One of the joys of mathematics is the discovery of a numbers list that mirrors patterns found in nature. These instances create a sense of belonging in the universe, a sense of some grand cosmic interconnectedness — it’s practically like magic, but rooted firmly in science and mathematics. The Fibonacci Sequence is one such example. But what is It? How is it defined, and how is it created? What are some examples of it in nature, and what is it used for? Let’s discuss all there is to know about these fascinating numbers listed below.

## What is the Fibonacci Sequence? Complete Explanation

The Fibonacci Sequence (*F _{n}*) is a numbers list that follows an interesting pattern: Starting with 0, then 1, then 1, then 2, then 3, and so on, each subsequent number in the sequence is the sum of the two preceding numbers added together. It’s defined by what’s known as the recurrence relation, the formula for which is

*F*= 0,

_{0}*F*= 1, and

_{1}*F*=

_{n }*F*+

_{n – 1 }*F*for

_{n – 2 }*n*> 1. (In older iterations, 0 was skipped and the formula began at

*F*=

_{1}*F*= 1, with

_{2}*F*=

_{n }*F*+

_{n – 1 }*F*being true for

_{n – 2 }*n*> 2. Fibonacci himself began the formula at

*F*= 1 and

_{1}*F*= 2.)

_{2}First discovered in Sanskrit Indian mathematics as far back as 200 BC, the Fibonacci Sequence eventually got its name from the Italian mathematician Leonardo of Pisa — a.k.a. Fibonacci — who detailed the formula in his book *Liber Abaci *(1202). In his book, the Fibonacci Sequence was used for describing the growth pattern of the rabbit population, where the sum of the formula was used for hypothesizing about a rabbit’s breeding pattern.

What’s so fascinating about this concept is that the formula often appears out of the blue in mathematics, often unexpectedly and often without trying to find it in the first place. It even appears in nature, such as in the pattern of branching in trees or the placement of a stem’s leaves.

## The Fibonacci Sequence: An Exact Definition

Simply put, the Fibonacci Sequence is a series of numbers where each proceeding number is the sum of the two previous numbers. While the sequence begins with some simple addition, you’ll need a calculator before too long. The first twenty numbers are as follows:

0, 1, 1, 2, 3, 5, 8, 13, 21, 34, 55, 89, 144, 233, 377, 610, 987, 1597, 2584, 4181, 6765.

## How Does the Fibonacci Sequence Work?

It works by the rules of a closed-form expression. This means that it’s defined by a linear recurrence that has constant coefficients. As the Fibonacci numbers continue, the ratio between the numbers converges. As they go on, they get incredibly close to the Golden Ratio — however, it’s not an exact match. The Fibonacci Sequence can be proved with a calculator via combinatorial arguments.

## How Do You Create the Fibonacci Sequence?

To create the Fibonacci Sequence, take a calculator and begin by adding 1 + 1 to get 2. Then, add 1 + 2 to get 3. Then add 2 + 3 to get 5. Continue for as long as your calculator can handle it — the numbers get quite big quite fast. Even after the numbers exceed the calculator’s abilities, the sequence theoretically continues infinitely.

## Who Created the Fibonacci Sequence?

The Fibonacci Sequence was first detailed not by mathematicians, but by Sanskrit prosody — a form of ancient poetry used as far back as 1200 BC. In this ancient poetic form, all patterns of long syllables were given two units of duration, while short syllables were given one unit of duration. Counting these patterns of long and short syllables resulted in the first discovery of a Fibonacci Sequence.

Ancient Indian poet and mathematician, Pingala, told of this formula as far back as 450-200 BC. He referred to it as “misrau cha,” or “the two are mixed.” Scholars took this to mean that the long and short syllables created a unique pattern of Fibonacci numbers. Talk of this concept didn’t emerge again until Indian mathematicians Virahanka in 700 AD and Hemachandra in 1150 AD.

Despite their early work with the sequence, the creation of the Fibonacci Sequence as we know it today is credited to Fibonacci’s aforementioned book *Liber Abaci*. It was first credited to Fibonacci by theorist Édouard Lucas in the 19th century — 3,000 years after its initial discovery in Sanskrit prosody.

## What Are the Applications of the Fibonacci Sequence?

There are several applications of this concept, both in mathematics and in nature. For one, the Fibonacci Sequence can be used to describe the totals of the shallow diagonals in Pascal’s Triangle. It can also be used to count {1, 2}-restricted compositions. The sequence also has a lot of significance in analyzing Euclid’s algorithm to determine computational run-time. In nature, the Fibonacci Sequence appears in a pineapple’s fruitlets, an artichoke’s flowering pattern, a fern’s unfurling method, a pine cone’s arrangement, and a honeybee’s family tree. Many flowers also demonstrate this concept in their blooming formation.

## Examples of the Fibonacci Sequence in the Real World

Beyond these applications of the Fibonacci Sequence in math and nature, there are several other instances of the Fibonacci Sequence in the real world.

### Finance

In the world of finance and economics, there are several appearances of the Fibonacci Sequence. The Fibonacci retracement is used to analyze stock market trading, describing a phenomenon where stock prices fluctuate in a dependable pattern. The concept also appears in the economic growth model of Brock-Mirman.

### Music

Composer Joseph Schillinger created compositions using the Fibonacci Sequence as applied to melodies, with the intervals between notes being determined by the formula. Schillinger’s compositions are distinct from Golden Ratio music, which follows a similar idea down a different path.

### Converting Miles to Kilometers

Because the conversion factor from miles to kilometers is very close to the Golden Ratio, the Fibonacci Sequence can be used to get a general idea of miles to kilometers over longer distances (so long as each proceeding Fibonacci number is replaced by its successor).