on September 12th, 2022

This year on 23 November the world is celebrating Fibonacci Day to acknowledge a famous Fibonacci Sequence.

It makes sense, the Fibonacci Sequence is a sequence of numbers that changed math and our understanding of the natural environment forever.

If you’re not an Engineering student or missed the day in class when Fibonacci numbers were discussed, here’s what you need to know.

### Nature’s Secret Code

Born to an Italian merchant in the late 1100s Leonardo Fibonacci traveled to North Africa with his father. Here the Italian learned the Hindu-Arabic numeral system.

The system included zero and limits itself to 10 symbols making it more agile and flexible compared to the Roman numeral system Fibonacci would have grown up with.

In 1202, Fibonacci published Liber Abaci which introduced Europe to the Hindu-Arabic system and his now-famous sequence of numbers. These numbers are a pattern of counting that influences math and technology even today.

The pattern is made up of numbers that sum the previous two numbers before them: 0, 1, 1, 2, 3, 5, 8, 13,21,34,55,89,144,233, 377, 610, 987, 1597, 2584, 4181, 6765, 10946, 17711, 28567, 46368, 75025, 121393, 196418, 317811 and so on.

The sequence proved to be everywhere in nature, from natural phenomena, biology and even in DNA. This led the Fibonacci sequence to be dubbed nature’s secret code.

### The Fibonacci Sequence Formula

The formula for the Fibonacci sequence is Fn= Fn-1+Fn-2.

• The first and second terms are 0 and 1, respectively.
• F0 = 0 and F1 = 1.
• F2 = F0 + F1 = 0+1 = 1 is the third term.
• F3 = F2+F1 = 1 + 1 = 2 is the fourth term.
• F4 = F3+F2 = 1+2 = 3 is the fifth term.

### How Fibonacci realized the numbers

History accounts often give the story that Fibonacci unriddled the following problem: Someone has placed a couple of rabbits in sort of an enclosure to find out how many couples are born within a year when a couple of rabbits bring another couple into the world every month, considering the rabbits begin to give birth when they are two months old.

In the first and second months, there is only one pair of rabbits. In the third month, it is two couples, because the original couple brings a new couple (a baby boy rabbit and a baby girl rabbit) after two months.

In the fourth month, there are three couples because it is only the original couple that still bears offspring.

In the fifth month, we have five couples. After all, two more couples are added to those that lived in the fourth month, because in the fifth month the couples that lived in the third month will give birth.

In the sixth month, except for the couples that already existed in the fifth month, three more couples will come in, because the fourth-month couples will give birth.

Fibonacci realized that the number of couples in a given month was the sum of the number of couples that lived in the previous month and the number of couples that already lived two months earlier.

He marked the number of couples in the n-th month as 𝑎𝑛. Then the rule he found could be written as 𝑎𝑛 = 𝑎𝑛−1 + 𝑎𝑛−2, with only 𝑛 > 2 allowed under this rule.

### The Fibonacci sequence in everyday life

The Fibonacci sequence is found everywhere. Here’s a short list of everyday items and experiences that have the sequence attached to it.

• Music: The sequence is used to gather numbers to create proportions.
• Coding systems: Distributing systems, interconnected parallels and computer algorithms all use the sequence.
• Quantum mechanics, high-energy physics, and cryptography use the sequence for calculations.
• The Fibonacci spiral: This arrangement is seen in the seeds and flower heads in most daisies and sunflowers.
• Pinecones show the Fibonacci spirals.
• Flower petals are arranged in a way that gives each one optimal sunlight and nutrients thanks to the sequence.
• In seashells, the logarithmic spiral formed by the golden rectangle shows the infinite spirals of seashells.
• Each arm of the Milky Way Galaxy is a logarithmic spiral.
• Spiral clouds of hurricanes follow a Fibonacci spiral.
• The lengths of bones in the human hand follow the Fibonacci sequence.
• The cochlea of the inner ear of humans and other mammals and animals forms a spiral that uses the sequence.
• The Fibonacci numbers are applied to 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.
• The Golden Ratio, the side of a regular pentagon to its diagonal, is also based on Fibonacci numbers. The five-point symmetry in a starfish is an example of the Golden Ratio.
• The Great Pyramids were built on the concept of the Golden Ratio, which is based on Fibonacci numbers.

While this shows the sequence’s importance in the physical world – it is equally important in physical sciences and engineering. Here are the applications it is used for:

• The Fibonacci sequence is widely used in engineering applications such as financial engineering trading algorithms; computer data structures and sorting algorithms; audio compression; and architectural engineering.
• In recent years, robots have migrated from factory shop floors (serving as industrial manipulators) to outer space (serving as interplanetary explorers), hospitals (serving as minimally invasive surgical assistants), homes (serving as vacuum cleaners and lawn mowers), and battlefield (serving as unmanned air-, underwater-, and ground vehicles).
• This activity exploits students’ fascination with robotics to expose them to the notion of sequences and develops their critical thinking skills.

References:

Fibonacci Sequence History and Modern Applications. Available from: https://www.researchgate.net/publication/359541696_Fibonacci_Sequence_History_and_Modern_Applications [accessed Aug 31 2022].

Siddiqui, S. (2022). Fibonacci Sequence Formula, Applications, With Solved Examples. Turito.

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