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.html François 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