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.