# Why is the Fibonacci sequence so important

## The Fibonacci sequence

### Table of Contents

2. INTRODUCTION

3. LEONARDO FIBONACCI

4. DERIVATION OF THE FIBONACCI SEQUENCE

5. RELATIONSHIP WITH THE GOLDEN CUT

6. EXAMPLES OF THE FIBONACCI SEQUENCE

7. LIST OF SOURCES

8. APPENDIX

### 2. Introduction

"Fibonacci, Leonardo (around 1170 to around 1240), Italian businessman and mathematician, who brought together the mathematical knowledge of the classical European, Arab and Indian cultures and supplemented it with contributions to algebra and number theory." (From " *Microsoft ® Encarta ® Encyclopedia - the 2005 © 1993-2004 "Microsoft Corporation (Virtual Lexicon), entry" Fibonacci ")*

You can find such an entry if you look up “Fibonacci” in a dictionary.

Who Fibonacci really was and what “contributions to algebra and number theory” he made, I would like to explain in more detail in the written elaboration of my GLF on the subject of “The Fibonacci sequence”.

At the beginning I will go into more detail on Fibonacci’s biography, then derive “his” famous Fibonacci sequence using the model of a rabbit population, clarify the connection to the golden ratio and finally go into a few examples of the application of the Fibonacci sequence today.

### 3. Leonardo Fibonacci

The Italian businessman and mathematician Leonardo Fibonacci ("figlio di Bonacci" means son of Bonacci) was born around 1170 in Pisa, which is why he is also called Leonardo of Pisa.

Since his father worked as a customs officer in what is now Bougie in Algeria, Leonardo was tutored by a Moorish private teacher. From him he learned the Arabic number system and the basics of commercial arithmetic. He traveled a lot and finally wrote his “Liber abaci” (“Book of Arithmetic”) in 1202, a work in which he explained this number system and its calculation.

This publication is still of great importance today, as it led to the suppression of the Roman numerals at that time and is very similar to today's number system. However, Fibonacci became famous mainly through the "Fibonacci sequence", which the French mathematician Edouard Lucas only later named after him.

The Fibonacci sequence is

(0), 1, 1, 2, 3, 5, 8, 13, 21, 34, 55,…

So it forms each new successor member from the sum of the two previous members.

Fibonacci published a few more works, of which only a few still exist, and he also succeeded in solving the equation of the third degree x, for example^{3} + 2x^{2} + 10x = 20

Leonardo Fibonacci died in Pisa around 1240.

### 4. Derivation of the Fibonacci sequence

Fibonacci introduced “his sequence” using a rabbit population as an example. The “rabbit problem” published in his arithmetic book “Liber abaci” was as follows: It is assumed that

I. There is a pair of rabbits (a "Parent Couple") in a fenced area on January 1st of each year

II. Another pair of (children) rabbits litters on February 1st and on each additional first day of the month.

III. It is also assumed that each new (child) rabbit couple grows to sexual maturity for one month, i.e. becomes a “parent couple” itself and then also throws a (child) rabbit pair in the third month of life and on each subsequent first day of the month .

Fibonacci then asked the question "How many pairs of rabbits (children and parents) live after n months if there was a young couple at the beginning?"

The growth stock after one year (n = 13) can be shown in a table:

Figure not included in this excerpt

Result: After one year there would be 233 (parent) rabbit pairs, 144 (child) rabbit pairs, i.e. 377 pairs (754 rabbits) in total. To answer the question of how many rabbits live after n months, you have to find a general rule, i.e. a sequence that enables the calculation of each individual successor element: As already said, it forms each new successor element from the sum of the two previous elements.

Figure not included in this excerpt

The following applies to the initial terms f0 = 0 and f1 = 1.

Sequences with this recursion rule are called general Fibonacci sequences.

From this it follows: Fibonacci sequence Fn = 0, 1, 1, 3, 5, 8, 13, 21, 34, 55, ...

The terms of the Fibonacci sequence are called Fibonacci numbers.

Edouard Lucas (see 2. Leonardo Fibonacci) “baptized” this sequence “Fibonacci sequence”. In honor of this act, the "Lucas sequence" was named after him, a sequence with the same recursion rule, in which the initial terms f0 = 1 and f1 = 3.

From this it follows: Lucas sequence Ln = 1, 3, 4, 7, 11, 18, 29, 47, ...

With these recursive rules, however, the terms of the sequence can only be calculated using the previous ones.

Therefore one looked for an explicit sequence that describes this growth:

Figure not included in this excerpt

The following applies: [Figure not included in this excerpt]

This explicit sequence was not found until 1843; it is called the “Binet formula” after the French mathematician Jacques-Phillipe-Marie Binet (1786 - 1856).

### 5. Relationship with the golden ratio

If you divide two consecutive terms of the Fibonacci sequence, you will quickly notice that the quotient for large n approaches the value 1.61803398…. This value is also called the golden ratio Φ.

According to the definition, "The golden ratio (lat. *sectio aurea*) [is] a certain ratio of two numbers or quantities. It's about 1: 1.618. [...] Other terms used are continuous division and divine division (lat. * proportio divina*). "(Source: www.wikipedia.org)

Here the quotients from the first 12 (or 13) Fibonacci numbers

Figure not included in this excerpt

*(Values from www.wikipedia.org, entry "golden ratio")*

The golden ratio Φ is an irrational number. It can be most easily represented by dividing two consecutive Fibonacci numbers:

Figure not included in this excerpt

*(Graphics from www.wikipedia.org, entry "Fibonacci sequence")*

### 6. Examples of the Fibonacci sequence

The connection between the golden ratio and the Fibonacci sequence just described results in an astonishing number of examples of the Fibonacci sequence in our everyday lives.

The golden ratio can be found almost everywhere in what we consider “correct”, “standardized” proportions. One often comes across our Fibonacci numbers.

Here is a selection of examples:

- In architecture, the golden ratio can be traced back to ancient times. The proportions of areas in the Great Pyramid of Cheops and on the front of the Parthenon temple on the Athens Acropolis correspond, according to today's calculations, to the golden ratio. "

Figure not included in this excerpt

Parthenon temple with five assumed golden rectangles arranged in the manner of a golden spiral "(source www.wikipedia.org)

- Astronomy: The orbital times of some planets and their moons share in the ratio of the golden section. This is supposed to stabilize the orbit.

- In music we only perceive tone sequences, ie melodies, as constant and clear when the relationship between the oscillation frequencies is correct (= "interval"). These are also partly determined by the golden ratio.

- In the figurative sense, art consists almost entirely of the golden ratio. It is extremely influenced by the artist's aesthetics and perception of proportion. On top of that, art history was written through the deliberate break with the golden ratio!

- Botany shows most of the examples of the golden section. Inflorescences and leaf arrangement run in so-called "Fibonacci spirals".

Figure not included in this excerpt

Sunflower with 34 and 55 Fibonacci spirals. (Source www.wikipedia.org)

Figure not included in this excerpt

“Arrangement of leaves at the distance of the golden angle viewed from above. The sunlight is optimally used "(source www.wikipedia.org)

Figure not included in this excerpt

Spruce cones with 5, 8 and 13 Fibonacci spirals. (Source

- The Fibonacci numbers can also be found in the human body! Leonardo da Vinci already researched the proportions of the human body. To this day, they shape our image of beauty more than any fashion trend can!

Figure not included in this excerpt

Human proportions according to Vitruvius by Leonardo da Vinci (1492) (www.wikipedia.org)

Now we know that the Fibonacci numbers are an integral part of our lives, but what benefit can we get from this discovery?

- The Fibonacci numbers and the golden ratio are deliberately used in instrument construction; for example, they are intended to give the resonance body of string instruments a special sound.

- Both the composer Béla Bartóks and the rock band Tool used the Fibonacci sequence in their compositions. They let the motifs (Bartóks) and syllables of the verse texts (Tool) sound in the appropriate rhythm.

- In the technical area, the proportions of the golden ratio are used, among other things, for paper format ("DIN A4"), for television images and PC monitors.

- A controversial application of the Fibonacci numbers is currently being discussed in the stock market. The Fibonacci business cycles are supposed to not only explain past fluctuations, but also to predict future fluctuations!

- The Fibonacci sequence can also be used for encryption. However, I do not know the exact principle. In this function, the Fibonacci sequence also makes its big appearance in Dan Brown's bestseller “The Da Vinci Code”!

### 7. List of sources

- "Fibonacci and Lucas Numbers", Verner E. Hoggatt, Jr, 1969

- „ *Microsoft ® Encarta ® Encyclopedia - the 2005 © 1993-2004* "Microsoft Corporation (Virtual Lexicon), entry" Fibonacci "

- *www.wikipedia.org*, Entry "Fibonacci sequence" & entry "The golden ratio"

- "FIBONACCI AND THE GOLDEN CUT", Johannes Becker, February 2004 (from *www.uni-giessen.de/~g013*)

- *http://www.bogen-gmbh.de/unser_know-how_.html* ("Example of a Fibonacci cycle")

- *http: //www.boerse- online.de/wissen/lexikon/boersenlexikon/index.html?action=descript&buchstabe=F& concept = Fibonacci +% 281% 29*

- *http://www.sdk.org/lexikon_inhalt.php?id=11* ("Fibonacci Business Cycles")

- *http://www.math-edu.de/Mathegarten/Mathe_Musik/mathe_musik.html#Fibonacci- numbers% 20in% 20der% 20Musik* (Fibonacci numbers in music)

- “The Da Vinci Code,” Dan Brown, 2004, Illustrated Edition

### 8. Appendix

Mathematics GLF

"The Fibonacci Sequence"

From Bettina Munz

Structure:

Short biography of the Fibonacci sequence

Relationship with the golden ratio Examples of the Fibonacci sequence

Short biography

- Born in Pisa in 1170 ➔ "Leonardo of Pisa"

- Moorish private teacher ➔ Arabic number system & commercial calculations "Learning by traveling"

- 1202 "Liber abaci" ("Book of arithmetic")

- Consequences: displacement of Roman numbers, similarity to today's number system

Fibonacci sequence

It is believed that

IV. There is a pair of rabbits (a "parent pair") in a fenced area on January 1st, which

V. throws another pair of (children) rabbits on February 1st and on each additional first day of the month.

VI. It is also assumed that each new (child) rabbit couple grows to sexual maturity for one month, i.e. becomes a “parent couple” itself and then also throws a (child) rabbit pair in the third month of life and on each subsequent first day of the month .

VII.

Figure not included in this excerpt

Mathematics GLF

"The Fibonacci Sequence"

From Bettina Munz

Leonardo Fibonacci

- Born in Pisa in 1170 ➔ "Leonardo of Pisa"

- Moorish private teacher ➔ Arabic number system (1,2,3, ...) & commercial calculations

- "Learning by traveling"

- 1202 "Liber abaci" ("Book of arithmetic")

- Consequences: displacement of Roman numbers (I, II, III, IV, ...), number system with similarities in today's number system

2. Fibonacci sequence

➔ Example "rabbit problem" (see assignment on the back)

a. Recursive sequence:

[Figure not included in this excerpt], (General Fibonacci Arc)

With initial terms f0 = 0 and f1 = 1 ➔ Fibonacci sequence Fn = 0, 1, 1, 3, 5, 8, 13, 21, 34, 55, ... With initial terms f0 = 1 and f1 = 3 ➔ Lucas sequence Ln = 1, 3, 4, 7, 11, 18, 29, 47, ...

b. Explicit consequence: ("Binet's formula" according to Jacques-Phillipe-Marie Binet)

Figure not included in this excerpt

Relationship to the golden ratio

The quotient of two consecutive Fibonacci numbers strives for the value 1.61803…. ➔ corresponds to the "golden ratio" Φ = 1.61803398 ... (= best. Ratio of two numbers or sizes)

Examples of the Fibonacci sequence and the golden ratio

In everyday life (through proportions):

Figure not included in this excerpt

- architecture

- astronomy

- music

- Art

- botany

- Application:

- Instrument making

- compositions in music, images in art

- Technical area (TV, photo, monitor formats)

- Stock market forecasts

- encryption

Work order:

Fill in the table with the corresponding values for the number of parent rabbit pairs, child rabbit pairs and the number of all rabbit pairs using the recursive Fibonacci sequence.

Why would the value f0 = 0 for the number of parent rabbit pairs make no sense?

Figure not included in this excerpt

Result: After one year there would be _____ pairs of rabbits, i.e. _____ rabbits!

Question: The rabbit population example is illustrative, but why is it unrealistic?

Fibonacci sequence

Figure not included in this excerpt

The following applies to the initial terms f0 = 0 and f1 = 1.

Sequences with this recursion rule are called general Fibonacci sequences. From this it follows: Fibonacci sequence Fn = 0, 1, 1, 3, 5, 8, 13, 21, 34, 55, ...

The same recursion rule, but initial terms f0 = 1 and f1 = 3 From this it follows: Lucas sequence Ln = 1, 3, 4, 7, 11, 18, 29, 47, ...

Explicit result: 1843, it is called "Binet's formula" (Jacques-Phillipe-Marie Binet)

Figure not included in this excerpt

Relationship to the golden ratio

According to the definition, "The golden ratio (lat. *sectio aurea*) [is] a certain ratio of two numbers or quantities. It's about 1: 1.618. [...] Other terms used are continuous division and divine division (lat. *proportio divina*). "(Source: www.wikipedia.org)

Figure not included in this excerpt

(Values from www.wikipedia.org, entry "golden ratio")

16

Examples of the Fibonacci sequence

1. Architecture

2. Astronomy

3. Music (= "interval")

4. Art

Figure not included in this excerpt

"Parthenon temple with five assumed golden rectangles arranged like a golden spiral" (source www.wikipedia.org)

botany

Figure not included in this excerpt

Spruce cones with 5, 8 and 13 Fibonacci spirals. (Source www.wikipedia.org)

Figure not included in this excerpt

Sunflower with 34 and 55 Fibonacci spirals. (Source www.wikipedia.org)

Figure not included in this excerpt

“Arrangement of leaves at the distance of the golden angle viewed from above. The sunlight is used optimally "(source www.wikipedia.org)

5. human body

Figure not included in this excerpt

Human proportions according to Vitruvius by Leonardo da Vinci (1492) (www.wikipedia.org)

6. Applications:

7. Instrument making

8. Compositions ("Tool")

9. technical area

(A4, TV, PC monitors)

10. Exchange business

11. Encryption

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