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**Some arithmetic problems raised by rabbits, cows and the Da Vinci Code**

December 1, 2009 ISLAMABAD PFAN COMSATS Some arithmetic problems raised by rabbits, cows and the Da Vinci Code Michel Waldschmidt Université P. et M. Curie (Paris VI) Version révisée le 22 novembre 2009

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**Saxophones: Daniel Kientzy Realization: Michel Waldschmidt**

Narayana’s Cows Music: Tom Johnson Saxophones: Daniel Kientzy Realization: Michel Waldschmidt Narayana's Cows, inspired by an Indian mathematician of the 14th century, and playable on any combination of instruments, is written on three staves: the complete melody, the reduced bass melody, and the drone. The present multi-track saxophone version is probably as rich and energetic as any of the large ensemble versions. The melody is played by threeoverdubbed sopranino saxophones in unison, the bass line is played by three baritones, and the drone is played by three altos. Pogus CD, $14.00 US

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**Narayana was an Indian mathematician in the 14th**

Narayana was an Indian mathematician in the 14th. century, who proposed the following problem: A cow produces one calf every year. Begining in its fourth year, each calf produces one calf at the begining of each year. How many cows are there altogether after, for example, 17 years? While you are working on that, let us give you a musical demonstration.

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**The first year there is only the original cow and her first calf.**

1 Original Cow Second generation Total 2 long-short

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**The second year there is the original cow and 2 calves.**

1 2 Original Cow Second generation Total 3 + = long -short -short

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**The third year there is the original cow and 3 calves.**

1 2 3 Original Cow Second generation Total 4 + = long -short -short -short

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**long - short - short - short - long - short**

The fourth year the oldest calf becomes a mother, and we begin a third generation of Naryana’s cows. Year 1 2 3 4 Original Cow Second generation Third generation Total 6 long - short - short - short - long - short

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**Year 1 2 3 4 = + long-short long-short-short long-short-short-short**

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**The fifth year we have another mother cow and 3 new calves.**

1 2 3 4 5 Original Cow Second generation +1 Third generation +2 Total 6 9 +3 + =

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Year 2 3 4 5 = +

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**The sixth year we have 4 productive cows, 4 new calves, and a total herd of 13.**

2 3 4 5 6 Original Cow Second generation Third generation Total 9 13 + =

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The sixth year 4 productive cows = 4 long 9 young calves = 9 short Total: 13 cows = 13 notes

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Year 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 Original Cow Second generation Third generation 21 28 36 45 55 66 78 91 105 Fourth generation 20 35 56 84 120 165 220 286 Fifth generation 70 126 210 330 Sixth generation Seventh generation Total 19 41 60 88 129 189 277 406 595 872

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17th year: 872 cows

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**Saxophones: Daniel Kientzy Realization: Michel Waldschmidt**

Narayana’s Cows Music: Tom Johnson Saxophones: Daniel Kientzy Realization: Michel Waldschmidt Narayana's Cows, inspired by an Indian mathematician of the 14th century, and playable on any combination of instruments, is written on three staves: the complete melody, the reduced bass melody, and the drone. The present multi-track saxophone version is probably as rich and energetic as any of the large ensemble versions. The melody is played by threeoverdubbed sopranino saxophones in unison, the bass line is played by three baritones, and the drone is played by three altos. Pogus CD, $14.00 US

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**Narayana was an Indian mathematician in the 14th**

Narayana was an Indian mathematician in the 14th. century, who proposed the following problem: A cow produces one calf every year. Begining in its fourth year, each calf produces one calf at the begining of each year. How many cows are there altogether after, for example, 17 years? While you are working on that, let us give you a musical demonstration.

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**The first year there is only the original cow and her first calf.**

1 Original Cow Second generation Total 2 long-short

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**The second year there is the original cow and 2 calves.**

1 2 Original Cow Second generation Total 3 long -short -short

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**The third year there is the original cow and 3 calves.**

1 2 3 Original Cow Second generation Total 4 long -short -short -short

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**long - short - short - short - long - short**

The fourth year the oldest calf becomes a mother, and we begin a third generation of Naryana’s cows. Year 1 2 3 4 Original Cow Second generation Third generation Total 6 long - short - short - short - long - short

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**The fifth year we have another mother cow and 3 new calves.**

1 2 3 4 5 Original Cow Second generation +1 Third generation +2 Total 6 9 +3

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**The sixth year we have 4 productive cows, 4 new calves, and a total herd of 13.**

2 3 4 5 6 Original Cow Second generation Third generation Total 9 13

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**Archimedes cattle problem**

The rabbit breeding problem that caused Fibonacci to write about the sequence in Liber abaci may be unrealistic but the Fibonacci numbers really do appear in nature. For example, some plants branch in such a way that they always have a Fibonacci number of growing points. Flowers often have a Fibonacci number of petals, daisies can have 34, 55 or even as many as 89 petals! Finally, next time you look at a sunflower, take the trouble to look at the arrangement of the seeds. They appear to be spiralling outwards both to the left and the right. There are a Fibonacci number of spirals! It seems that this arrangement keeps the seeds uniformly packed no matter how large the seed head.

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Archimedes The cattle problem of Archimedes asks to determine the size of the herd of the God Sun. The sun god had a herd of cattle consisting of bulls and cows, one part of which was white, a second black, a third spotted, and a fourth brown. The rabbit breeding problem that caused Fibonacci to write about the sequence in Liber abaci may be unrealistic but the Fibonacci numbers really do appear in nature. For example, some plants branch in such a way that they always have a Fibonacci number of growing points. Flowers often have a Fibonacci number of petals, daisies can have 34, 55 or even as many as 89 petals! Finally, next time you look at a sunflower, take the trouble to look at the arrangement of the seeds. They appear to be spiralling outwards both to the left and the right. There are a Fibonacci number of spirals! It seems that this arrangement keeps the seeds uniformly packed no matter how large the seed head. Among the bulls, the number of white ones was one half plus one third the number of the black greater than the brown.

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**Archimedes Cattle Problem**

The number of the black, one quarter plus one fifth the number of the spotted greater than the brown. The number of the spotted, one sixth and one seventh the number of the white greater than the brown. Among the cows, the number of white ones was one third plus one quarter of the total black cattle. The number of the black, one quarter plus one fifth the total of the spotted cattle; The number of spotted, one fifth plus one sixth the total of the brown cattle

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**Archimedes cattle problem**

There are infinitely many solutions. The smallest one has digits. This problem was almost solved by a german mathematician, A. Amthor, in 1880, who commented: « Assume that the size of each animal is less than the size of the smallest bacteria. Take a sphere of the same diameter as the size of the milked way, which the light takes ten thousand years to cross. Then this sphere would contain only a tiny proportion of the herd of the God Sun. » The rabbit breeding problem that caused Fibonacci to write about the sequence in Liber abaci may be unrealistic but the Fibonacci numbers really do appear in nature. For example, some plants branch in such a way that they always have a Fibonacci number of growing points. Flowers often have a Fibonacci number of petals, daisies can have 34, 55 or even as many as 89 petals! Finally, next time you look at a sunflower, take the trouble to look at the arrangement of the seeds. They appear to be spiralling outwards both to the left and the right. There are a Fibonacci number of spirals! It seems that this arrangement keeps the seeds uniformly packed no matter how large the seed head.

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**Number of atoms in the known finite universe**

When I was young: 1060 atoms (1 followed by 60 zeroes) A few years later (long back): 1070 The rabbit breeding problem that caused Fibonacci to write about the sequence in Liber abaci may be unrealistic but the Fibonacci numbers really do appear in nature. For example, some plants branch in such a way that they always have a Fibonacci number of growing points. Flowers often have a Fibonacci number of petals, daisies can have 34, 55 or even as many as 89 petals! Finally, next time you look at a sunflower, take the trouble to look at the arrangement of the seeds. They appear to be spiralling outwards both to the left and the right. There are a Fibonacci number of spirals! It seems that this arrangement keeps the seeds uniformly packed no matter how large the seed head. Nowadays: ?

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**Solution of Archimedes Problem**

A. Amthor "Das Problema bovinum des Archimedes » Zeitschrift fur Math. u. Physik. (Hist.-litt.Abtheilung) Volume XXV (1880), pages H.C. Williams, R.A. German and C.R. Zarnke, Solution of the cattle problem of Archimedes, Math. Comp., 19, (1965).

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**How many ancesters do we have?**

Sequence: 1, 2, 4, 8, 16 …

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Bees genealogy

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**Bees genealogy Number of females at a given level =**

total population at the previous level Number of males at a given level= number of females at the previous level Bees genealogy Sequence: 1, 1, 2, 3, 5, 8, … 3 + 5 = 8 2 + 3 = 5 1 + 2 = 3 1 + 1 = 2 0 + 1 = 1 1 + 0 = 1

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**Fibonacci (Leonardo di Pisa)**

Pisa (Italia) ≈ Liber Abaci ≈ 1202 0, 1, 1, 2, 3, 5, 8, 13, 21, 34, 55,… Each term is the sum of the two preceding ones: 21+34=55 Fibonacci played an important role in reviving ancient mathematics and made significant contributions of his own. Liber abaci introduced the Hindu-Arabic place-valued decimal system and the use of Arabic numerals into Europe. The Fibonacci Quarterly is meant to serve as a focal point for interest in Fibonacci numbers and related questions, especially with respect to new results, research proposals, challenging problems, and innovative proofs of old ideas.

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**Modelization of a population**

First month Adult pairs Young pairs Second month Third month Fourth month Fifth month Sixth month Sequence: 1, 1, 2, 3, 5, 8, …

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**The Fibonacci sequence**

1, 1, 2, 3, 5, 8, 13, 21, 34, 55, 89, 144, 233, 377, 610, … 1+1=2 1+2=3 2+3=5 3+5=8 5+8=13 8+13=21 13+21=34 21+34=55

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Narayana cows (back) année 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 Total 19 28 41 60 88 129 189 277 406 595 872 Each term is the sum of the preceding one and the one before the penultimate: 872= Pénultième = avant dernier, ante-penultième = avant avant dernier

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**Ajout de l’année n = ajout de l’année n-1 + ajout de l’année n-2**

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**Theory of stable populations (Alfred Lotka)**

Assume each pair generates a new pair the first two years only. Then the number of pairs who are born each year again follow the Fibonacci rule. Arctic trees In cold countries, each branch of some trees gives rise to another one after the second year of existence only.

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**Exponential sequence First month Second month Third month Fourth month**

Number of pairs: 1, 2, 4, 8, …

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**Size of Narayana’s herd after 60 years: 11 990 037 126 (11 digits)**

Number of Fibonacci rabbits after 60 months: (13 digits) Number of pairs of mice after 60 months: (19 digits) Exponential growth 59 ème année: nombre de vaches de Narayana: Différence: Beetle larvas Bacteria Economy

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Phyllotaxy Study of the position of leaves on a stem and the reason for them Number of petals of flowers: daisies, sunflowers, aster, chicory, asteraceae,… Spiral patern to permit optimal exposure to sunlight Pine-cone, pineapple, Romanesco cawliflower, cactus

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Leaf arrangements

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Université de Nice, Laboratoire Environnement Marin Littoral, Equipe d'Accueil "Gestion de la Biodiversité"

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Phyllotaxy Marguerite, choux Romanesco, pomme de pin, cactus - aussi: ananas, tournesol Peter S. Stevens. Les formes dans la nature. Seuil, 1978. Disposition des feuilles

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Phyllotaxy J. Kepler (1611) uses the Fibonacci sequence in his study of the dodecahedron and the icosaedron, and then of the symmetry of order 5 of the flowers. Stéphane Douady and Yves Couder Les spirales végétales La Recherche 250 (Jan. 1993) vol. 24.

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**Plucking the daisy petals**

En effeuillant la marguerite I love you A little bit A lot With passion With madness Not at all Je t’aime Un peu Beaucoup Passionnément À la folie Pas du tout He loves me He loves me not Sequence of the remainders of the division by 6 of the Fibonacci numbers He loves me. He loves me not. The chant continues as one-by-one the daisy petals are plucked from the flower and dropped to the ground. At game's end the last petal tells all; whether or not the person of their desire also cares for them in return. 1, 1, 2, 3, 5, 2, 1, 3, 4, 1, 5, 0, 5,… First multiple of 6 : 144

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**Division by 6 of the Fibonacci numbers**

8 = 6 I love you: A little bit : A lot : With passion : +4 With madness : +5 Not at all : 13 = 6 21 = 6 34 = 6 55 = 6 89 = 6 144 = 6

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**Division by 2 of the Fibonacci numbers**

8 = 2 He loves me : He loves me not +0 13 = 2 21 = 2 34 = 2 55 = 2 89 = 2 144 = 2

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**The Da Vinci Code Five enigmas to be solved**

In the book written by Dan Brown in one finds some (weak) crypto techniques. 1 The first enigma asks for putting in the right order the integers of the sequence This reordering will provide the key of the bank account. 2 An english anagram O DRACONIAN DEVIL, OH LAME SAINT The rabbit breeding problem that caused Fibonacci to write about the sequence in Liber abaci may be unrealistic but the Fibonacci numbers really do appear in nature. For example, some plants branch in such a way that they always have a Fibonacci number of growing points. Flowers often have a Fibonacci number of petals, daisies can have 34, 55 or even as many as 89 petals! Finally, next time you look at a sunflower, take the trouble to look at the arrangement of the seeds. They appear to be spiralling outwards both to the left and the right. There are a Fibonacci number of spirals! It seems that this arrangement keeps the seeds uniformly packed no matter how large the seed head. 3 A french anagram SA CROIX GRAVE L’HEURE

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**The Da Vinci Code Five enigmas to be solved (continued)**

4 A french poem to be decoded : élc al tse essegas ed tom xueiv nu snad eétalcé ellimaf as tinuér iuq sreilpmet sel rap éinéb etêt al eélévér ares suov hsabta ceva cinq épreuves de codage Sophie Neveu 5 An old wisdom word to be found. Answer for 5: SOPHIA (Sophie Neveu)

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**The Da Vinci Code the bank account key involving eight numbers**

The eight numbers of the key of the bank account are: These are the eight first integers of the Fibonacci sequence. The goal is to find the right order at the first attempt. The right answer is given by selecting the natural ordering: The rabbit breeding problem that caused Fibonacci to write about the sequence in Liber abaci may be unrealistic but the Fibonacci numbers really do appear in nature. For example, some plants branch in such a way that they always have a Fibonacci number of growing points. Flowers often have a Fibonacci number of petals, daisies can have 34, 55 or even as many as 89 petals! Finally, next time you look at a sunflower, take the trouble to look at the arrangement of the seeds. They appear to be spiralling outwards both to the left and the right. There are a Fibonacci number of spirals! It seems that this arrangement keeps the seeds uniformly packed no matter how large the seed head. The total number of solutions is 20 160

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**Primitive languages Given some letters, how many words does one obtain**

if one uses each letter exactly once? With 1 letter a, there is just one word: a. With 2 letters a,b, there are 2 words, namely ab, ba. With 3 letters a,b,c : select the first letter (3 choices), once it is selected, complete with the 2 words involving the 2 remaining letters. Hence the number of words is 3 ·2 ·1=6, namely abc, acb, bac, bca, cab, cba. The rabbit breeding problem that caused Fibonacci to write about the sequence in Liber abaci may be unrealistic but the Fibonacci numbers really do appear in nature. For example, some plants branch in such a way that they always have a Fibonacci number of growing points. Flowers often have a Fibonacci number of petals, daisies can have 34, 55 or even as many as 89 petals! Finally, next time you look at a sunflower, take the trouble to look at the arrangement of the seeds. They appear to be spiralling outwards both to the left and the right. There are a Fibonacci number of spirals! It seems that this arrangement keeps the seeds uniformly packed no matter how large the seed head.

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**Six words 3 ·2 ·1=6 3 2 1 Three letters: a, b, c First letter Second**

Third Word c Six words abc b c b a c b c a acb bac a c 3 ·2 ·1=6 bca cab a b cba 3 2 1

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**Four letters: a, b, c, d a b c d b c d a c d a b d a b c c d abcd d c**

abdc c b d acbd d d b acdb b c adbc c b adcb a a c d bacd d c badc c a d bcad b d d a bcda a c c bdac a bdca b d cabd c a d b cadb b a d cbad d d a cbda a b b cdab d a cdba b a c c dabc b b a dacb c c dbac c a a dbca b b dcab a dcba

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The sequence In the same way, with 8 letters, the number of words is 8· 7 ·6 ·5 ·4 ·3 ·2 ·1= Here the digit 1 occurs twice, this is why the number of orderings is only half: 20 160 The rabbit breeding problem that caused Fibonacci to write about the sequence in Liber abaci may be unrealistic but the Fibonacci numbers really do appear in nature. For example, some plants branch in such a way that they always have a Fibonacci number of growing points. Flowers often have a Fibonacci number of petals, daisies can have 34, 55 or even as many as 89 petals! Finally, next time you look at a sunflower, take the trouble to look at the arrangement of the seeds. They appear to be spiralling outwards both to the left and the right. There are a Fibonacci number of spirals! It seems that this arrangement keeps the seeds uniformly packed no matter how large the seed head.

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**The Da Vinci Code 2 An english anagram DRACONIAN DEVIL, OH LAME SAINT**

THE MONA LISA LEONARDO DA VINCI 3 A french anagram SA CROIX GRAVE L’HEURE LA VIERGE AUX ROCHERS The rabbit breeding problem that caused Fibonacci to write about the sequence in Liber abaci may be unrealistic but the Fibonacci numbers really do appear in nature. For example, some plants branch in such a way that they always have a Fibonacci number of growing points. Flowers often have a Fibonacci number of petals, daisies can have 34, 55 or even as many as 89 petals! Finally, next time you look at a sunflower, take the trouble to look at the arrangement of the seeds. They appear to be spiralling outwards both to the left and the right. There are a Fibonacci number of spirals! It seems that this arrangement keeps the seeds uniformly packed no matter how large the seed head.

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**The Da Vinci Code (continued)**

4 A french poem to decode: élc al tse essegasedtom xueiv nu snad eétalcé ellimaf as tinuér iuq sreilpmet sel rap éinéb etêt al eélévér ares suov hsabta ceva dans un vieux mot de sagesse est la clé qui réunit sa famille éclatée la tête bénie par les Templiers avec Atbash vous sera révélée The rabbit breeding problem that caused Fibonacci to write about the sequence in Liber abaci may be unrealistic but the Fibonacci numbers really do appear in nature. For example, some plants branch in such a way that they always have a Fibonacci number of growing points. Flowers often have a Fibonacci number of petals, daisies can have 34, 55 or even as many as 89 petals! Finally, next time you look at a sunflower, take the trouble to look at the arrangement of the seeds. They appear to be spiralling outwards both to the left and the right. There are a Fibonacci number of spirals! It seems that this arrangement keeps the seeds uniformly packed no matter how large the seed head. « utiliser un miroir pour déchiffrer le code » « use a mirror for decoding»

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**Geometric construction of the Fibonacci sequence**

8 5 3 2 1 1 2

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**This is a nice rectangle**

A square 1 1 -1

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**Golden Rectangle Sides: 1 and Condition:**

-1 Sides: 1 and Condition: The two rectangles with sides 1 and for the big one, -1 and1 for the small one, have the same proportions. Proportion of the big one: 1 = Proportion of the small one: 1 -1

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**Golden Rectangle Sides: 1 and =1.618033… 1 -1 = Condition:**

(-1)=1 1 -1 = Equation: 2--1 = 0 Solution: is the Golden Number …

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**To go from the large rectangle to the small one:**

divide each side by 1 -1 = The Golden Rectangle 1 -1 1 1 2 1 3

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**Ammonite (Nautilus shape)**

L’enroulement régulier d’une ammonite se fait selon une spirale logarithmique

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Spirals in the Galaxy

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**The Golden Number in art, architecture,… aesthetics**

The architectural theory of Luca Pacioli: ''De divina proportione'’ Ned May has generated some beautiful pictures based on Fibonacci Spirals using Visual Basic (an example is shown here on the right).

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**Kees van Prooijen http://www.kees.cc/gldsec.html**

Kees van Prooijen of California has used a similar series to the Fibonacci series - one made from adding the previous three terms, as a basis for his art.

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**Regular pentagons and dodecagons**

nbor7.gif =2 cos(/5)

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**Penrose non-periodic tiling patterns and quasi-crystals**

Cerf volant et flèche

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Thick rhombus G G/M= Thin rhombus M

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proportion=

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**Diffraction of quasi-crystals**

diffraction des quasi-cristaux.

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**Doubly periodic tessalation (lattices) - cristallography**

Géométrie d'un champ de lavande J'ai pris cette photographie sur le plateau de Valensole (04) en février 1981, à l'aide d'un objectif à grand angle (90° environ). Les pieds de lavande sont disposés aux sommets d'un réseau Z² du plan horizontal enneigé. L'image donne de ce réseau une vue en perspective, qui est sa projection sur le plan du film à partir d'un point (le centre optique de l'objectif). Observer et expliquer : - la ligne d'horizon - les "points de convergence" à l'infini, qui correspondent aux droites de pentes 0, 1, 2, 3, 4 du réseau. Distinguez-vous celles de pente -1/2, 1/2, 3/2 ? - les nombreuses divisions harmoniques, formées par trois pieds alignés consécutifs et le point à l'infini de la droite qui les porte. Géométrie d'un champ de lavande François Rouvière (Nice)

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**The Golden Number and aesthetics**

The architectural theory of Luca Pacioli: ''De divina proportione''

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**The Golden Number and aesthetics**

The architectural theory of Luca Pacioli: ''De divina proportione''

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**Marcus Vitruvius Pollis (Vitruve, 88-26 av. J.C.)**

Léonard de Vinci (Leonardo da Vinci, ) The architectural theory of Luca Pacioli: ''De divina proportione''

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**Music and the Fibonacci sequence**

Dufay, XVème siècle Roland de Lassus Debussy, Bartok, Ravel, Webern Stockhausen Xenakis Tom Johnson Automatic Music for six percussionists Citer Yves Hellegouarch Tom Johnson: Automatic Music for six percussionists, 221 mesures de 5 notes ici, c'est à dire 1205 notes,

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**Example of a recent result**

Y. Bugeaud, M. Mignotte, S. Siksek (2004): The only perfect powers (squares, cubes, etc.) in the Fibonacci sequence are 1, 8 and 144.

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**The quest of the Graal for a mathematician**

Open problems, conjectures. Example of an unsolved question: Are there infinitely many primes in the Fibonacci sequence?

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**Some applications of Number Theory**

Cryptography, security of computer systems Data transmission, error correcting codes Interface with theoretical physics Musical scales Numbers in nature

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**Some arithmetic problems raised by rabbits, cows and the Da Vinci Code**

December 1, 2009 ISLAMABAD PFAN COMSATS Some arithmetic problems raised by rabbits, cows and the Da Vinci Code Michel Waldschmidt Université P. et M. Curie (Paris VI)

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