RabbitsCowsDaVinci

Slide1: 

Simon Fraser University, Vancouver Campus (Harbour Center) Segal Center SFU Vancouver 515, West Hastings (at Richards) Some arithmetic problems raised by rabbits, cows and the Da Vinci Code Michel Waldschmidt Université P. et M. Curie (Paris VI) July 12, 2006 CNTA9 http://www.math.jussieu.fr/~miw/ Version révisée le 14/07/2006

Slide2: 

http://www.pogus.com/21033.html Narayana’s Cows Music: Tom Johnson Saxophones: Daniel Kientzy Realization: Michel Waldschmidt http://www.math.jussieu.fr/~miw/

Slide3: 

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.

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

The first year there is only the original cow and her first calf. long-short

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

The second year there is the original cow and 2 calves. long -short -short

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

The third year there is the original cow and 3 calves. long -short -short -short

The fourth year the oldest calf becomes a mother, and we begin a third generation of Naryana’s cows.: 

The fourth year the oldest calf becomes a mother, and we begin a third generation of Naryana’s cows. long - short - short - short - long - short

Slide8: 

Year = +

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

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

Slide10: 

Year 2 3 4 5 = +

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

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

Slide12: 

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

Slide16: 

17th year: 872 cows

Slide17: 

http://www.pogus.com/21033.html Narayana’s Cows Music: Tom Johnson Saxophones: Daniel Kientzy Realization: Michel Waldschmidt http://www.math.jussieu.fr/~miw/

Slide18: 

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.

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

The first year there is only the original cow and her first calf. long-short

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

The second year there is the original cow and 2 calves. long -short -short

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

The third year there is the original cow and 3 calves. long -short -short -short

The fourth year the oldest calf becomes a mother, and we begin a third generation of Naryana’s cows.: 

The fourth year the oldest calf becomes a mother, and we begin a third generation of Naryana’s cows. long - short - short - short - long - short

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

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

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

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

Slide26: 

Archimedes cattle problem

Archimedes : 

Archimedes The cattle problem of Archimedes asks to determine the size of the herd of the God Sun. This problem amounts to find two integers x and y such that the square of x minus a suitable multiple of the square of y is 1 x2 - 410 286 423 278 424 y2 =1

Archimedes cattle problem : 

Archimedes cattle problem There are infinitely many solutions. The smallest one has x with 206 545 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. »

Number of atoms in the known finite universe : 

Number of atoms in the known finite universe When I was young: 1060 atoms A few years later (long back): 1070 Nowadays: ?

Solution of Archimedes problem : 

Solution of Archimedes problem H.C. Williams, R.A. German and C.R. Zarnke, 1965

Pell-Fermat equation : 

Pell-Fermat equation x2 - d y2 =1 Brahmagupta (628) x2 - 92 y2 =1 Smallest solution 11512 - 92 · 1202 =1 Bhaskhara II (1150) x2 - 61 y2 =1 Smallest solution 17663190492 - 61 · 2261539802 =1 Narayana (XIVth Century) x2 - 103 y2 =1 Smallest solution 2275282 - 103 · 224192 =1 Fermat: 1601 (?)- 1665 1151 ·1151 = 1324801 92 · 120 ·120 = 1324800

Fibonacci (Leonardo di Pisa): 

Fibonacci (Leonardo di Pisa) Pisa (Italia) ≈ 1175 - 1250 Liber Abaci ≈ 1202

Modelization of a population: 

Modelization of a population Third month Fifth month Sixth month Second month Fourth month Adult pairs Young pairs Sequence: 1, 1, 2, 3, 5, 8, …

The Fibonacci sequence: 

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

Theory of stable populations (Alfred Lotka): 

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.

The Da Vinci Code  Five enigmas to be solved : 

The Da Vinci Code Five enigmas to be solved 1 The first enigma asks for putting in the right order the integers of the sequence 1 3 - 3 - 2 - 2 1 - 1 - 1 - 8 - 5. This reordering will provide the key of the bank account. 2 An english anagram O DRACONIAN DEVIL, OH LAME SAINT 3 A french anagram SA CROIX GRAVE L’HEURE In the book written by Dan Brown in 2003 one finds some (weak) crypto techniques.

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

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 5 An old wisdom word to be found. Answer for 5: SOPHIA (Sophie Neveu)

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

The Da Vinci Code the bank account key involving eight numbers 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: 1 - 1 - 2 - 3 - 5 - 8 - 1 3 - 2 1 The total number of solutions is 20 160 The eight numbers of the key of the bank account are: 1 3 - 3 - 2 - 2 1 - 1 - 1 - 8 - 5

Primitive languages: 

Primitive languages 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. 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.

Slide42: 

Three letters: a, b, c Six words 3 ·2 ·1=6 abc acb bac bca cab cba 3 2 1 First letter Second Third Word

Slide43: 

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

The sequence 1 3 - 3 - 2 - 2 1 - 1 - 1 - 8 - 5: 

The sequence 1 3 - 3 - 2 - 2 1 - 1 - 1 - 8 - 5 In the same way, with 8 letters, the number of words is 8· 7 ·6 ·5 ·4 ·3 ·2 ·1= 40 320. Here the digit 1 occurs twice, this is why the number of orderings is only half: 20 160

The Da Vinci Code: 

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 Da Vinci Code (continued): 

The Da Vinci Code (continued) 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 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 « utiliser un miroir pour déchiffrer le code » « use a mirror for decoding»

Exponential sequence : 

Exponential sequence Second month Third month Fourth month Number of pairs: 1, 2, 4, 8, …

Number of Fibonacci rabbits after 60 months:  1 548 008 755 920 (13 digits): 

Number of Fibonacci rabbits after 60 months: 1 548 008 755 920 (13 digits) Beetle larvas Bacteria Economy Exponential growth Number of pairs of mice after 60 months: 1 152 921 504 606 846 976 (19 digits) Size of Narayana’s herd after 60 years: 11 990 037 126 (11 digits)

How many ancesters do we have? : 

How many ancesters do we have? Sequence: 1, 2, 4, 8, 16 …

Bees genealogy : 

Bees genealogy

Bees genealogy: 

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 Sequence: 1, 1, 2, 3, 5, 8, …

Phyllotaxy: 

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

Leaf arrangements: 

Leaf arrangements

http://www.unice.fr/LEML/coursJDV/tp/tp3.htm: 

http://www.unice.fr/LEML/coursJDV/tp/tp3.htm Université de Nice, Laboratoire Environnement Marin Littoral, Equipe d'Accueil "Gestion de la Biodiversité"

Phyllotaxy: 

Phyllotaxy

Phyllotaxy: 

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 et Yves Couder Les spirales végétales La Recherche 250 (janvier 1993) vol. 24.

Plucking the daisy petals: 

Plucking the daisy petals I love you A little bit A lot With passion With madness Not at all Sequence of the remainders of the division by 6 of the Fibonacci numbers 1, 1, 2, 3, 5, 2, 1, 3, 4, 1, 5, 0, 5,… First multiple of 6 : 144 En effeuillant la marguerite Je t’aime Un peu Beaucoup Passionnément À la folie Pas du tout He loves me He loves me not

Division by 6 of the Fibonacci numbers: 

Division by 6 of the Fibonacci numbers 8 = 6  1 + 2 13 = 6  2 + 1 21 = 6  3 + 3 34 = 6  5 + 4 55 = 6  9 + 1 89 = 6  14 + 5 144 = 6  24 + 0 I love you: +1 A little bit : +2 A lot : +3 With passion : +4 With madness : +5 Not at all : +0

Division by 2 of the Fibonacci numbers: 

Division by 2 of the Fibonacci numbers 8 = 2  4 + 0 13 = 2  6 + 1 21 = 2  10 + 1 34 = 2  17 + 0 55 = 2  27 + 1 89 = 2  44 + 1 144 = 2  72 + 0 He loves me : +1 He loves me not +0

Geometric construction of the Fibonacci sequence: 

Geometric construction of the Fibonacci sequence

Slide63: 

This is a nice rectangle A N I C E R E C T A N G L E A square 1 1  -1

Slide64: 

Golden Rectangle 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.

Slide65: 

Golden Rectangle Sides: 1 and  Solution:  is the Golden Number 1.618033… (-1)=1 =1.618033… Equation: 2--1 = 0

The Golden Rectangle: 

The Golden Rectangle  1 1  1 3 -1 To go from the large rectangle to the small one: divide each side by  1 2

Ammonite (Nautilus shape): 

Ammonite (Nautilus shape)

Spirals in the Galaxy: 

Spirals in the Galaxy

The Golden Number in art, architecture,… aesthetics: 

The Golden Number in art, architecture,… aesthetics

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

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

Regular pentagons and dodecagons: 

Regular pentagons and dodecagons nbor7.gif  =2 cos(/5)

Penrose non-periodic tiling patterns and quasi-crystals: 

Penrose non-periodic tiling patterns and quasi-crystals

Slide75: 

G/M= Thick rhombus G Thin rhombus M

Slide76: 

proportion=

Diffraction of quasi-crystals: 

Diffraction of quasi-crystals

Géométrie d'un champ de lavande http://math.unice.fr/~frou/lavande.htmlFrançois Rouvière (Nice): 

Géométrie d'un champ de lavande http://math.unice.fr/~frou/lavande.html François Rouvière (Nice) Doubly periodic tessalation (lattices) - cristallography

The Golden Number and aesthetics: 

The Golden Number and aesthetics

The Golden Number and aesthetics: 

The Golden Number and aesthetics

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

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

Music and the Fibonacci sequence: 

Music and the Fibonacci sequence Dufay, XVème siècle Roland de Lassus Debussy, Bartok, Ravel, Webern Stockhausen Xenakis Tom Johnson Automatic Music for six percussionists

Slide83: 

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.

Slide84: 

The quest of the Graal for a mathematician Example of an unsolved question: Are there infinitely many primes in the Fibonacci sequence? Open problems, conjectures.

Some applications of Number Theory: 

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

Slide86: 

Simon Fraser University, Vancouver Campus (Harbour Center) Segal Center SFU Vancouver 515, West Hastings (at Richards) Some arithmetic problems raised by rabbits, cows and the Da Vinci Code Michel Waldschmidt Université P. et M. Curie (Paris VI) July 12, 2006 http://www.math.jussieu.fr/~miw/ CNTA9