God's Algebraic Geometry: From Babylon to the XXI Century.

Plimpton 322

* 2000 BC - 600 BC: Babylonian Era in the Mesopotamian Valley. Developments: Writing, wheel, metal works... Number system using base sixty... Squares of numbers represented wealth. Area is a product of the length and the width of a field, and this is where squares come in. E.g. A farmer who owned a field of 25 square units of land could swap it for 2 square fields (25 (5² ) = 9 (3² ) + 16 (4 ²))... The Babylonians were obsessed with tables. Because of the clay tablets' durability, many of them survive today. Eg. Plimpton 322 at Columbia University. This tablet suggest 15 Pythagorean triplets such as 75²= 45² + 60².

* In Babylon, Pythagoras came in contact with mathematicians and likely became aware of their studies of numbers now named after him, i.e. the Pythagorean triplets (e.g. 5² = 2² + 3²). When Pythagoras moved to Italy (Crotona), he founded a secret society dedicated to the study of numbers. The Pythagoreans developd a subtantial body of mathematical knowledge all in complete secrecy. Their motto: 'Number is everything.' They worshipped numbers and believed them to have magical properties (e.g. Perfect Numbers, or numbers that are the sum of their multiplicative factors---> 6 = 3 x 2 x 1); a concept that continued to be pursued in the Middle Ages and in mystical systems such as the Kabbalah...

* The symbol of the Pythagorean order was the five-pointed star embedded in a pentagon. This symbol has some properties which the Pythagoreans believed were mystical. The ratio of the entire diagonal to the larger segment is exactly the same as the ratio of the larger segment to the smaller one. This same ratio exists in all smaller and smaller diagonals, and is called the Golden Section. The Golden Section is an irrational number: 1.618..., and appears in natural phenomena as well as proportions that the human eye perceives as beautiful (e.g. Fibonacci numbers)... The Pythagoreans thought that the basic ratios in music (''All in heaven is musical scale and numbers") involved only the numbers 1, 2, 3, and 4, whose sum is 10. They represented 10 as a triangle called Tetraktys:

000

0000 . Tetraktys was considered holy by the Pythagoreans who, also, swore oaths by it... Much of what we know about Greek mathematics comes from Euclid's Elements (300 BC). It's believed that the first two volumes are all from the works of Pythagoras and his disciples. The Greeks developed an entire theory of geometry, and it is this theory, mostly unchanged, that is taught in schools today.

* Early 800s: A a House of Wisdom is established in Baghdad. One of his menbers was Mohammed Ibn Musa Al-Khowarizmi (visit---> Muslim Founders of Mathematics: http://muslimheritage.com/topics/default.cfm?TaxonomyTypeID=12 ). Al-Khowarizmi wrote books on arithmetic and algebra. The Word 'algorithm' is derived from his name, and 'algebra' is derived from the title of his book Al Jbr Wa'l Mugabalah. This book is concerned with solutions of equations of first and second degree... The Arabs were insterested in finding Pythagorean triples giving an area of a right-triangle that is also an integer. By 1225, Leonardo of Pisa, better known as Fibonacci ('son of Bonaccio') showed the same interest when writing the book Liber Quadratorum. Fibonacci was born in Pisa, visited Sicily, Siria, Egypt, and related with the elite of Mediterranean society, bringing him into contact with Arab mathematical ideas, as well as Greek and Roman culture...

* Fibonacci is best known for the sequence of numbers named after him: the Fibonacci Numbers. In this sequence, each term after the first is obtained by adding together the numbers that precede it: 1, 1, 2, 3, 5, 8, 13, 21, 34, 55, 89, 144... This sequence has significant properties: For start, the ratio of 2 successive numbers (e.g. 2/1, 3/2... 144/89) in the sequence tends to the Golden Section (1.618...) The Fibonacci sequence appears anywhere in nature. Leaves on a branch grow at angles from one another that correspond to this sequence. In most flowers the number of petals is one of 3, 5, 8, 13, 21, 34, 55, or 89. In sunflowers, the florets that become seeds are arranged in 2 sets of spirals, one clockwise and the other counter-clockwise. The number of spirals in the clockwise orientation is 34, or 55, or 89; and in the counter-clockwise is 55, or 89, or 144...

* If a rectangle is is drawn with sides in the Golden Section ratio to each other, then the rectangle can be divided into s square and another rectangle. This rectangle is similar to the large one in that it, too, has a ratio of sides equal to the Golden section. The smaller rectangle can now be divided into a square and a remaining rectangle, also in the Golden ratio, and so on. A spiral through successive vertices of the sequence of rectangles that can be drawn is one that appears often in shells, in the arrangement of sunflower florets, and in that of leaves on a branch...

* The Golden section appears in such places as the Athenian Parthenon. The ratios of the height of the Parthenon to its lenght is the Golden Section. The Great Pyramid at Giza has a ratio of height of a face to half the side of the base also in the Golden Section. The Egyptian Rhind Papirus refers to a 'sacred ratio.' Ancient statues as well as Renaissance painting followed the Golden Section or Divine Ratio (1.618...)

THE LIFE AND NUMBERS OF FIBONACCI: CLICK HERE

To be continued...

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A NOVEL BY (C) DANIEL YANEZ 2006

(c) Daniel Yanez Gonzalez-Irun 2005

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