The Fibonacci SequenceDemonstrating the Magic of Math

By Kimberly Rivera

The Fibonacci Sequence

The Fibonacci Sequence begins with a 1, followed by another 1. Later

terms are found by adding together the two previous terms.

an=an-1+an-2 for a1=1 and a2=1

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

4181, 6765, 10946, 17711, 28657, 46368, 75025, 121393, 196418, 317811,

…

The Fibonacci numbers are a complete sequence. This means that any positive

integer can be expressed as the sum of various Fibonacci numbers, without

repeating any of the Fibonacci numbers.

The Golden Ratio

Any term in the Fibonacci sequence divided by the

previous has a quotient of approximately

1.618034…. That is, an/an-1≈1.618034. For the first

few terms, this is a very loose approximation, but

as the term number (n) increases, the quotient

coincides more exactly with this irrational value.

The ratio between 1 and 1.618034 is known as the

Golden Ratio (abbreviated 𝛗), and a rectangle with

a width to height ratio of 1:1.618034 is known as

the Golden Rectangle.

Rectangles and Spirals

Each term of the Fibonacci sequence can

be represented with a square whose

sides have a length equal to the value of

the corresponding term. If one takes all

of the squares from the beginning of the sequence to any point along it, the squares

can be arranged into a rectangle. As more squares are added to the rectangle, the

ratio of its width to its height approaches the Golden Ratio, and the rectangle

approaches the dimensions of the Golden Rectangle. Furthermore, the squares within

the rectangle can be arranged to form a spiral pattern if one traces from the largest

square to the smallest square.

Divisibility Patterns

When examining the Fibonacci sequence, it is interesting to note:

● Every third term is even.

● Every fourth term is a multiple of 3.

● Every fifth therm is a multiple of 5.

● (Violet represents multiples of both 2 and 3. Cyan represents multiples of both 2

and 5. Orange represents multiples of both 3 and 5.)

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

4181, 6765, 10946, 17711, 28657, 46368, 75025, 121393, 196418,

Applications to Art and Architecture

The Fibonacci spiral

is can be used as a

guide to placing

features in art and

architecture, in order

to create a pleasing

visual effect.

Applications to Biology

The Fibonacci numbers can be seen throughout nature.

The number of spirals on

pinecones, pineapples, and certain

flowers is always a Fibonacci

number.

The number of branches or

leaves present at certain

heights on a plant is often a

Fibonacci number.

The lengths of the bones in the

human finger are proportionate

to Fibonacci numbers.

More Applications to Biology

The sequence is also seen in the inheritance tree of the human X chromosome,

the population growth of rabbits, and the lineage of a male bee.

Applications to Business

Following a significant change in the value of the

stock market, it is expected to retrace a certain

portion of the increase (advance) or decrease (decline) before stabilizing or

reversing trend. This is called retracement. Quite often, the amount of pullback is

23.6%, 38.2%, or 61.8% of the original advance or decline. These values are ratios

generated from the Fibonacci sequence, in which an/an+3≈.236, an/an+2≈.382, and

an/an+1≈.618. For this reason, a retracement of 23.6%, 38.2%, or 61.8% of the

original change is called a Fibonacci retracement, and a retracement of 61.8% is

knows as the golden retracement. This knowledge allows us to anticipate a trend

reversal in the stock market when the retracement has reached one of these

levels.

Applications to

Computer Algorithms

The Fibonacci sequence is used in the following computer science-related

algorithms and processes:

● Euclid’s algorithm, which determines the greatest common divisor of two

integers

● The pseudorandom number generator, which creates a set of numbers

with similar properties to those of a random set of numbers

● Planning poker, a process used in developing computer software that uses

Scrum methodology

Pseudorandom Number Generator

More Applications to

Computer Algorithms

● The polyphase merge sort algorithm, which divides a set of terms into two

lists, whose numbers of terms are two consecutive Fibonacci numbers

● The Fibonacci heap data structure

● The Fibonacci search technique, which operates more quickly than the

binary search technique, by finding possible positions of the desired item

within a sorted array

● The Fibonacci cube, a graph used in parallel computing

Fibonacci Heap Data Structure

Works Cited

"Fibonacci Number." Wikipedia. Wikimedia Foundation, 22 Apr. 2017. Web. 12 Apr. 2017.

"Fibonacci Sequence." Math Is Fun. N.p., n.d. Web. 12 Apr. 2017.

Patki, Omkar. "What Is the Application of Fibonacci Series in Stock Markets?" Quora. N.p., 6 May 2014. Web. 12 Apr. 2017.

Parveen, Nikhat. "Fibonacci in Nature." Fibonacci in Nature. N.p., n.d. Web. 12 Apr. 2017.

Photo Credits

Balaji. "Fibonacci Series Program in PHP." Innovsystems Blogs. N.p., 24 Aug. 2013. Web. 12 Apr. 2017.

Belmonte, Curtis, and Conor Papas. "Painting by Numbers." IEEE Spectrum 47.3 (2010): n. pag. Sciencia Review. Web.

"Fibonacci Number." Wikipedia. Wikimedia Foundation, 22 Apr. 2017. Web. 12 Apr. 2017

"Fibonacci Sequence." Math Is Fun. N.p., n.d. Web. 12 Apr. 2017.

Imms, Daniel. "Fibonacci Heap." Growing with the Web. N.p., 31 July 2016. Web. 12 Apr. 2017.

Patki, Omkar. "What Is the Application of Fibonacci Series in Stock Markets?" Quora. N.p., 6 May 2014. Web. 12 Apr. 2017.

Parveen, Nikhat. "Fibonacci in Nature." Fibonacci in Nature. N.p., n.d. Web. 12 Apr. 2017.

Pit-Claudel, Clement. "How Random Is Pseudo-random?" Code Crumbs. Word Press, n.d. Web. 12 Apr. 2017.