Leonardo Pisano was the first great mathematician of medieval Christian Europe. He played an important role in reviving ancient mathematics and made great contributions of his own. After his death in 1240, Leonardo Pisano became known as Leonardo Fibonacci. Leonardo Fibonacci was born in Pisa in about 1180, the son of a member of the government of the Republic of Pisa.
When he was 12 years old, his father was made administer of Pisa’s trading colony in Algeria. It was in Algeria that he was taught the art of calculating. His teacher, who remains completely unknown seemed to have imparted to him not only an excellently practical and well-rounded foundation in mathematics, but also a true scientific curiosity. In 1202, two years after finally settling in Pisa, Fibonacci produced his most famous book, Liber a baci (the book of the Calculator).
The book consisted of four parts, and was revised by him a quarter of a century later (in 1228).
It was a thorough treatise on algebraic methods and problems which strongly emphasized and advocated the introduction of the Indo-Arabic numeral system, comprising the figures one to nine, and the innovation of the “zephirum” the figure zero. Dealing with operations in whole numbers systematically, he also proposed the idea of the bar (solidus) for fractions, and went on to develop rules for converting fraction factors into the sum of unit factors. At the end of the first part of the book, he presented tables for multiplication, prime numbers and factor numbers.
Book Review: International Conflict Resolution
The first chapter of the book talks about the peace-making and de-escalation strategies. It discusses the theoretical approaches of statism, pluralism and populism in reference to the international conflict. The author of the book convinces the reader that effective conflict resolution should make use of convincing power and positive sanctions. Therefore, he argues that win-win solutions are ...
In the second part he demonstrated mathematical applications to commercial transactions. He began the sequence with 0, 1, … and then calculated each successive number from the sum of the previous two. This sequence of numbers is called the Fibonacci Sequence. The Fibonacci numbers are interesting in that they occur throughout both nature and art. Especially of interest is what occurs when we look at the ratios of successive numbers.
Fibonacci numbers and ratios have has a curios influence on art and architecture for many centuries. There seems to be a visually pleasing quality to these numbers and their relationship to each other that has appealed to humanity’s sense of beauty since recorded history. Today 3 x 5 and 5 x 8 index cards and booklets are extremely popular. This obviously shows the appeal of Fibonacci numbers that characterize such things as playing cards, writing pads, windows, mirrors, calculators, and credit cards, to mention only a few items from the endless list of Fibonacci an shapes.
In fact these shapes are close approximations of a rectangle best known as the golden rectangle. It strikes people as quite perfect, being neither too fat and stubby nor to long and skinny. It lies somewhere between a square and a double square, but not exactly in the middle. When you construct a set of rectangles using the sequence (1, 1, 2, 3, 5, 8, 13, 21, ), a design found in nature is revealed (page of pictures #1).
Next, when you construct in each square an arc of a circle with a radius the size of the edge of each respective square (a quarter circle), the organic design, which can be found in a snail shell can be seen (page of pictures #2).
Throughout history the length to width ratio for rectangles was one to 1.
61803 39887 49894 84820. This ratio has always been considered most pleasing to the eye. This ratio was named the golden ratio by the Greeks. In the world of mathematics, the numeric value is called “phi”, named for the Greek sculptor Phidias.
The space between the columns form golden rectangles. There are golden rectangles throughout this structure which is found in Athens, Greece. He sculpted many things including the bands of sculpture that run above the columns of the Parthenon. Phidias widely used the golden ratio in his works of sculpture. The exterior dimensions of the Parthenon in Athens, built in about 440 BC, form a perfect golden rectangle.
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Kroger and Whole Foods are the two giants in the grocery industry; however, their capital structure and financial measures paint vastly different pictures. The liquidity ratios, which measure short term solvency of the company, were calculated for both companies. The current ratio for Kroger was calculated to be .76 compared to a current ratio for Whole Foods of 1.60. At a glance, Whole Foods is ...
Many artists who lived after Phidias have used this proportion. Piet Mondrian and Leonardo da Vinci both thought that art should manifest itself in continuous movement and beauty. Therefore, they both expressed movement by incorporating the golden rectangle into their paintings. The golden ratio expresses movement because it keeps on spiraling to infinity.
They showed beauty in their paintings by using the golden ratio because it is pleasing to the eye. To express the Fibonacci Sequence in art one must pay close attention to beauty, proportions, and continuous rhythm. Leonardo Da Vinci dubbed this proportion the “divine proportion.” If you draw a rectangle around Mona Lisa’s face, you would find that the rectangle is in the golden proportion. He did an entire exploration of the human body and the ratios of the lengths of various body parts. A modern day artist that used the golden ratio in a numerous amount of paintings was Mondrian. Piet Mondrian avoided any suggestion of reproducing the material world.
Instead using horizontal and vertical black lines that outline blocks of pure white, red, blue or yellow, he expressed his conception of ultimate harmony and equilibrium. His style, and its underlying artistic principles, he called neoplasticism. Here is an example of one of his angular paintings which employ the proportion (page of pictures #3).
Bicknell, M. ; and Hogg att, V. E.
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The Curves of Life Being and Account of Spiral Formations and Their Application to Growth in Nature, To Science and Art. New York: Dover, 1979. Candy, H. and Rowlett, A. Mathematical Models, 3 rd edition. Stradbroke, England: Tarquin Pub.
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The Essay on The golden proportion
Among many proportions there is one having unique properties. If the ratio of the sum of the quantities to the larger quantity is equal to the ratio of the larger quantity to the smaller one, then the two quantities are in golden proportion, denoted by the Greek letter phi (? ). And this proportion can be called in different terms such as the golden mean, golden section or the golden proportion ...
Williams, R. The Geometrical Foundation of Natural Structure: A Source Book of Design. New York: Dover, p. 53, 1979..
