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Math in 12th c England

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Joined: April 2009
Location: Truro, Cornwall, UK

Post by Ken » Sat March 5th, 2011, 1:04 pm

[quote=""annis""]The use of Arabic numerals (or more strictly Hindu-Arabic numerals) was known in North-Eastern Spain from the 10th century, and Italian merchants who moved freely through the Byzantine Empire knew of them. Fibonnaci was the son of an Italian merchant based in North Africa, and he wrote about them in the 12th century. Fibonnaci, btw, was a favourite with the enlightened Holy Roman Emperor Frederick II, who was considered a heretic and arch-enemy of the Church by various popes. Anything coming out of the east was automatically suspect and tainted with the possibility of heresy according to Church understanding.[/quote]

Annis, I have studied some of Fibinacci’s works particularly in relation to his sequence of numbers that provide the ‘Golden Ratio;’ a ratio that medieval (and earlier) builders new well and used when deciding on the form of their buildings.
In mathematics, the Fibonacci numbers are the numbers in the following integer sequence:
0, 1, 1, 2, 3, 5, 8, 13, 21, 34, 55, 89, 144, 233, 377, 610, 987, 1597, 2584, 4181, 6765, 10946 ….. where by definition, the first two Fibonacci numbers are 0 and 1, and each subsequent number is the sum of the previous two.

Now, if one draws a line divided into two parts a and b, with total length = a + b, then a + b is to the length of the longer segment a as the length of a is to the length of the shorter segment b. This is shown graphically in Wikipedia or other publications on line, making it easier to understand.
The golden ratio is derived from this geometric relationship that arises from this and can be expressed algebraically as:
a+b/a = a/b = Φ
This equation has one positive solution in the set of algebraic irrational numbers:
Φ = 1/2 + √5/2 = 1.6180339887
So the Golden Ratio is 1.6180339887 …….

Now, if you take the Fibonacci sequence above and, ignoring the lesser numbers divide each number by its preceding number you will get the following:

6765/10946 = 1.618033996 ….. a close approximation to the golden ratio!

At least since the Renaissance, many artists and architects have proportioned their works to approximate to the golden ratio—especially in the form of the golden rectangle, in which the ratio of the longer side to the shorter is the golden ratio—believing this proportion to be aesthetically pleasing.

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