# The Ultimate Resource on the Fibonacci Sequence

Named after the Leonardo of Pisa, more commonly known as Fibonacci, the Fibonacci Sequence is defined mathematically by the relation Fn=Fn-1+Fn-2 with seed values F0=0 and F1=1. In layperson’s terms, the Fibonacci Sequence is a linearly recurring sequence in which the next number is produced by adding the previous two numbers together. Thus, the sequence is as follows: 0, 1, 1, 2, 3, 5, 8, 13, 21

Although this is a seemingly simple sequence of numbers, its occurrence in nature and its relationship to the Golden Number have lured many mathematicians, scientists, and artists to its mysterious properties. This article provides fun and advanced facts about the Fibonacci Sequence, the Fibonacci Sequence in popular culture, and sites and resources about the Fibonacci Sequence.

### Fun Facts about the Fibonacci Sequence

• Before the Fibonacci Sequence was known in Europe, it was used in ancient India for the metrical sciences, also known as prosody (the study of poetic meter).
• If you divide a Fibonacci number by the number before it, (as n approaches infinity) the ratios produced settle into what is known as the Golden Ratio or the Golden Number (approximately 1.618034).
• It was used by Fibonacci to illustrate an idealization of rabbit population growth. It has also been applied to cow population and honey bee populations.
• Two consecutive Fibonacci numbers have been found in tree branches, the number of leaves on a stem, the structure of pineapples and artichokes, etc.
• Sometimes the Fibonacci numbers are called pine cone numbers because of their application to the structure of pine cones.
• If you construct a set of rectangles in a spiral formation using Fibonacci numbers as unit lengths, the resulting spiral is very similar to the spirals on snail, nautilus, and other shells. Click here for an example.
• The Fibonacci Sequence has been used in the visual arts because it is believed to produce aesthetically appealing images. One of the most famous artists (and mathematicians!) who used the Fibonacci Sequence in his art is Leonardo da Vinci.
• It has also been used in music, most notably by Mozart.

• It is a divisibility sequence. Every k th number of the sequence is divisible by F k , meaning that every third number is a multiple of two or F(3), every fourth number is a multiple of three or F(4), every fifth number is a multiple of four or F(5), etc. It satisfies the divisible property ged (F m , F n ) = F ged(m.n).
• The above fact means that no neighboring Fibonacci numbers have common factors.
• The last digit of each Fibonacci number in sequence forms a 60-digit long pattern that repeats over and over again throughout the Fibonacci Sequence. A 300-digit long pattern can be found in the last two digits of each Fibonacci number in sequence. A 1,500-digit long pattern can be found in the last three digits of each Fibonacci number in sequence. A 15,000-digit long pattern in the last four digits, a 150,000-digit long pattern in the last five digits, etc.
• Every number is a factor of a Fibonacci number.
• If a Fibonacci number is a factor of both F(n) and F(m) then it is also a factor of F(n+m) because F(n+m) = F(m-1) F(n) + F(m) F(n+1).
• Any Fibonacci number that is a prime number must also have a subscript that is a prime number (with the exception of the Fibonacci number 3, whose subscript is 4. This property can be used to discover larger prime numbers.
• In prime Fibonacci numbers, at least one of the prime factors of each of the numbers will never have appeared before in a previous Fibonacci number. This is known as Carmichael ‘s Theorem and has only four exceptions: F(1), F(2), F(6), and F(12).

### The Fibonacci Sequence in Pop Culture and Other Fun Stuff

• “Astronomy (8 th Light) by Black Star : On their album MosDef& TalibKweli are Black Star,Black Star uses the Fibonacci Sequence in their lyrics for the song “Astronomy (8 th Light)”Now everybody hop on the one, the sounds of the two / It’s the third eye vision, five side dimension / The 8 th Light is gonna shine bright tonight.”
• Fibonacci Quilt: Here is a quilt whose design is based on Fibonacci numbers!
• Fibonacci Sequence” Music Video
: A music video for the song “Fibonacci Sequence” by BT, using experimental time changes and different filters.
• Fibonacci Sequence on the Ukulele
: The Fibonacci Sequence can be used in musical composition!
• “Lateralus” Music Video : The music video “Lateralus” by Tool uses images from the Hubble telescope along with information about the use the Fibonacci Sequence in the song and in their album.
• Leonardo da Vinci : Leonardo da Vinci used the Fibonacci Sequence in his paintings due to their aesthetic appeal.
• Martina Schettina: Schettina is an artist who uses Fibonacci numbers in her paintings, which have been popularized as “Mathemagic paintings”.
• Mozart : Did Mozart use the Fibonacci Sequence in his musical compositions? This website has information about the Fibonacci Sequence’s relationship to music.
• The World’s Best Photos of the Fibonacci Sequence
: Look at pictures of the occurrence of the Fibonacci Sequence in nature!
• Sunflower : This photo depicts the Fibonacci Sequence in the seeds of a sunflower.

### Sites and Resources about the Fibonacci Sequence

• Dave Sabine
: This article has information on the relationship between the Fibonacci Sequence, number theory, and music.
• Fibonacci Numbers in Nature : This page has information on Fibonacci numbers in botany.
• Golden Number : This website has information on the relationship between the Fibonacci Sequence and the Golden Number (also known as the Golden Ratio).
• Oracle ThinkQuest: ThinkQuest provides a basic explanation of the Fibonacci Sequence and the Golden Ratio, applications of the Fibonacci Sequence and Golden Ratio, and biographies of important Fibonacci mathematicians. It also includes a forum and links for more resources.
• Platonic Realms : Platonic Realms gives a basic explanation of the Fibonacci Sequence and contains information on the application of the sequence to Ancient Greek aesthetics and nature.

• PTG Media : This link provides information on the Fibonacci Sequence as an STL Sequence.
• Surrey University Maths Department : This website has the most comprehensive information on the Fibonacci Sequence. It includes information on Fibonacci numbers in nature, Fibonacci puzzles, the mathematics of the Fibonacci Sequence and the Golden Ratio, and other information relating to the Fibonacci Sequence.
• Fibonacci Numbers and Golden Section in Nature : This page is an introduction to the Fibonacci Sequence and includes information on its relationship to the Golden Ratio, shell spirals, plants, and fingers. It also includes “things to do” as classroom applications.
• The Mathematical Magic of the Fibonacci Numbers : This page has information on the mathematical properties and applications of the Fibonacci numbers.

• Textism : This website provides a visual aid to the concept of the Fibonacci Sequence.
• University of Minnesota Geometry Center : This website provides interactive information on the Fibonacci Sequence and its geometric relationship to the structure of spiral shells.
• Wikipedia: Fibonacci Number: Wikipedia provides a comprehensive overview of the Fibonacci Sequence and its applications in nature and in math.
• Wolfram Math World : Wolfram Math World provides information on the Fibonacci Sequence and its various mathematical properties as well as various fun facts.
• World Mysteries : World Mysteries has information on Fibonacci Sequence and the Golden Number along with their application to nature and the art of Ancient Greece. It also provides a list of online and other resources.