The woman warrior Deborah (Old Testament /Jewish Tanach). âQueen Beeâ. 4. Islam at .... Archaeologists insist this is the best preserved Ancient Greek Theatre.
On the Origins of the Fibonacci Sequence
T.C. Scott and P. Marketos
vLeonardo de Pisa (Fibonacci) writes Liber Abaci. vConsiderable influence on Europe. vHindu-Arabic numbers based on a positional decimal system: Revolution for computation. vAlgebra. vInfluence of Muḥammad ibn Mūsā al-Khwārizmī. vThe Fibonacci sequence.
Fibonacci posed the problem as follows: "A certain man put a pair of rabbits in a place surrounded on all sides by a wall. How many pairs of rabbits can be produced from that pair in a year if it is supposed that every month each pair begets a new pair which from the second month on becomes productive? ''
• The sequence begins as follows:
• The sequence begins with one. Each subsequent number is the sum of the two preceding numbers.
• Unfortunately, his application had little practical bearing to nature, since incest and immortality was required among the rabbits to complete his problem. • Also: Rabbits reproduce like rabbits! • The Fibonacci sequence has far more applications than immortal rabbits! • Fibonacci numbers have numerous naturally-occurring applications, ranging from the very basic to the complex geometric.
• Many aspects of nature are grouped in bunches equaling Fibonacci numbers. • For example, the number of petals on a flower tend to be a Fibonacci number.
• 3 petals: lilies • 5 petals: buttercups, roses • 8 petals: delphinium • 13 petals: marigolds • 21 petals: black-eyed susans • 34 petals: pyrethrum (Chrysanthemum) • 55/89 petals: daisies
• Leaves are also found in groups of Fibonacci numbers. • Branching plants always branch off into groups of Fibonacci numbers.
• • • • • • •
Think about yourself. You should have: 5 fingers on each hand 5 toes on each foot 2 arms 2 legs 2 eyes 2 ears
• 2 sections per leg • 2 sections per arm I could go on, but I think you get the point.
• Fibonacci numbers have geometric applications in nature as well. • The most prominent of these is the Fibonacci spiral.
• The Fibonacci spiral is constructed by placing together rectangles of relative side lengths equaling Fibonacci numbers.
•A spiral can then be drawn starting from the corner of the first rectangle of side length 1, all the way to the corner of the rectangle of side length 13.
Cauliflower
Pine Cone
• Music involves several applications of Fibonacci numbers.
• A full octave is composed of 13 total musical tones, 8 of which make up the actual musical octave.
Fibonacci Ratio
Calculated Frequency
Tempered Frequency
Note in Scale
Musical Relationship
1/1
440
440.00
A
Root
2/1
880
880.00
A
Octave
2/3
293.33
293.66
D
Fourth
2/5
176
174.62
F
Aug Fifth
3/2
660
659.26
E
Fifth
3/5
264
261.63
C
Minor Third
3/8
165
164.82
E
Fifth
5/2
1,100.00
1108.72
C#
Third
5/3
733.33
740
F#
Sixth
5/8
275
277.18
C#
Third
8/3
1173.33
1174.64
D
Fourth
8/5
704
698.46
F
Aug. Fifth
• One of the most significant applications of the Fibonacci sequence is a number that mathematicians refer to as Phi (F).
• F refers to a very important number that is known as the golden ratio.
Ratio of two consecutive Fibonacci numbers :
This implies that F’s reciprocal is smaller by 1. It is .618034, also known as phi (f).
• F is defined as the limit of the ratio of the i-th Fibonacci number and its predecessor. • Mathematically, this number is:
or approximately 1.618034.
F
• Remember how flowers have leaves and petals arranged in sets of Fibonacci numbers? • This ensures that there are F leaves and petals per turn of the stem, which allows for maximum exposure to sunlight, rain, and insects.
• Is there anything mathematically definitive about F when used in geometry? You bet there is. • A rectangle whose sides are in the golden ratio is referred to as a golden rectangle. • When a golden rectangle is squared, the remaining area forms another golden rectangle!
• How about your body? • You have NO IDEA how many segments of the human body are related in size to each other by F!
• The human arm:
• The human finger:
• Botany and Zoology. • Formation of spirals. E.g. Galaxies. • Electronic networks, etc. . . • Optimization rules: nature is “lazy” but efficient.
• Ancient Parthenon in Athens • Violins of Stradivarius • Paintings of Leonardo da Vinci • Music e.g. : Bartok and Debussy.
Proportion through use of the Golden section F is embedded in his masterpiece "The Birth of Venus" .
Many books on painting point out that it is better to position objects not in the centre of the picture but to one side or "about one-third" of the way across, and to use lines which divide the picture into thirds. This seems to make the picture design more pleasing to the eye and relies again on the idea of the golden section F being "ideal".
• When used in dimensioning objects, it has always been thought that F produces the most visually appealing results. • Many marketers have used F in their products over the years to make them more attractive to you. • An extremely basic example: 3 x 5 greeting cards.
• Very striking in comparison to the banality by which Leonardo de Pisa presented this number sequence. • Historians have wondered as to the real inspiration that lead to the Fibonacci sequence. • Leonardo de Pisa himself admits to influences from Islamic culture.
1. Show a standard conjecture as to how Leonardo de Pisa “discovered” these numbers and the reasons both historical and mathematical for putting this conjecture in doubt. 1. Present our own conjecture and its justifications, both mathematical and historical. 1. Examine the consequences of our conjecture.
Examine Pascal's Triangle
Put coefficients in Triangular Form
Entries of the Triangle are Binomial Coefficients
Now arrange them in “flush-left” mode (Note: indices of lines and columns start at zero.
Added up the diagonals
Added up the diagonals of Original Triangle
When the numbers of Pascal’s triangle are added on the diagonal → get exactly the Fibonacci sequence
• Invention of the triangle : attributed to Blaise Pascal (16231662 CE). • ... but invented 500 years earlier by the Chinese. (Chinese triangle). • The great Persian Poet and Mathematician Omar Al-Khayyam (1048-1131 CE) would have known the triangle.
• Through contacts with Muslim scholars, Leonardo de Pisa would have learned about the (Chinese) triangle through the works of Al-Khayyam . . . • . . . and from the triangle: • deduced the Fibonacci sequence.
Historical Reasons: Roshdi Rashed (1993 CE) examined a collection of around 100 algebraic problems in Liber Abaci.
• 22 problems borrowed from al-Khwārizmī's ( Al-jabr wa’l muqabala, 830 CE). • and 53 problems borrowed from Abu-Kamil’s (Kitab fi al-jabr wa’lmuqabala, 912 CE). • Exactly the same exercises/problems with only a few changes in the numerical coefficients. • 25 problems remain unidentified but follow the models of AlKhwarizmı and Abu-Kamil.
• Moreover Leonardo de Pisa remains contemporary of Muslim scholarship of the 9th and 10th centuries. e.g. : Consider the cubic polynomial :
posed as a challenge to Leonardo by John of Palermo.
• Al-Khayyam (1048-1131 CE) gave exact solution using methods of algebraic geometry. • Leonardo gets a best an approximate solution (Flos or “The Flower”, 1225 CE) . According to Rashed: "Leonardo de Pisa was not up to date on the most recent Mathematical developments of Muslim."
• Gerard of Cremona (1114-1187 CE): Important Figure within the “school” of translation (from Arabic and Hebrew into Latin), based in Toledo Spain. • Associated with the Archdiocese of Toledo (Raymond-12th century, Rodrigo Jiminez de la Rada -13th century). • Conjecture: Translation of the works of Abu-Kamil into Latin before the end of the 12th century.
• In hindsight : the works of Fibonacci and even those of Roger Bacon are works of translation. • Unavoidable: Islam was considerably more advanced than Europe in the middle-ages. • The middle-ages: period of superstition, witch hunts, etc . . .
• “Magic” and Scientific fact: “Prehistory of Science”: Kepler was an astronomer and an astrologer. • E.g. Alchemy: "Spirit of Salt". • Often translation results had to be disguised. • Legend claims Adelard of Bath disguised himself as a Muslim to break into Cordova and stole a copy of Euclid's elements to have it translated it into Latin!
Mathematical Reasons: Even if Leonardo de Pisa knew the Chinese triangle: It is very unlikely that he could have inferred his rabbit reproduction model from the triangle.
Follow the minimalist principles of Ockham’s razor 1. Locate Leonardo de Pisa in place and history. 2. Where was he before writing his book? 3. What was his environment ?
From the great Donald Knuth - The Art of Computer Programming Fibonacci sequence already known in India 1. Gopala, 1135 CE. 2. Hemachandra, circa 1150 CE. 3. 600-800 CE, Singh, Parmanand (1985), "The So-called Fibonacci numbers in ancient and medieval India", Historia Mathematica, 12 (3): 229–44.
• Mercantile Culture of Bejaia in North Africa. • Golden age of the Berber Dynasties: i. The Almoravids (11th- 12th centuries) and ii. The Almohads (12th- 13th centuries). • Cultural Peak: Considerable intellectual, artistic and mercantile (bourgeoisie) elite.
• Wax exported all around the Mediterranean. • In great demand by the (Christian) clergy: Torches or candles made of animal fat were less interesting (with good reason). • Produced by the bee keepers of the Kabyle (Amazigh) tribes in their mountains and exported in the ports of Bejaia.
• Leonardo's rabbit reproduction model has no correspondence with nature . . . • BUT the Fibonacci numbers perfectly describe the reproduction of bees. • (Only) the Queen lays eggs. If the eggs are: fertilized i.e. workers → (females). NON fertilized i.e. drones →
(males).
• Leonardo de Pisa was the son of a notary and a member of a Pisan Trade colony stationed in Bejaia. • Note: Liber Abaci contains many problems related to commerce (e.g. exchange rates). • Later, Bejaia falls into decline: Spanish conquest and later Turkish domination. • Loss of many scholarly works.
1. The male drone has one parent, a female. 2. He also has 2 grand-parents, since his mother had 2 parents, a male and a female. 3. He has 3 great-grand-parents: his grand-mother had 2 parents but his grand-father had only one, and so forth . . .
• In tracing the number of ancestors at each generation: You get exactly the Fibonacci sequence . . .
• Number of ancestors vs. generation for the females is merely a
Fibonacci sequence “shifted” in relation to that of the male drones. Note: only difference between the Queen and the workers : the Queen is fed with “royal jelly”.
• Number of ancestors vs. generation for the female bees is merely a "shifted" Fibonacci sequence in relation to that of the male drones.
• The ratio between two consecutive generations for sexual reproduction (mammals including rabbits) is 2.
•For bees, this ratio is that of 2 consecutive Fibonacci numbers and thus F in the limit.
• This reproduction model is much more realistic insofar as the Fibonacci numbers are concerned.
• Correspondence between nature and mathematical model is perfect.
• Rabbit reproduction model can be viewed as a variation of the same problem.
However, we need to establish that: 1.The culture of Bejaia was sufficiently advanced to understand the reproduction bees and work out their family/ancestry trees. 2. Indications that Leonardo de Pisa would have taken these numbers from the culture of Bejaia (Bougie).
•During the period of the “Reconquista”, the taking of Toledo captured "booty" of valuable books in Arabic.
•Community of Muslims (Arabs and Berbers), Jews and Christians living side by side: Special geographical/social situation.
•Foundation of a “ school” of translation. Gerard of Cremona and “socii” and others . . .
• Translation of Scientific works (e.g. Aristotle) : by Michael Scotus or “Michael Scot” (1175-1235 CE) at Toledan School of Translation.
• Interested in Mathematics, Astronomy (Astrology), Medicine and Alchemy.
• Good relations with the clergy and the pope
...until branded a "wizard" and condemned to hell in Dante's inferno.
• Essential Knowledge: unfertilized bee egg produces a (male) drone bee. Tabulating the bee ancestries is child’s play.
• Parthenogenesis
(literally from the Greek : “Virgin Birth”).
•Recognized conventional history in the 18th century by Charles Bonnet (1720-1793 CE).
• Fibonacci and Scot together at the court of Frederick II, Ruler of the “Holy Roman Empire”.
• In 1228 CE, Liber Abaci is revised and dedicated to Scot with a very flattering preface. Why ? 2 possible reasons:
1. A (rich) patron providing financial assistance. 2. A person inspiring the Mathematical problem.
• Scot wrote Liber Introductorius which presents an extension of
Aristotle’s Meterologica: identifying a honey produced by the bee’s digestive process. Fair observation for its time.
•Note: Specific books by Aristotle in Zoology : that do not (re)appear in Europe before translations of Scot (not before 1220 CE).
•However, asexual reproduction recognized early by Aristotle : very interested in Apiculture. Historia animalium (History of Animals).
•We have to examine some beekeeping ourselves. Note: Beekeeping can be traced to ancient Egypt (2400 BCE).
•If you want a long-lasting productive beehive, you have to know how bee reproduction works!
•However, he does admit : “Others again assert that these insects (bees) copulate, and that drones are male".
A LOT
•He distinguished the 3-member “cast” system: “workers”, “slaves” (drones) and a “ruler”.
• Identifies much which is confirmed today: (e.g. bees use odor (chemical trails) to find honey).
•However, he makes crucial mistakes. He believes that:
1. The bee “ruler” is a King and not a Queen! (this idea was still alive in the time of Shakespeare). 2. The “workers” are male (soldiers) and the “slaves” (drones) are female.
• Also Aristotle doesn’t understand the fabrication of honey. • This is coming from a guy who thinks unforgivably that women have fewer teeth than men. • If you want to know how many teeth a person: just open that person's mouth and count them!
• When the Queen dies or gets old she can no longer mate with the (male) drones.
• If the Queen dies: worker "pseudo-Queen” but Her eggs cannot be fertilized.
• Number of (male) bee drones increases and the number of (female) workers diminishes: The beehive is condemned!
•Goddess Melissa identified as Queen Bee who annually killed
her male consort (much as the bee drone dies at copulation). Her priestesses were called Melissae.
• Notion of Parthenogenesis. E.g.: 1. Hera → Hephaistos (particular version). • Gaia (Earth) → Ladon
•Bejaia where Leonardo Fibonacci was in the world of Islam. •According to Toufy: What we know of beekeeping in Islam of
the middle-ages is expressed in the Koran in the section known as the “Surah an-Nahl” ref. (16 : 68 -69), (“The Bee”)
And your Lord inspired the bee, (Saying), “Take for yourself dwellings in hills, on trees and in what they (mankind) build. Then eat of all fruits.” From their bellies comes a drink of varied colors, beneficial to men. This is a meaningful sign for thinkers.
• We recover the origin of Michael Scot’s notion: the role of the bee’s digestive process in making honey and ....
•In the Original Arabic: the bees are female!
•From (Hudhayli tribe of the Arab peninsula). poet and
contemporary of Mohammed, wrote about the power of the Queen in the “city of the bees”.
•Arab biologist : Al-Jahiz (776-868 CE) rediscovers the works of Aristotle and writes his own Kitab al-hayawan (“Book of Animals”). Describes a powerful bee King.
• Continuation by later Arabic Scholars: 1. Al-Qazwını (d. 1283 CE). 2. Al-Damırı (d. 1405 CE). 3. Al-Maqrızı (d. 1442 CE).
• Confusion/Contradictions. Took a while but gradual Transition to Powerful Queen Bee and POTENT wax production at last.
1. Greek Mythology (before Aristotle). 2. A passage in the writings of the Indian Veda called Prashnopanishada, (500 BCE). 3. The woman warrior Deborah (Old Testament /Jewish Tanach) “Queen Bee”. 4. Islam at its beginning (and much later)...
•From the first prince/first dynasty, all Pharaohs have the principal Royal Title of “Sons of the Bee”. Maternal Ancestor: the bee.
•Symbol of “Neith” , great goddess of the North of Egypt (Delta)
originating from: Libya, “born” (Egyptian and Greek Mythology) on the shores of Lake Triton (Tunisia and Algeria).
•According to the story of the Titans, which precede the Greek and
Egyptian Gods (Greek Mythology): a God originating from Tritonis: Aristeios (Agreos) taught humans the domestication of the Hunting Dog, Agriculture and Apiculture.
•Term used for Honey: “TAMMENT”, (feminine) and is the same for the
Archaic Egyptian and the Berber of yesterday and now (all North Africa).
• "virgin birth" is a subject of the Exultet or Easter Proclamation, a hymn of praise sung before the Paschal (or Easter candle) during the Easter Vigil. • Made around the 12th century, the Exultet Roll of Salerno includes: "The Praise of the Bees'' describing beekeeping in the Middle Ages and the "reputed chastity" of the skillful hard-working bees.
• The Barberini Exultet roll in the Benedictine abbey of Monte Cassino (Italy) shows the "praise to the bees" and is dated at around 1087 (picture shown). • One possibility believed by some is a tradition descending from Pope Augustine, himself a native of North-Africa.
• Fibonacci dedicates his Liber Abaci to Michael Scot even though Frederick II is his patron (a debt towards Scot?). • Wax “Technology” of Bejaia was well developed at that time. Important Export. Pisan trade colony. • Intellectual/Mercantile Culture of Bejaia sufficiently developed to deduce the bee ancestries in the middle-ages.
• Ancestral Reproduction Model of bees matches perfectly the Fibonacci sequence although the latter does NOT represent the real reproduction of rabbits. • Many of the algebraic problems shown Liber Abaci are (thinly disguised) translations of the work of Muslim scholars. • Scot wrote about bee apiculture and his sources are Muslim and Aristotelian.
• A major “school” of translation existed in the time of Leonardo de Pisa in Toledo, Spain where Scot worked (before 1220 CE). In particular, he translated Aristotle’s works in Biology. • Essential Notions of Muslim bee keepers during the middle-ages are expressed right in the Koran which was translated into Latin by Marcus Toledanus (Marc of Toledo), a colleague of Scot. • Beeswax was in great demand for candles by the Christian Clergy. Scot and Marcus Toledanus were both associated with the Cathedral of Toledo.
End Part 1……
Beginning Part 2…..
• The reproduction mechanism of bees could have been found out a long time ago, as far back as the period of Aristotle. • Fibonacci Numbers are present in Nature and thus have always been “observable”. • The Ancient Greeks and even Other Cultures could have also known the Fibonacci numbers.
• A gold bracelet belonging to a Queen of Djer found in the tomb of a King of Abydos. • Presently in the Egyptian Museum of Cairo. • 1st - 2nd dynasty (“Archaic period”). • Centerpiece (modern watch like design) floral Gold rosette, probably a daisy → with exactly 21 rays!
• Egyptian Mathematics was limited. • Babylonian mathematics was better. • Ancient Greek Mathematics was the best. • Experts deny the builders of the Great Pyramid knew the Golden number F. • We make no bold claims only that Egyptians may have noticed natural patterns.
• The ancient Greek Mathematician Thales (~600 BCE) visited the Great Pyramid in Egypt. • Thales measured the height of the pyramid by measuring its shadow. • What was the right time to measure the shadow? He used a stick and reasoned that when the length of the stick's shadow equaled the height of the stick, the height of the pyramid would equal the length of its shadow
• Once he got his answer, he believed the Great Pyramid embedded what is called "The Egyptian Pyramid". • This triangle embeds the golden number as f² = f + 1 in a geometric form following the theorem of Pythagoras. It imparts a slope of arctan √ F
• Experts deny that the builders of the Great Pyramid knew the Golden number. • Proportion can also be realized intuitively.
• No written records but . . . • Architectural Evidence • Ancient Theatre of Epidaurus located in Argolis, Greece. • Goes back to the Hellenistic period.
• Archaeologists insist this is the best preserved Ancient Greek Theatre. • Constructed in 2 stages : Finalized by 2nd century BCE. • Tradition : Optimally built (great acoustics, location) by Polykleitos the Younger. • Connection with the School of Pythagoras.
• Two Levels of seats: 34 seats (first level) 21 seats (2nd level). • According to Dimitris Tsimpourakes: the builders wanted to inject “harmony” by using F (e.g. Parthenon) by the approximation:
• Meticulous analysis by Arnim von Gerkan and Wolfgang Muller-Wiener in 1950s. • Extrapolation of the lines defining the aisles joining the rows of seats of the theatre to its center reveals 2 back-to-back Golden triangles i.e. balanced by F.
• Each Golden triangle is an isosceles triangle where the apex angle is arccos (F/2) = p/5 (or 36°). • This construction by Gerkan takes into account slight irregularities and asymmetries likely caused by earth tremors and ground movements over the last 2500 years.
• There is also the theatre of Dodona: 34 seats: 19 + 15 21 seats (upper level). • but plan reveals that upper 21 seats is separated by a wider "gangway" and has more intermediate staircases. • Maybe 19/15 ≈ √ F ?
The Golden number F is the root of:
Multiply eq. 12 by F:
Multiply eq. 13 by F:
Note: Result is always linear in F:
Multiply again and again by F:
We see that is expressed in terms of
Multiplying eq. 12 by gives:
Combining eqs. (15) and (16) yields:
and similarly for:
• HOWEVER this analysis was algebraic. • The Muslims could have done this during the Middle Ages. • But what the ancient Greeks? • Could they have that exercise in terms of Geometry ? • Yes by using the GNONOM
• Since Hipassos (450 BCE): for Mathematics and even for Art and Architecture, one wanted to know if a given quantity (number) was commensurable (rational). • Symbol of the School of Pythagoras (5-th century BCE) was the Pentagram which embedded the Golden Number.
• Triangle ABC is isosceles: BC=1 and AB=AC=f • Angles ABC=ACB= 2a and BAC=a=p/5 (or 36°). We draw a line from C to D by a distance f. • Since AC=CD=f: the triangle ACD is also isosceles and similar to the first triangle. • Let's look at the diagram and analyze it.
• If BC = 1, then CD = f. • since CD/BC = f and • BD = BC + CD = 1+f or
• The exterior triangle becomes a "new'' triangle ABC (same proportions). • AB = AC = f² = f + 1 and BC = f • Repeat exercise but with CD = f² = f + 1.
We get:
BD = f × CD = f × (f² ) = f³ → f³ = 2f+1 Iterate a 3rd time to get:
Repeat: exercise:
Iterate again:
• Obtain 34 and 21 in 3 more iterations. • This is called the Gnomon. • So the Ancient Greeks could have done with their geometry what the Muslims would be able to do with algebra in the Middle-Ages.
• Zenon's paradox of the problem of Achilles and the Tortoise (5-th century BCE): Ancient Greek had the concept of recursion but a problem with limits. • Make No pretense that the Ancient Greeks knew the Fibonacci numbers beyond 21 and 34 but reaching those numbers is an achievement.
• Who invented the Fibonacci numbers? • Definitely not Leonardo de Pisa. • These numbers like F are so universal in nature, in art, music and applications, can be "discovered" independently by many cultures but their knowledge can also spread by cultural influences.
• According to earliest records (Knuth reference) • See “The so-called Fibonacci Numbers in Ancient and Medieval India” by Parmanand Singh, Historia Mathematica, 12 (1985) 229-244. • Fibonacci numbers and multinomial coefficients - since 600 CE. • D. Knuth gives a reference viz-a-viz Sanskrit prosody (music) – could known in India as early as 200 BCE.
• Earliest record of the Fibonacci sequence was in India in Sanskrit prosody around 200 BCE.
• Ancient Greeks likely knew them: if you know F, you eventually discover the Fibonacci sequence and vice-versa. • Maybe Egypt?
• North-Africa in Fibonacci's time served as a "corridor" of consumer goods and ideas from many sources. • Islam back then was experiencing a peak in culture. • Leonardo likely got his numbers from the culture of Bejaia in what is now modern day Algeria.
http://www.mcs.surrey.ac.uk/Personal/R.Knott/Fibonacci/
http://library.thinkquest.org/27890/goldenRatio2p.html
http://www-groups.dcs.st-and.ac.uk/~history/Publications/fibonacci.pdf
http://www-history.mcs.st-andrews.ac.uk/Biographies/Scot.html
单击此处添加标题
Thank you for your kind attention!
