Have you ever counted the number of petals on a flower?
If you started counting them on a flower whose petals are all intact, you’d most likely notice
that the number of petals is a Fibonacci number, that is, a number from the Fibonacci sequence.
Many flowers have 3,5,8, 13, 21 or more petals. The leaves of cacti and seeds of sunflowers are
arranged in both left- and right-handed spirals. The numbers of seeds or leaves in these spirals
are generally in the Fibonacci sequence. We have 2 hands, both of which have 5 fingers,
and each finger is divided into 3 parts. All of these numbers are Fibonacci numbers.
Moreover, the lengths of bones in hand are all Fibonacci numbers too.
So what’s the Fibonacci sequence anyway? And why is it seemingly everywhere?
The Fibonacci sequence is a series of numbers in which a given number
is the addition of the two numbers before it. So, if you start with 0, the next number
will be 1, followed by 1, followed by 2, followed by 3 and so on.
(not to be spoken in audio) 0, 1, 1, 2, 3, 5, 8, 13, 21, 34, 55…..
As you can see, every number in this series or sequence is obtained by adding the two preceding
numbers. This simple series of numbers is referred to as the Fibonacci sequence for
Fibonacci series. And individual numbers in this sequence are often called Fibonacci numbers.
Mathematically, the fibonacci sequence is represented with this formula
F(n) = F(n-1) + F(n-2) (not to be spoken in audio)
where n>1 You can use this expression
to find any ‘n’th digit in the sequence. This fascinating sequence is widely associated
with the mathematician, Leonardo Pisano, aka Fibonacci. He hailed from the Republic of Pisa,
which is why he is also known as Leonardo of Pisa and was known as one of the most
talented mathematicians of the Middle ages. At the time, Europeans were still using Roman
numbers while Hindu-Arabic mathematicians were using a different number system
which was more robust and efficient. Fascinated by the brilliance of the Hindu-Arabic numeral system,
Fibonacci brought them to the western world in 1202 through his now-famous book Liber Abaci.
In the book, he reviewed and compared the Hindu-Arabic numeral system with other systems,
such as Roman numerals, and described how using the Hindu-Arabic system made
calculations faster and easier. Although his book contains the
earliest known description of the Fibonacci sequence outside of India,
it has been described in ancient Indian texts by mathematicians as early as the 6th century.
So, what makes the Fibonacci sequence so special? To the uninitiated, it may just seem like a series
of numbers, but the Fibonacci sequence has been discovered and rediscovered in various forms,
not only in mathematics, but also in nature and our everyday lives.
There’s another exciting offshoot of the Fibonacci sequence - the golden ratio.
Suppose you have two quantities, A and B, wherein A is greater than B.
Now, add A and B and divide the sum by A. If this ratio comes out to be equal
to the ratio of A and B, then you’d say that A and B have a golden ratio.
It’s represented by the Greek letter phi ( ϕ ). Write down the Fibonacci sequence on a piece
of paper and calculate the ratio using this formula. You will notice that all
Fibonacci numbers have the golden ratio, the value of which is close to 1.618033…
In geometry, when the golden ratio is applied as a growth factor, you get a special kind of
logarithmic spiral known as a golden spiral. Simply put, a golden spiral gets wider by a
factor of phi for every quarter turn it makes. You’ll find examples and manifestations
of golden ratio and golden spirals in countless places in your everyday life.
Seashells are one of the most common examples of the golden spiral in nature. Ocean waves,
hurricanes, flower buds, snail shells, spider webs are some of the many naturally-occurring examples.
Many artists use the golden ratio or spiral in their creative works. Legendary painter
Salvador Dali explicitly used the ratio in his masterpiece The Sacrament of the Last Supper.
Architects often use the golden ratio in designing buildings and massive other massive structures.
Popular Swiss-French architect Le Corbusier, who is widely acclaimed as one of the pioneers
of modern architecture, explicitly used the golden ratio in his Modulor system
for the scale of architectural proportion. From snail shells to flowers, from bananas to
the inside of the human ear, from large, iconic buildings to galaxies, the golden
ratio can be seen everywhere. This is probably why it’s also known as the divine proportion.
There are so many places where you can find the golden ratio or the Fibonacci numbers that it’s
impossible to list all of them in one place and be done with it, because the list is ever-growing.
What other examples of the Fibonacci sequence or the golden ratio have you observed in your life?
Tell us in the Comments below!