When Fibonacci met Reinfried Marass, but who is Fibonacci ?
Fibonacci (Leonardo Pisano Bigollo c. 1170 – c. 1250), was an Italian mathematician, considered by some “the most talented western mathematician of the Middle Ages.
The Fibonacci numbers are a sequence of integers, starting with 0, 1 and continuing 1, 2, 3, 5, 8, 13, …, each new number being the sum of the previous two. The Fibonacci numbers, and in conjunction the golden ratio, are a popular theme in culture. They have been mentioned in novels, films, television shows, and songs. The numbers have also been used in the creation of music, visual art, and architecture.
Chimney of Turku Energia, Turku, Finland featuring Fibonacci sequence in 2m high neon lights. By Italian artist Mario Merz for an environmental art project (1994)
- The sequence has been used in the design of a building, the Core, at the Eden Project, near St Austell, Cornwall, England.
- Along with the golden rectangle and golden spiral, the Fibonacci sequence is mentioned in Darren Aronofsky’s independent film Pi (1998). They are used to find the name of God.
- In The Da Vinci Code, the numbers are used to unlock a safe. They are also placed out of order in a message to indicate that the message is also out of order (anagram).
- In Mr. Magorium’s Wonder Emporium (2007), Magorium hires accountant Henry Weston (Jason Bateman) after an interview in which he demonstrates knowledge of Fibonacci numbers.
- In a strip of FoxTrot by Bill Amend, characters Jason and Marcus take one nacho from a bowl, one more nacho, then two nachos, three nachos, five nachos, eight nachos, etc., calling it ‘Fibonacho.’
- In the strip “Alone” of the online comic xkcd by Randall Munroe, a male and female stick figure are seen together in an intimate situation. The male voices over, explaining his obsessive tendency to count numbers and find patterns. When he realizes that she is touching him in a pattern corresponding to the Fibonacci Sequence, his appreciation for her increases tremendously.
- In a strip of Frazz by Jef Mallett, Frazz and a student are discussing her knitted hat. The student says, “Mom sewed one sparkly here and here. Two sparklies here. Three sparklies. Five sparklies. Eight sparklies. Thirteen…” To which Frazz replies, “Fibonacci sequins, of course.”
John Waskom postulated that stages of human development followed the Fibonacci sequence, indicating that the unfolding psychology of human life would ideally be a “living proof” of the Golden Mean. This theory was developed and published by Norman Rose in two articles. The first article laid out the general theory: Design and Development of Wholeness: Waskom’s Paradigm. The second article laid out the applications and implications of the theory to the topic of moral development: Moral Development: The Experiential Perspective.
- The Fibonacci sequence plays a small part in the bestselling novel and film The Da Vinci Code.
- In Philip K. Dick’s novel VALIS, the Fibonacci sequence (as well as the Fibonacci constant) are used as identification signs by an organization called the “Friends of God”.
- In the collection of poetry alfabet by the Danish poet Inger Christensen, the Fibonacci sequence is used to define the number of lines in each poem.
- It was briefly included (and recognized by Charles Wallace Murry) in the television film adaptation of A Wrinkle in Time.
- The Fibonacci sequence is frequently referenced in the 2001 book The Perfect Spiral by Jason S. Hornsby.
- A youthful Fibonacci is one of the main characters in the novel Crusade in Jeans (1973). He was left out of the 2006 movie version, however.
- The Fibonacci sequence and golden ratio are briefly described in John Fowles’s 1985 novel A Maggot
- The Fibonacci sequence is explored in Emily Gravett’s 2009 book The Rabbit Problem
- hip hop duo Black Star’s song “Astronomy (8th Light)” from the 1998 album Mos Def & Talib Kweli are Black Star, features the Fibonacci sequence in the chorus:
Now everybody hop on the one, the sounds of the two
It’s the third eye vision, five side dimension
The 8th Light, is gonna shine bright tonight
- Tool’s song “Lateralus” from the album of the same name features the Fibonacci sequence symbolically in the verses of the song. The syllables in the first verse count 1, 1, 2, 3, 5, 8, 5, 3, 13, 8, 5, 3. The missing section (2, 1, 1, 2, 3, 5, 8) is later filled in during the second verse.The time signatures of the chorus change from 9/8 to 8/8 to 7/8; as drummer Danny Carey says, “It was originally titled 9-8-7. For the time signatures. Then it turned out that 987 was the 16th number of the Fibonacci sequence. So that was cool.”
Fibonacci intervals (counting in semitones) in Bartók’s Sonata for Two Pianos and Percussion, 3rd mov. (1937). Play
- Ernő Lendvaï analyzes Béla Bartók’s works as being based on two opposing systems, that of the golden ratioand the acoustic scale. In the third movement of Bartok’s Music for Strings, Percussion and Celesta, the opening xylophone passage uses Fibonacci rhythm as such: 1:1:2:3:5:8:5:3:2:1:1.
- The Fibonacci numbers are also apparent in the organisation of the sections in the music of Debussy’sImage, Reflections in Water, in which the sequence of keys is marked out by the intervals 34, 21, 13 and 8.
- Polish composer Krzysztof Meyer structured the values in his Trio for clarinet, cello and piano according to the Fibonacci sequence.
- Fibonacci’s name was adopted by a Los Angeles-based art rock group The Fibonaccis, that recorded from 1981 to 1987.
- American musician BT also recorded a song titled “Fibonacci Sequence”. The narrator in the song goes through all the numbers of the sequence from 1 to 21 (0 is not mentioned). The track appeared on a limited edition version of his 1999 album Movement in Still Life, and is also featured on the second disc of the Global Underground 013: Ibiza compilation mixed by Sasha.
- Voiceover and recording artist Ken Nordine described Fibonacci numbers in a word jazz piece called “Fibonacci Numbers” on his album A Transparent Mask.
- American musician Doctor Steel has a song titled “Fibonacci Sequence” on his album People of Earth.
- Artist Mario Merz made the Fibonacci sequence a recurring theme in his work. Examples are the Chimney of Turku Energia, in Turku, Finland, featuring the start of the Fibonacci sequence in 2m high neon lights, and the representation of the first Fibonacci numbers with red neon lights on one of the four-faced dome of the Mole Antonelliana in Turin, Italy, part of the artistic work Il volo dei Numeri (“Flight of the numbers”).
- Fibonacci numbers have also been used in knitting to create aesthetically appealing patterns.
- The artist Martina Schettina uses Fibonacci numbers in her paintings. Her “Mathemagic paintings” were shown at the Museumsquartier Vienna in 2010.
- The scientist character Walter Bishop in the television show Fringe recites the Fibonacci sequence to fall asleep. It is later revealed to be the key sequence identifying a series of safe deposit boxes he had maintained.
- Square One Television‘s Mathnet series had a storyline that featured a parrot belonging to a deceased individual who was fascinated by the Fibonacci numbers. When “1, 1, 2, 3” is said in the parrot’s presence, it responds “5, eureka!” This proves to be the key to case; tiles in a garden wall are found to follow the Fibonacci sequence, with a secret compartment hidden behind the lone misplaced tile.
- The Criminal Minds episode “Masterpiece” in season 4 features a serial killer who uses Fibonacci sequences to choose both the amount of victims at a given time and the location of their hometowns.
- Aliens use Fibonacci’s sequence in the Taken episode “God’s Equation”.
- In the Disney Channel TV show So Weird, the Fibonacci sequence is used to build a house. The house becomes a nexus for lost spirits, Fiona is given a choice to use it to free her father as well as the builder of the house, but ultimately chooses to free the spirits and destroys the nexus.
- The Fibonacci sequence is a main plot theme in the 2012 television show Touch, produced by Fox Network and starringKiefer Sutherland.
- In the CBS show Numb3rs, the episode entitled Thirteen uses a Fibonacci sequence embedded in a numeric code left behind by a serial killer.
Fibonacci sequences appear in biological settings, in two consecutive Fibonacci numbers, such as branching in trees,arrangement of leaves on a stem, the fruitlets of a pineapple, the flowering of artichoke, an uncurling fern and the arrangement of a pine cone. In addition, numerous poorly substantiated claims of Fibonacci numbers or golden sections in nature are found in popular sources, e.g., relating to the breeding of rabbits, the seeds on a sunflower, the spirals of shells, and the curve of waves. The Fibonacci numbers are also found in the family tree of honeybees.
Przemysław Prusinkiewicz advanced the idea that real instances can in part be understood as the expression of certain algebraic constraints on free groups, specifically as certain Lindenmayer grammars.
A model for the pattern of florets in the head of a sunflower was proposed by H. Vogel in 1979. This has the form
where n is the index number of the floret and c is a constant scaling factor; the florets thus lie on Fermat’s spiral. The divergence angle, approximately 137.51°, is the golden angle, dividing the circle in the golden ratio. Because this ratio is irrational, no floret has a neighbor at exactly the same angle from the center, so the florets pack efficiently. Because the rational approximations to the golden ratio are of the form F(j):F(j + 1), the nearest neighbors of floret number n are those at n ± F(j) for some index j which depends on r, the distance from the center. It is often said that sunflowers and similar arrangements have 55 spirals in one direction and 89 in the other (or some other pair of adjacent Fibonacci numbers), but this is true only of one range of radii, typically the outermost and thus most conspicuous.
The bee ancestry code
Fibonacci numbers also appear in the description of the reproduction of a population of idealized honeybees, according to the following rules:
- If an egg is laid by an unmated female, it hatches a male or drone bee.
- If, however, an egg was fertilized by a male, it hatches a female.
Thus, a male bee will always have one parent, and a female bee will have two.
If one traces the ancestry of any male bee (1 bee), he has 1 parent (1 bee), 2 grandparents, 3 great-grandparents, 5 great-great-grandparents, and so on. This sequence of numbers of parents is the Fibonacci sequence. The number of ancestors at each level, Fn, is the number of female ancestors, which is Fn−1, plus the number of male ancestors, which is Fn−2. This is under the unrealistic assumption that the ancestors at each level are otherwise unrelated.