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Aside from 144 being only Fibonacci number. It is also the 12th Fibonacci number. Note that 12 is the square root of 144. the square The first seven digits of the golden ratio (1618033) concatenated is prime!
❖Boethius and his Quadrivium ❖Nicomachu’s Introduction to Arithmetic ❖Leonardo Pisano Bigollo (Fibonacci) ❖Thomas Bradwardine ❖Nicole Oresme ❖Giovanni di Casali
Medieval Mathematics much mathematics and astronomy available in the 12th century was written in Arabic, the Europeans learned Arabic. By the endofthe12th century the best mathematics was done in Christian Italy. During this century there was a spate of translations of Arabic works to Latin. Later there were other translations. Arabic → Spanish Arabic → Hebrew (→ Latin) Greek → Latin. Europe had fallen into the Dark Ages, in which science, mathematics and endeavour stagnated. Scholastic scholars only valued studies in the humanities, such as philosophy and literature, and spent much of their energies quarrelling over subtle subjects in metaphysics and theology, such as "How many angels can stand on the point of a needle?" almost all intellectual
❖Boethius and his Quadrivium Boethius was one of the most influential early medieval philosophers. His most famous work, The Consolation of Philosophy, was most widely translated and reproduced secular work from the 8th century until the end of the Middle Ages Quadrivium (plural: quadrivia) is the four subjects, or arts (namely arithmetic, geometry, music taught after teaching the trivium. The word is Latin, meaning four ways, and its use for the four subjects has been attributed Cassiodorus in the 6th century. and astronomy), to Boethius or
❖Nicomachu’s Introduction to Arithmetic Nicomachus of Gerasa was an important ancient mathematician best known for his works Introduction to Arithmetic and Manual of Harmonics in Greek. He was born in Gerasa, in the Roman province of Syria, and was strongly influenced by Aristotle. He was a Neopythagorean,who wrote about the mystical properties of numbers.
❖Leonardo Pisano Bigollo (Fibonacci) Leonardo Pisano is better known by his nickname Fibonacci. Fibonacci was an Italian mathematician from the Republic of Pisa, considered to be "the most talented Western mathematician of the Middle Ages". The name he is commonly called, Fibonacci, was made up in 1838 by the Franco-Italian historian Guillaume Libri and is short for filius Bonacci .
Fibonacci sequence is a series of numbers in which the next number is calculated by adding the previous two numbers. It goes 0,1,1,2,3,5,8,13,21,34,55 and so on. Though the sequence had been described in Indian Mathematics long ago, it was Leonardo Fibonacci who introduced the sequence to Western European mathematics. The sequence starts with F1=1 in Leonardo Liber Abaci but it can also be extended to 0 and negative integers like F0=0, F1=1, F2=2, F3=3, F4=4, F5=5 and so on. Example: The common difference of 1,3,5,7,9,11,13,15 is 2. The 2 is found by adding the two numbers before it (1+1) The common 2,5,8,11,14,17,20,23 is 3. The 3 is found by adding the two numbers before it (1+2). difference of
The Rule The Fibonacci Sequence can be written as a "Rule" (Sequences and Series). First, the terms are numbered from 0 onwards like this: So term number 6 is called x6(which equals 8). So we can write the rule: The Rule is xn= xn-1+ xn-2 where: •xnis term number "n" •xn-1is the previous term (n-1) •xn-2is the term before that (n-2)
Makes A Spiral When we make squares with those widths, we get a nice spiral: Do you see how the squares fit neatly together? For example 5 and 8 make 13, 8 and 13 make 21, and so on. This spiral is found in nature!
➢ FIBONNACI SEQUENCE WAS THE SOLUTION OF A RABBIT POPULATION PUZZLE IN LIBER ABACI In Liber Abaci, Leonardo considers a hypothetical situation where there is a pair of rabbits put in the field . They mate at the end of one month and by the end of the second month the female produces another pair. The rabbit never die , mate exactly after a month and the females always produces a pair (one male, one female). The puzzle that the Fibonacci posed was: how many pair will there be in one year? If one calculates then one will find that the number of pairs at the end of the nth month would be Fn or the nth Fibonacci number. Thus the number of rabbit pairs after 12 months would be F12 or 144
The Fibonacci numbers occur in the sums of “shallow” diagonal in Pascal's triangle starting with 5 , every second Fibonacci number is the length of the hypotenuse of a right triangle with integers sides. Fibonacci number are also an example of a complete sequence. This means that every positive integers can be written as a sum of Fibonacci numbers, where any one number is used once at most. Fibonacci sequence is used in computer science for several purpose like the Fibonacci search technique , which is a method of searching a sorted array with aid from the sequence.
Two quantities are said to be in golden ratio if (a+b)/a=a/b where a>b>0. its value is (1=root5)/2 or 1.6180339887…Golden ratio can be found in patterns in nature like the spiral arrangement of leaves which is why it is called divine proportion. The proportion is also said to be aesthetically pleasing due to which several artists and architects. The Fibonacci Sequence and the golden ratio are intimately interconnected. The ratio of consecutive Fibonacci numbers converge and golden ratio and the closed from expression for the Fibonacci sequence involves the golden ration, Example:
The Actual Value The Golden Ratio is equal to: 1.61803398874989484820... (etc.) The digits just keep on going, with no pattern. In fact the Golden Ratio is known to be an Irrational Number, and I will tell you more about it later. Formula We saw above that the Golden Ratio has this property: ab = a + ba We can split the right-hand fraction like this: ab = aa + ba ab is the Golden Ratio φ, aa=1 and ba=1φ, which gets us: φ = 1 + 1φ So the Golden Ratio can be defined in terms of itself! Let us test it using just a few digits of accuracy: φ =1 + 11.618 =1 + 0.61805... =1.61805... With more digits we would be more accurate.
➢ FIBONACCI NUMBERS CAN BE FOUND IN SEVERAL BIOGICAL SETTING Apart from drone bees, Fibonacci sequence can be found in other places in nature like branching in trees, arrangement of leaves on a stem, the fruitlets of a pineapple, the flowering of artichoke, an uncurling fern and the arrangement of pine cone. Also on many plants, the number of petals is a Fibonacci number. Many plants including butter cups have 5 petals; lilies and iris have 3 petals; some delphiniums marigolds have 13 petals; some asters have 2 whereas daisies can be found with 34,55 or even 89 petals. have8;corn
THOMAS BRADWARDINE Thomas Bradwardine, (born c. 1290—died Aug. 26, 1349, London), archbishop of Canterbury, theologian, and mathematician. Bradwardine studied at Merton College, Oxford, and became a proctor there. About 1335 he moved to London, and in 1337 he was made chancellor of St. Paul’s Cathedral. He became a royal chaplain and confessor to King Edward III. In 1349 he was made archbishop of Canterbury but died of the plague soon afterward during the Black Death. Bradwardine’s most famous work in his day was a treatise on grace and free will entitled De causa Dei (1344), in which he so stressed the divine concurrence with all human volition that his followers concluded from it a universal determinism. Bradwardine also wrote works on mathematics. In the treatise De proportionibus velocitatum in motibus (1328), he asserted that an arithmetic increase in velocity corresponds with a geometric increase in the original ratio of force to resistance. This mistaken view held sway in European theories of mechanics for almost a century.
THOMAS BRADWARDINE Bradwardine’s most famous work in his day was a treatise on grace and free will entitled De causa Dei (1344)
❖Nicole Oresme Nicole Oresme also known as Nicolas Oresme was a significant philosopher of the later Middle Ages. Oresme was a determined opponent of astrology, which he attacked on religious and scientific grounds. In De proportionibus proportionum (On Ratio of Ratios) Oresme first fixed examined raising rational number to rational powers before extending his work to include irrational power. Significantly, Oresme developed the first proof of the divergence(is an infinite series that is not convergent)of the harmonic series (is the divergent infinite series: σ?=1 His proof, requiring less advanced mathematics that current “standard” tests for divergence( for example, the integral test (is method used to test infinite series of non-negative terms for convergence), begins by nothing that for any n that is a power of 2, there are n/2-1 terms in the series between 1/(n/2) and 1/n. ? − 1 = 1 +1 2+1 3+1 ∞ 4+ ⋯
❖Giovanni di Casali Giovanni (or Johannes) di Casali (or da Casale; c. 1320- after 1374) was a friar in the Franciscan Order, a natural philosopher and a theologian , author of works on theology and science , and a papal legate. About 1346 he wrote a treatise De velocitate motus alterationis( on the Velocity of the Motion of the Alteration) which was subsequently printed in Venice in 1505. In it he presented a graphical analysis of the motion of accelerated bodies. mathematics physics influenced University of Padula and it is believed may have ultimately influenced the similar ideas presented over two centuries by Galileo Galilie. His scholars teaching in at the
REFERENCES • https://Leonardo-newtonic.com/fibonnaci-facts • Giovanni da Casale’,Enciclopedie on line, Treccani • Maarten van der Heijden and Bert Roest,’Franaut-j’,Franciscan AUTHORS.13th-18thCentury:A Catalogue in Progress • Marshall Claget. The Science of Mechanics in the Middle Ages.(Madison:Univ.of Wisconsin 391.644 Pr.,1959).pp 332-3,382-
