What is the Fibonacci Sequence?
The Fibonacci sequence is a series of numbers that appears in surprisingly many aspects of nature, from the branching of trees to the spiral shapes of shells. This series is named after the Italian mathematician Leonardo Fibonacci.
The Fibonacci sequence is simple to calculate. The series starts with 0 and 1, and each subsequent number is the sum of the two preceding numbers and goes on infinitely. So, the first few numbers in the sequence are 0, 1, 1, 2, 3, 5, 8, 13, 21, 34, and so on.
The next number in the sequence is the sum of the last two numbers: 21 + 34 = 55.
Then, 34 + 55 = 89.
Fibonacci Sequence Formula
To calculate the nth term of the Fibonacci sequence, we can use a formula. This calculation requires us to number the terms as shown below—n is the term and Fn is the corresponding Fibonacci number.
The Fibonacci sequence formula applies for any term after the initial 0 and 1 (i.e., n > 1):
Fn = Fn-1 + Fn-2
Where F0 = 0 and F1 = 1, and n is any positive integer > 1.
Again, you just add the last two numbers to get the next number.
So, what makes the Fibonacci sequence interesting? It’s simple formula might hide its importance! Read on!
Surprising Places where the Fibonacci Sequence Appears
The Fibonacci sequence has remarkable properties and appears in many unexpected places in nature, art, and mathematics. This series of numbers represents a fundamental mathematical pattern present in many natural phenomena.
Moreover, the Fibonacci sequence has practical applications in various fields, such as stock market analysis and population growth modeling. Its appearance in so many diverse areas speaks to the universality and importance of mathematics in understanding the world around us.
Fibonacci Sequence in Nature
The Fibonacci sequence appears in many natural spiral patterns, such as the arrangement of leaves on a stem, the spiral patterns of sunflower seeds, and the shells of mollusks. These spirals follow a pattern of adjacent numbers in the sequence. For example, in sunflowers, the number of spirals in each direction is usually a pair of adjacent Fibonacci numbers, such as 21 and 34.
The Fibonacci sequence appears elsewhere in nature, specifically in population growth. The series can model the growth of a population in which each generation is proportional to the sum of the previous two generations. Biologists observe this type of growth in some species of rabbits.
Golden Ratio
The golden ratio, denoted by the Greek letter phi (φ), is a mathematical constant approximately equal to 1.618034. The golden ratio appears in many places in art, architecture, and nature, and it is intimately connected with the Fibonacci sequence. In fact, the proportion of any two adjacent numbers in the series approaches the golden ratio as the numbers get larger. For example, the ratio of 21 to 13 is approximately 1.615, which is very close to the golden ratio.
In the output to the right, you can see that proportions of adjacent Fibonacci terms quickly converge on the Golden Ratio!
Fibonacci numbers increase at an exponential rate equaling the golden ratio, making it an exponential distribution.
Learn more about Exponential Distributions: Uses, Parameters & Examples.
Pascal’s Triangle
Pascal’s triangle is an array of numbers where each value is the sum of the two numbers immediately above it. The Fibonacci sequence appears in Pascal’s triangle in several ways. For example, the sum of the numbers in the nth row of Pascal’s triangle equals the n+1th Fibonacci number. Additionally, the Fibonacci sequence is related to the diagonals of Pascal’s triangle, as the nth diagonal contains the Fibonacci numbers.
I’ve written about Pascal’s triangle separately, which also has an intriguing number pattern with multiple uses. It’s not surprising that these two fascinating phenomena intersect!
Learn more about Pascal’s Triangle.
Stock Market Analysis
The Fibonacci sequence has also found its way into stock market analysis. Market analysts use the Fibonacci retracement tool, which identifies potential levels of support and resistance for a stock’s price. The analysis calculates the retracement levels by dividing the vertical distance between two points on a price chart by the key Fibonacci ratios of 61.8%, 38.2%, and 23.6%. These ratios are calculated by dividing numbers in the series by the subsequent number, two numbers later, and three numbers later, respectively.
Typically, modern stock trading systems automatically draw in horizontal lines at these locations. Market analysts use these lines to predict where a stock’s price movement is likely to change direction.
Music
The Fibonacci sequence has even influenced the world of music. Italian composer, Gioachino Rossini, used the number series in his William Tell opera. In the overture, a section features a melody repeated several times, with a different instrument playing each repetition. The number of repetitions for each instrument follows the Fibonacci sequence.
Art
The Fibonacci sequence has also inspired many artists, such as the Dutch painter Piet Mondrian. Mondrian used the number series in his abstract paintings to determine the proportions of the rectangular shapes and lines that make up his compositions.
The Fibonacci sequence is a fascinating pattern in numbers that has captured the imagination of mathematicians, scientists, artists, and musicians for centuries. Its appearance in so many unexpected places highlights the beauty and elegance of mathematics and its role in understanding the natural world.
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