The Golden Ratio and The Fibonacci Numbers
Please tell me about the Golden Ratio (or Golden Mean), the Golden Rectangle, and the relation between the Fibonacci Sequence and the Golden Ratio. The Golden Ratio () is an irrational number with several curious properties. . in which the Fibonacci Numbers occur naturally in relation to the Golden Ratio. the golden ratio ϕ, which is related to the shapes of pineapples, The connection between the Fibonacci numbers Fnand the golden ratio ϕis.
Geometry has two great treasures: Mathematics and architecture In a recent book, author Jason Elliot speculated that the golden ratio was used by the designers of the Naqsh-e Jahan Square and the adjacent Lotfollah mosque.
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The authors note, however, that the areas where ratios close to the golden ratio were found are not part of the original construction, and theorize that these elements were added in a reconstruction. The Swiss architect Le Corbusierfamous for his contributions to the modern international stylecentered his design philosophy on systems of harmony and proportion.
Le Corbusier's faith in the mathematical order of the universe was closely bound to the golden ratio and the Fibonacci series, which he described as "rhythms apparent to the eye and clear in their relations with one another. And these rhythms are at the very root of human activities.
They resound in man by an organic inevitability, the same fine inevitability which causes the tracing out of the Golden Section by children, old men, savages and the learned. He saw this system as a continuation of the long tradition of Vitruvius, Leonardo da Vinci 's Vitruvian Manthe work of Leon Battista Albertiand others who used the proportions of the human body to improve the appearance and function of architecture. In addition to the golden ratio, Le Corbusier based the system on human measurementsFibonacci numbersand the double unit.
He took suggestion of the golden ratio in human proportions to an extreme: The villa's rectangular ground plan, elevation, and inner structure closely approximate golden rectangles.
Several private houses he designed in Switzerland are composed of squares and circles, cubes and cylinders. In a house he designed in Origliothe golden ratio is the proportion between the central section and the side sections of the house.
While we see that the Fibonacci series emerges naturally in the evaluation of the powers of the Golden Ratio, this does not necessarily make it clear why the ratio of the members of the Fibonacci series should approach the Golden Ratio as a limit. As it happens, the connection can be illustrated through the technique of Continued Fractions, which is a device for reducing non-repeating decimals to fractions, i.
The technique for reducing repeating decimals to fractions has been discussed elsewhere. With non-repeating decimals, the integer part of the number is successively removed, and the reciprocal is taken of the remaining decimal, producing a new integer, which is then removed, and the process repeated.
This can be continued until the desired accuracy is attained or the capacity of the calculator is exceeded -- since I would assume that most people today would be using a calculator to get the reciprocals it is not a very convenient procedure otherwise.
Math Forum: Ask Dr. Math FAQ: Golden Ratio, Fibonacci Sequence
Once enough integers are obtained, the corresponding fraction with all the embedded fractions can then be solved for a simple integer fraction. For instanceat right is a continued fraction for the ratio between the the length of the lunar synodic month This tells us the number of lunar months per solar year; and, in integer form, the fractional part would tell us how many extra lunar months more than 12 per year would need to be added in a certain period of solar years to approximate the true ratio.
The true ratio is Removing the 12 and then successively taking the reciprocal and removing the integer part again gives us the integers, after 12, 2, 1, 2, 1, 1, and 17 at least. Successive approximations can be made by stopping at each new integer. More important, however, was the fraction two steps further.
Adding 7 lunar months every 19 years was a device adopted for the Babylonian Calendar. The rule was inherited and used even today by the Jewish Calendar and for the Christian reckoning of Easter. The continued fraction for is given in a footnote.What is the relationship between the Fibonacci Sequence and the Golden Ratio
The Golden Ratio has the unique property that its reciprocal always produces the same decimal and the reciprocal of the decimal will always produce the integer 1.