you are aware of golden ratio But what is wrong with that

Because I dont think it has anything to do with nature It is the number 16 and people think that if rooms or halls are made with the length to breadth ratio equal to this number then they look nice But those are man made structures

Well golden ratio is not just 16 it has a complicated formula that gives it a value of around 1618 up to three decimal places and then there are more But more importantly it is not only used for man made designs A man is considered handsome if the height and width of his face follow the golden ratio rule Many natural objects follow this too But since you are not interested let us discuss Fibonacci

What is Fibonacci Name of a soup

Fibonacci sequence is a series of numbers that starts with first two numbers as 0 and 1 Then onward every subsequent number is the sum of the two previous numbers

So it goes like 0 1 1 2 3 5 8 11 No wait it is not 11 it is 13

Absolutely approved his sister But why did you say it is a soup

I saw a meme in social media that showed a sign on a restaurant It said they sell Fibonacci soup I did not quite understand it then But now I could figure it out It is the mixture of yesterdays soup and day before yesterdays soup that is the unsold soups from previous days

They laughed out so loudly that a few sparrows flew away from the nearby peepal tree

But what does it have to do with nature didi asked the boy

Arohi silently sat down beside her brother Picking up a twig she started drawing something on the ground Then she said Look If you keep drawing squares with sides equal to subsequent Fibonacci numbers and arrange them as I am doing here one side of every third square gets perfectly aligned with the sides of the two previous squares because it is the sum of the previous two sides

Abir seemed interested But how does it help you prove your point he asked

Coming to that Now see as you move from the smaller cells towards the larger ones is it not giving taking a spiral shape

That is interesting It resembles something I cannot remember

It resembles a lot of things bro A conch shell a sea wave just before it breaks the galaxies around us

Abir was amazed He realized that this shape indeed resembles a lot of common objects of nature But even the galaxies

She continued Even if we forget the spiral shape what about the individual Fibonacci numbers themselves Do you know that the number of florets in each spiral in a sunflower is a Fibonacci number such as 21 34 and 55 You can find similar patterns on pineapples or pine cones too

Abir came closer to his didi obviously excited by the flooding information Arohi continued Now about the golden ratio that you do not like Do you know that it is closely related to the Fibonacci sequence

Abir was confused he did not have any clue that the innocent golden ratio that he always knew and ignored has some relation with the great Fibonacci sequence that shaped up the vast galaxies

Arohi asked Tell me the ratio of two subsequent numbers in Fibonacci sequence Start with 1 1

Abir calculated mentally 1 to 1 is 1 next two numbers are 1 and 2 so the ratio is 2 Next 2 and 3 the ratio being 15

Well let us skip first few of them What about the numbers 8 13 21 34

After some time Abir came back with the answers 13 by 8 is 1625 then 21 by 13 is 1616 and 34 divided by 21 gives 1619 But why are you making me do all these divisions

Because my dear brother Fibonacci sequence is infinite and if you continue doing this from smaller to bigger numbers you gradually approach the golden ratio of 1618 See you have already reached 1619 Arohi announced proudly

Wow That is fantastic Abir looked quite impressed His misconception had been cleared off already Even the human perception of beauty had been defined by mathematics

Didi can you tell me more such things about mathematics in nature

Of course Yesterday father called some locals to break open the honeycomb on our mango tree Did you notice how the honeycombs look like

Yes they were made of very large number of small cells each of hexagonal shape

Good Now can you think of a reason why they are all hexagons and not any other shape

I can understand that it cannot be circular because then there will be lot of gaps wasted between the circles Abir answered promptly But then I do not understand why they cannot be triangular or square or at least rectangular

The walls in a honeycomb are made of wax and wax comes from honey To produce each ounce of wax bees need to spend eight ounces of honey This makes wax expensive to the little bees In order to make the maximum honey they need to find out some shape that gives you maximum volume inside a cell with minimum area of wall

Let us think in two dimensions instead of three Abir appealed desperately as he cannot visualize volumes but he can easily think of a shape that can be drawn on a paper

That is fine So let us talk about areas and perimeter Area within the shape indicates how much honey can be stored so it has to be as large as possible On the other hand perimeter represents the wall between two cells Larger perimeter means larger walls hence more wax So bees need to find a shape with maximum area but with minimum perimeter It can be seen that between a triangle and a square of equal perimeters the square has more area within itself And between a square and a hexagon again with equal perimeter the hexagon has more area From that context octagons are even better than hexagons So more the number of sides better is the utilization of wax Now can you tell me where do we land up if we keep increasing the number of sides indefinitely

A circle Abir shouted as he could now clearly see through it

Excellent But then comes wastage of space between shapes as you already pointed out It is seen that out of all shapes or polygons of equal sides only triangles squares and hexagons can be drawn in such a way that their walls overlap exactly without any wastage of space in between

So that is why they fixed it at six Wonderful Bees know math

Not only bees Let us look at the weaver birds nest Unlike honeycombs they do not need multiple cells placed side by side Instead they just need one enclosure that is as large as possible made with minimum surface area to reduce raw materials needed So the wastage of space due to inter cellular gap is not an issue for them Which of the shapes that we discussed is best for them

The circle again answered Abir

You mean the sphere That is what the weaver birds make Although it ultimately becomes oval due to gravity

Abir was completely astonished

If you are interested you can look online to study about various patterns found in nature that will make you think as if they came directly out of your geometry box

Like what

Like the symmetry both reflectional and rotational mentioned Arohi

I know Reflectional symmetry is where you see the same thing when looked through a mirror And rotational symmetry is when you rotate the object by a particular angle the result still looks the same Right

Absolutely Now tell me some objects you see in nature that display such symmetry

Most animals exhibit reflectional symmetry humans butterfly even cats or frogs Many leaves And about rotational symmetry I can think mostly of the flowers

Right So that is about geometrical symmetry There is another thing called fractals have you heard about that

No what are they

Fractals are patterns that are repeated For example you can keep drawing triangles and squares even some curves in a particular repetitive way and get beautiful patterns

That is awesome And where do we find these in nature asked Abir

Think about a large tree If you take the largest branch that again will resemble a complete tree without the root which is anyway not visible If you take out another branch from it the result is another tree like structure It continues until you reach a leaf

That is correct Abir thought a little Then he asked Are spider webs also examples of fractals

Maybe Arohi was not sure but there are better examples Look at the snowflakes they look so perfect

Arohi took out her phone searched the internet for a while and showed some images to her brother

Wow That is so beautiful Now I am convinced that math too like all other subjects that we study at school has deep roots in nature

I am happy that you changed your impression Now it is late already can we go home

Yes didi certainly Abir completely enchanted now got up from where he sat for the entire afternoon and held his sisters hand

As darkness approached the siblings started their journey back home On his way unknowingly though Abir kept thinking about the work of the great Italian mathematician Leonardo Fibonacci who connected some dots between nature and mathematics almost a thousand years ago.