At school, we all become familiar with certain types of numbers such as prime numbers, composite numbers, triangular numbers, etc. But these aren’t all of the types of special numbers. There are numbers out there with some remarkable properties and often very imaginative names. They may not have any importance in our day-to-day lives, but they are beautiful and worth looking at for this reason alone. Among these different types of numbers, Fibonacci numbers are the most popular ones.
What is Fibonacci Series?
The Fibonacci sequence was first discovered by Leonardo Pisano. He was known by his nickname, Fibonacci.
The Fibonacci sequence is a set of numbers that starts with a one, followed by a one, and proceeds based on the rule that each number (called a Fibonacci number) is equal to the sum of the preceding two numbers.
The numbers in Fibonacci sequence are defined by the recursive relation F(n) = F(n – 1) + F(n – 2), for all n ≥ 3, where F(1) = 1 and F(2) = 1.
The Fibonacci sequence can elaborately be written as 1, 1, 2, 3, 5, 8, 13, 21, 34, 55, 89, 144, 233, … .
One of the most common experiments dealing with the Fibonacci sequence is his experiment with rabbits. Fibonacci put one male and one female rabbit in a field. Fibonacci supposed that the rabbits lived infinitely and every month a new pair of one male and one female was produced. Fibonacci asked how many rabbits would be formed in a year? Following the Fibonacci sequence, perfectly the rabbits’ reproduction was determined – 144 rabbits.
Though unrealistic, the rabbit sequence allows people to attach a highly evolved series of complex numbers to an everyday, logical, comprehendible thought.
Fibonacci Sequence in Nature
Fibonacci series is not only limited to coding for kids but also in nature. Fibonacci sequences of numbers can be found in nature not only in the famous rabbit experiment but also in beautiful flowers. On the head of a sunflower, the seeds are packed in a certain way so that they follow the pattern of the Fibonacci sequence. This spiral prevents the seed of the sunflower from crowding themselves out, thus helping them with survival. The petals of flowers and other plants may also be related to the Fibonacci sequence in the way that they create new petals.
Petals on Flowers
Probably most of us have never taken the time to examine very carefully the number or arrangement of petals on a flower. If we were to do so, we would find that the number of petals on a flower that still has all of its petals intact and has not lost any, for many flowers is a Fibonacci number.
1 petal: white calla lily
3 petals: lily, iris
5 petals: buttercup, wild rose, larkspur, columbine (aquilegia)
8 petals: delphiniums
13 petals: ragwort, corn marigold, cineraria
21 petals: aster, black-eyed susan, chicory
34 petals: plantain, pyrethrum
55, 89 petals: michaelmas daisies, the asteraceae family
The Fibonacci numbers are found in the arrangement of seeds on flower heads. Inside the fruit of many plants, we can observe the presence of Fibonacci order.
There are 55 spirals spiraling outwards and 34 spirals spiraling inwards in most daisy or sunflower blossoms. Pine cones clearly show the Fibonacci spirals.
A fibonacci spiral can be found in cauliflower.
The Fibonacci numbers can also be found in Pineapples and Bananas. Bananas have 3 or 5 flat sides and Pineapple scales have Fibonacci spirals in sets of 8, 13, and 21.
Fibonacci spirals are also found in the snail, seashells.
The Fibonacci numbers are also found in waves, combinations of colours; roses, etc.
Fibonacci in Organs of Human Body
Humans exhibit Fibonacci characteristics. Every human has two hands, each one of these has five fingers and each finger has three parts which are separated by two knuckles. All of these numbers fit into the sequence. Moreover, the lengths of bones in a hand are in Fibonacci numbers.
Fibonacci in Music
The Fibonacci sequence of numbers is manifested in music widely. The numbers are present in the octave, the foundational unit of melody and harmony. Stradivarius used the Fibonacci sequence to make the greatest string instruments ever created. Howat’s research on Debussy’s works shows that the composer used the Fibonacci numbers to structure his music. The Fibonacci Composition reveals the inherent aesthetic appeal of this mathematical phenomenon. The intervals between keys on a piano of the same scales are Fibonacci numbers.
Fibonacci in Pascal’s Triangle
The Fibonacci Numbers are also applied in Pascal’s Triangle. Entry is the sum of the two numbers on either side of it but in the row above. Diagonal sums in Pascal’s Triangle are the Fibonacci numbers.
Golden ratio, also known as the Golden Section, Golden Mean, or Divine Proportion, in mathematics, the irrational number (1 + √5)/2, often denoted by the Greek letter ɸ, which is approximately equal to 1.6180339887.
How did 1.6180339887 come from?
Let’s look at the ratio of each number in the Fibonacci sequence to the one before it:
|1/1 = 1||2/1 = 2||3/2 = 1.5||5/3 = 1.666…||8/5 = 1.6|
|13/8 = 1.625||21/13 = 1.61538…||34/21 = 1.61905…||55/34 = 1.61764…||89/55 = 1.61861…|
If we keep doing it, we get an interesting number which mathematicians call “phi”, denoted by ɸ.
If you take three successive terms of Fibonacci series as a, b and (a + b), then (b/a) ≈ (a + b)/b ≈ (a/b) + 1This means, ɸ = (1/ɸ) + 1 => ɸ2 – ɸ – 1 = 0 => ɸ = (1 + √5)/2 ≈ 1.618.
Applications of Golden Ratio
Leonardo da Vinci showed that in a ‘perfect man’ there were lots of measurements that followed the Golden Ratio. The Golden Ratio is known to mankind for thousands of years.
The Golden Ratio is widely used in geometry. It is the ratio of the side of a regular pentagon to its diagonal. And also, the diagonals cut each other in the Golden Ratio. Pentagram describes a star that forms parts of many flags.
The eyes, fins and tail of the dolphin fall at Golden sections along the body.
Even DNA exhibits Golden proportion.
Golden Ratio in Architecture
The Golden Ratio is frequently seen in architecture. It can be found in the great pyramid in Egypt. The perimeter of the pyramid, divided by twice its vertical height is the value of ɸ.
The Golden section appears in many of the proportions of the Parthenon in Greece. The front elevation is built on the Golden section.
Fibonacci in Distance
Take any two consecutive numbers from this series as example 13 and 21 or 34 and 55.
Now the smaller number is in miles = the other one in Kilometer or the bigger number is in Kilometers = the smaller one in Miles (The other way around).
34 Miles = 54.72 Kilometers = 55 Kilometers
21 Kilometers = 13.05 Miles = 13 Miles
For distances that are not exact Fibonacci values, you can always proceed by splitting the distance into two or more Fibonacci values.
As example, for converting 15 km into miles we can proceed as following:
15 km = 13 km + 2 km
13 km -> 8 mile
2 km -> 1 mile
15 km -> 8+1 = 9 mile
Another example, for converting 170 km into miles we can proceed as:
170 km = 10*17 km
17 km = 13 km + 2 km + 2 km = 8 + 1 + 1 miles = 10 miles (approximately)
Now, 170 km = 10*10 miles = 100 miles (approximately)
Fibonacci in Coding
Recently Fibonacci sequence and Golden Ratio are of great interest to researchers in many fields of science including high Energy Physics, Quantum Mechanics, Cryptography, and Coding. It has been found that communication may be secured by the use of Fibonacci numbers. A similar application of Fibonacci in Cryptography is described here by a simple illustration.
Suppose that the original message “CODE” is encrypted. It is sent through an unsecured channel. The security key is chosen based on the Fibonacci number. Anyone character may be chosen as the first security key to generate ciphertext and then the Fibonacci sequence can be used.