Leonardo Pisano (Fibonacci) PDF

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Leonardo Pisano (Fibonacci) Heather Risley Introduction Leonardo Pisano, also known as Leonardo of Pisa or Fibonacci, may have been the greatest mathematician of the middle ages. He wrote, among many books and published works, Liber Abaci, which is one of the most important books on mathematics of t...

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Leonardo Pisano (Fibonacci) Heather Risley

Introduction Leonardo Pisano, also known as Leonardo of Pisa or Fibonacci, may have been the greatest mathematician of the middle ages. He wrote, among many books and published works, Liber Abaci, which is one of the most important books on mathematics of the Middle Ages. It effected how all of Europe looked at mathematics because it acted as a transition from Roman numerals and the use of the abacus, to the Arabic-Hindu number system and the use of computation, calculation, and algebra. A curriculum based on Leonardo’s Liber Abaci was taught in Tuscany schools of abaco for over three centuries. These schools, which had a merchant based curriculum with a foundation in mathematics without use of the abacus, were normally attended by boys intending to be merchants or by others desiring to learn mathematics. Other very accomplished mathematicians wrote books of abaco for use in these schools, but none of these books were even comparatively as comprehensive and theoretical as the work of Leonardo in Liber Abaci (Sigler, 2002). Leonardo’s work, theories, discoveries, and problems are still in application today, particularly his rabbit problem. Some of the uses in modern mathematics and science include everything from Fibonacci-based pseudo-random number generators, to analysis of Euclid’s algorithm. Leonardo’s work can be used to explain ideas like the beauty seen in nature and the description of signals appearing in telecommunication systems. Because Leonardo’s work influenced a significant amount of mathematical and scientific life in the past and is continuing to influence mathematical and scientific life in the modern world, his work and his life are worth investigating further to understand how just one man made such leaps and bounds in the mathematical world.

Biography Leonardo Pisano was born in 1170 in the maritime city-state of Pisa in the province of Tuscany. Leonardo lived there, in which is now known as the state of Italy, until is death after 1240. Leonardo Pisano’s father was Guilielmo Bonaccio. This could be why Leonardo Pisano had the name Fibonacci attached to him. Contraction of filius Bonacci, Fibonacci means ‘son of Bonacci’. Even though Leonardo Pisano had the name Fibonacci associated with him, no person, including himself, ever referred to him as Fibonacci during his lifetime. It was only after his death when, in 1838, mathematic historian Guillaume Libri made the association of Leonardo Pisano to Fibonacci (Sigler, 1987). Bugia, a trading region associated with the city of Pisa, located on the Barbary Coast of Africa in the Western Muslim Empire, is where Leonardo was taught about mathematics as a child. Leonardos father, Guilielmo, was a teacher of the Arabic-Hindu numerals and the new style of calculation that came along with it. Leonardo very much enjoyed the teachings of the

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numerals. He liked studying them so much so that he continued to study Arabic-Hindu numerals and mathematics on his trips to Egypt, Syria, Greece, Sicily, Byzantium, and Provence. Leonardo learned from his father and Arabic scientists the Hindu numbers, their place system, and the algorithms for arithmetic operations (Sigler, 2002). Leonardo thought of Arabic-Hindu numerals to be of much greater use than Roman numerals because Hindu numerals can be used to do both calculations and write down answers. Roman numerals did not provide the appropriate environment to do computations; abaci were used to perform the computations needed and then answers were written down in Roman numerals. Arabic-Hindu numerals allowed computations to be completed without an abacus, which would free a mathematician, businessman, or merchant from using one. At the time of Leonardo’s first fascination with Arabic-Hindu numerals, most Eastern Muslim lands where Leonardo traveled, only mathematicians and scientists used Arabic numerals. Leonardo can later be accredited with introducing Arabic-Hindu numerals and their calculating methods into general business practice and bridging the gap between the mathematical knowledge of scientists to that of the common merchants in his native land.

Liber Abaci To bring the world’s best mathematics in usable form to the Italian people, Leonardo decided to write Liber Abaci upon his return to Pisa from his travels. The direct translation of the title means ‘free abacus’ or ‘free of the abacus’ which can be thought of how, with the Arabic-Hindu numerals that Leonardo would introduce, Italians would no longer need an abacus to do their computations. This meaning, although seeming to be significant, is not the translation that historians have used. Historians more commonly think of the title to translate to ‘the book of calculation’ because of how the word abaci, although derived from the word abacus, in the thirteenth century refered to calculation without the abacus as well (Sigler, 2002). Leonardo’s purpose of writing this book was to replace Roman numerals with the Hindu numerals among Italian scientists and merchants alike (Sigler, 2002). He achieved this goal more than he could have ever imagined because merchants spread the new mathematics and its methods wherever they went in the Mediterranean world. Liber Abaci was written in 1202 (and then a second version in 1228) and was written to be an introduction to Arabic-Hindu numerals, but turned out to be much more than just an introduction. It is what Leonardo believed to be the best of Hindu, Arabic, and Greek mathematical methods. Liber Abaci is an encyclopedic work that examines, interprets, and explains much of the known mathematics of the thirteenth century on arithmetic, algebra, and problem solving (Sigler, 2002). Liber Abaci is both a theoretical and practical work because it is meant for mathematical theory and higher algebraic thought as well as practical uses in the business world. The mathematical methods in Liber Abaci are based on geometric algebra and theoretical foundation found mainly in Book II of Euclid’s Elements (Sigler, 2002). Liber Abaci includes: the nine Arabic numerals (ten counting zero); calculation with these numerals; multiplication and addition of whole numbers; subtraction of whole numbers from larger numbers; division of whole numbers; addition, multiplication, and division by fractions; decompo2

sition of whole numbers into parts; barter, exchange, and rules for money; miscellaneous problems; methods of double false position; quadratic and cubic root extraction; analysis of quadratic equations, binomials, rules of proportion, algebraic rules, casting out nines, accounting, and progressions as well as many problems in applied algebra including problems in indeterminate analysis (Sigler, 1987). Tucked away in a rather large chapter, which proposes numerous problems, is the infamous rabbit problem. Leonardo did not spend a particularly large amount of time in Liber Abaci explaining any results or consequences of the rabbit problem. He states it and then moves on to his other problems.

The Rabbit Problem Problem 1. A man put one pair of rabbits in a certain place entirely surrounded by a wall. How many pairs of rabbits can be produced from that pair in a year, if the nature of these rabbits is such that every month each pair bears a new pair, which from the second month on becomes productive? (Burton, 1998) When continued indefinitely, the sequence encountered in the rabbit problem is as follows: Sequence 1. 1, 1, 2, 3, 5, 8, 13, 21, 34, 55, 89, 144, 233, 377, ... (Burton, 1998)

Figure 1: Leonardo Pisano Rabbit Problem Diagram (Blogupiel, 2009) This sequence found in the rabbit problem is called the Fibonacci sequence and its terms are the Fibonacci numbers. Leonardo proposed this problem in his book Liber Abaci, but he did not study this sequence of numbers himself, nor did he name this sequence of numbers. Edouard Lucas, 3

nineteenth century number theorist, studied this sequence and attached Fibonacci’s name to it (Burton, 1998). Each term in the sequence, after the second, is the sum of the two that immediately precede it. This is the first recursive sequence in mathematical work and was solidified as so in a published paper by Albert Girard (Burton, 1998). The Fibonacci sequence is formally defined as: Formula 1 (Fibonacci Sequence). u1 , u2 , u3 , ...∈ Z such that u1 =u2 =1 and un =un−1 +un−2 ∀ n ≥ 3 (Burton, 1998) There is no indication that Leonardo himself understood the consequences of this proposed problem. It could be assumed that such a brilliant mathematical mind must have explored the possibilities of his own problem, even if it was only quiet mental contemplation, but no record is found that Leonardo put much more thought into the rabbit problem past what is stated in Liber Abaci. It was his successors and mathematicians to follow Leonardo centuries later that found the incredibly large significance and the potential of the sequence produced by the rabbit problem. In fact, there is a nine-volume set of books that are compilations of papers produced in relation to the applications of Fibonacci numbers.

Major Publications Leonardo Pisano is known mostly because of this sequence that resulted from the rabbit problem and of his other work in Liber Abaci, but he also accomplished much else in his lifetime. He started his work with a solid foundation in the mathematics that was known and grew forward to expand upon that knowledge; he solidified this knowledge for mathematicians and the general educated public alike. Much of Leonardo’s work is based off of Euclid’s work and Euclid’s book entitled The Elements. Leonardo gave arguments and proofs in Euclidean tradition (Merzbach, 231). He became proficient in Euclids Elements, and the Greek mathematical method of definition, theorem, and proof. He referenced Euclid’s Elements many times to find a basis for a theory of the mathematics he was using and proving. He used The Elements and expanded on its theory to create a solid basis for what is used in mathematics today (Burton, 2002). In addition to Liber Abaci, Leonardo Pisano wrote Practica Geometriae (1223), Flos (1225), Epsistola ad Magistrum Theodorum (unknown year), Liber Quadratorum (1225), and now lost, a book on commercial arithmetic, Di Minor Quisa. Practica Geometriae deals with a number of geometrical subjects and is based upon both the geometry of Euclid and that of Heron of Alexandria. Included in Leonardo’s geometry are many practical problems and he did this because his main focus with much of his work was to educate the people of his country, not just to discover more advanced mathematics. The last section of the book is devoted to solution of indeterminate equations, which doesn’t have much to do with the geometry that is in the rest of his book. This interest in indeterminate equations is explained and investigated further in Liber Quadratorum (Burton, 1987).

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Liber Quadratorum Liber Quadratorum is Leonardo’s most advanced book and is his greatest work as a mathematician. It is primarily work on advanced arithmetic, which is referred to in modern times as the theory of numbers or number theory. Translated, Liber Quadratorum means The Book of Squares and it explores the relation of square numbers to sums of sequences of odd numbers. Leonardo takes this simple relation and builds a large amount of mathematical theory and results. He solves many problems building on the properties of squares as sums of odd numbers. Leonardo put together twenty-four propositions that further investigate the relation of square numbers to sums of sequences of odd numbers, the first of which, interestingly enough, gives solutions to a Pythagorean problem which is stated below. Problem 2. Find two square numbers which sum to a square number. (Sigler, 1987) Algebraically, ∀ a, b, c ∈ Z+ , a2 + b2 = c2 . The solution given in The Book of Squares is a way to find Pythagorean triples and Leonardo expands on this even further to give two other methods to find Pythagorean triples as well. Leonardo’s work need not be judged on the symbolism with which it was written; the mathematical symbolism that exists in modern times was not available for Leonardo to use. Nonetheless, the literature produced by Leonardo Pisano is excellent and rigorous mathematics; his books, including Liber Abaci and Liber Quadratorum, are serious mathematical work written by a superior creative mathematician (Burton, 2002).

Golden Ratio and Fibonacci Sequence The effects of Leonardo Pisano’s work and publications can be seen in mathematics, technology, and science throughout time. Take for example, the golden ratio: a highly known and used idea in many branches of mathematics. Expressed in an algebraic sense, a definition for the golden ratio, ϕ, is as follows: Definition 1. ∀ a, b ∈ R+ such that a > b,

a+b a




=

a b




= ϕ (PhiPoint, 2013)

Another way to find the golden ratio is by solving the equation x2 − x − 1 = 0. The ratio that surfaces is approximately ϕ = 1.618 (PhiPoint, 2013). This is the same ratio obtained by manipulating the Fibonacci sequence in a particular way. Take the ratio of two successive numbers in the Fibonacci series (reference Sequence 1) and divide each number before it to find the following sequence of numbers:






21 Sequence 2. 11 = 1, 21 = 2, 32 =1.5, 35 = 1.666..., 58 = 1.6, 13 8 = 1.625, 13 = 1.61538... (Beardon, 2014) It seems that as the terms of the Fibonacci sequence grow larger, the ratios get increasingly closer to approximately 1.618, the golden ratio. The Fibonacci sequence and the golden ratio 5

seem to have something of an intimate relationship because whenever one of them appears in mathematics, the other isn’t diffficult to find. Something as common and accepted in mathematics as the golden ratio has a relation to the work of Leonardo. Even though Leonardo did not study the Fibonacci sequence of numbers, he did pose the problem. He must have seen the potential in the sequence to have even posed the question at all. Little did he know that this one problem would have connections in a countless number of places, including, but not limited to, nature and the harmonious construction of architectural structures.

Natural Beauty in the Fibonacci Sequence The Fibonacci sequence occurs in nature in many hidden places. The spiral structure of the pedals of many flowers progress along a Fibonacci sequence. The spiral appears in the curve of the human ear, the seashell, and the pinecone. Also, and more commonly known, the sequence is found in the spiraled shell of the chambered nautilus (Henderson, 2007). To obtain this spiral from the Fibonacci sequence, the construction of a finite Fibonacci rectangle is necessary. The Fibonacci Rectangle Construction follows and it will produce a figure that looks like that in Figure 2. Algorithm 1 (Fibonacci Rectangle Construction). 1. Construct two unit squares (side length 1), which share a side, but are not the same square. 2. Construct a square of side length of the sum of the side lengths of the previous two squares. 3. Continue to some square with a side length fn where fi is a Fibonacci number and n is the number of squares desired (Beardon, 2014).

Figure 2: Fibonacci Rectangle for n=6 (Muller, 2011) It may not be simple to see the connection between this set of squares and the spiral structure seen in nature. The spiral can be constructed using the Fibonacci Spiral Construction Algorithm and it will produce a figure that looks like that in Figure 3. This constructed spiral is also known as the golden spiral because of the intimate connection between the Fibonacci sequence and the golden ratio. Photographers use computer technology to place a diagram of the spiral on photographs of people and when they use this method they achieve a new standard of ‘beauty’. They also use the golden ratio and the Fibonacci rectangles when lining up landscape or architectural photos to 6

achieve that standard of ‘beauty’ (Brandon, 2014). This ‘beauty’ that is referred to is what the human eye is more keen on seeing. There are all kinds of proportions in nature and man-made items that make these items more appealing to the human eye and many, if not majority, of the ratios that appear are Fibonacci or golden ratio related. In its essence, without being aware, Leonardo Pisano recognized that this sequence had some significance worth being investigated, and stumbled upon a series of numbers that puts an explanation to what the human mind considers beautiful. Algorithm 2 (Fibonacci Spiral Construction). 1. Using a compass, construct a half circle with radius 1 from the far corners of the two unit squares that are also the corners of the square of side length 2. 2. Construct a quarter circle with radius 2 from a corner where the half circle ended to the diagonal corner of the square that shares a corner with the square of side length 3. 3. Continue this process of constructing a quarter circle with the radius being the length of the side of the square you are constructing the quarter circle in. Be sure to use the corners of the squares that ‘continue’ the spiral on (Beardon, 2014).

Figure 3: Fibonacci Spiral for n=6 (Grange, 2013)

Random Number Generators Fibonacci numbers have been found to be useful in many aspects of mathematical applications. Peter G. Anderson introduced a Fibonacci-based pseudo-random number generator and although this particular number generator fails most of the tests that random number generators should pass, it does pass the uniformity test (Bergum, 1990). This test is arguably one of the most important tests for random number generators. The generator is as follows: Random Number Generator 1. If A and B are relatively prime integers, then the sequence of integers Sk = kA (mod B), k=0, 1, 2, ..., B-1 is a permutation of 0, 1, 2, ..., B-1. Choose A and B as two adjacent Fibonacci numbers, A = Fn , B = Fn+1 (Bergum, 1990). It has been proven that any two consecutive Fibonacci numbers are relatively prime, so it is ideal to use numbers from the Fibonacci sequence to find relatively prime integers easily and quickly. To obtain a better idea of how the pseudo-random number generator works, and example is provided. 7

Example. If A=8, B=13 then the pseudo-random permutation of 0, ..., 12 is (0, 8, 3, 11, 6, 1, 9, 4, 12, 7, 2, 10, 5). (Bergum 1990) This sequence repeats after B numbers are generated, and it could be used to generate a sequence of real numbers in the interval [0,1) by using the value Sk /B (Bergum, 1990). The applications of a Fibonacci-based pseudo-random number generator may not be immediately clear, but when put in the context of graphics software the advantage is abundantly clear. When graphics software developers put together graphics screen lines in a traditional way, it loads from left to right, top to bottom, just like reading from a book in western culture. This tends to be time consuming, but with the use of the random number generator, they can have the graphics screen lines load in a random order so an unclear image forms more quickly than traditionally. This is an advantage to graphics software developers because with the blurred picture, they can judge how the image will look earlier than before and make decisions about it more quickly (Bergum, 1990). On top of that, this random number generator is easy to remember and easy to code. Another type of random number generator that is Fibonacci based is lagged Fibonacci pseudo-random number generatora and one of these is generally defined below. There are many variations of this generator, but in its essence,...