The Golden Ratio: Natures Beautiful Proportion
At first glance of the title, many may wonder: What is the Golden Ratio? There are many names the Golden Ratio has been called including the Golden Angle, the Golden Section, the Divine Proportion, the Golden Cut, the Golden Number et cetera, but what is it and how is it useful for society today? One may have heard of the number π (Pi 3.14159265…) but less common is π’s cousin Φ (Phi 1.61803399…). Both Φ and π are irrational numbers, meaning they are numbers that cannot be expressed as a ratio of two whole numbers as well as the fact that they are never-ending, never-repeating numbers. The Golden Ratio is the ratio of 1:Φ (1.61803399…). The Golden Ratio is a surprising ratio that is based on the research of many composite mathematicians spanning over 2300 years, and it is found in many areas of everyday life including art, architecture, beauty and nature. Euclid, a Greek mathematician who taught in Alexandria around 300 B.C., was one of the first to discover and record the bases for the Golden Ratio back 2300 years ago. Euclid’s discovery was that if one takes a line and divide it into two unequal sections in such a way that the longer section is the same proportion to the shorter section as the entire line is to the longer section then one will get the Golden Ratio (Natural History. March 2003). On the line below, this is illustrated. If the segment BC is hypothetically the number 1, then the segment AB would be equivalent to Φ (1.61803399…) or if AB=1 then AC=Φ A
AB:BC is equal to AC:AB is equal to 1:1.618 (Φ)
One of the most interesting aspects of Φ is the connection it has with Fibonacci numbers. The Fibonacci sequence is a series of numbers that was discovered by the early 13th century Italian mathematician, Leonardo of Pisa. He was commonly known as Fibonacci, a shortened form of Filius Bonaccio, which literally means “son of Bonaccio”. Fibonacci, though Italian, was raised on the north coast of Africa due to his father’s work as a customs officer. As a result, Fibonacci was taught the Arabic system of numbers. When he returned to Italy, he published a book that introduced this Arabic number system to Europe and consequently gave Fibonacci the reputation of the most accomplished mathematician of his time (Dunlop, R. 2003). The Fibonacci sequence is a series of numbers where starting with 1, the previous two numbers are added together to get the next number in the sequence. Taking 0 and 1, that would be 0+1=1. Now the last two numbers in the sequence are 1 and 1, so 1+1=2, then 1+2=3, 2+3=5 etcetera so the sequence looks like: 1, 1, 2, 3, 5, 8, 13, 21, 34, 55, 89, 144, 233, 377, 610, 987, 1597, 2584... Fibonacci numbers are closely connected to the Golden Ratio, they are even sometimes called Golden Numbers. This is because if two progressive numbers are taken in the sequence that are right before/after each other, and divide the larger of the two by the smaller number, then a number very close Φ is achieved. This becomes more and more true, the larger the numbers you use. For example: 5 / 3 =1.666666667…, but 2584 / 1597 = 1.618033813… and Φ = 1.61803399… and this is only one of the many relationships we see between Fibonacci numbers and the Golden Ratio. Another important aspect of the Golden Ratio based on Fibonacci numbers is the Golden Rectangle. These rectangles are formed when the long side of the rectangle is divided by the short side and it equals Φ. This is because the lengths of the sides of a Golden Rectangle are composed of Fibonacci numbers.
Luca Pacioli was a mathematician born in Tuscany in 1445 who wrote many books which won him the title “The Father of Accounting”. After writing the book Summa de Arithmetic, Geometria, Proportioni et Proportionalita in 1494, Leonardo Da Vinci arranged for Luca to come and tutor him in mathematical perspective and proportion. Under the direct influence of Da Vinci, Luca wrote the masterpiece and...
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