1881 words - 8 pages

The Golden Ratio

The Golden Rectangle and Ratio

The Golden Rectangle and Golden Ratio have always existed in the physical universe. Nobody knows exactly when it was first discovered and applied to mankind. Many mathematicians assume that the Golden Rectangle has been discovered and rediscovered multiple times throughout history. This would explain why it is called many different names such as the Golden Mean, divine proportion, or the Golden Section.

The first person who is believed to have used the Golden Ratio is Phidias when he used it to design the statues inside of the Parthenon. This happened between 490 and 430 BC. In the early 300’s BC Plato used the Golden Ratio when he described the five platonic solids which are the tetrahedron, cube, octahedron, dodecahedron, and the icosahedron. Later in the 300’s Euclid gave the first written definition of the Golden Ratio which is an extreme and mean ratio. Then between 1170 and 1250 Fibonacci discovered a numerical series which had sequential elements that approaches the Golden Ratio asymptotically. Between 1445 and 1517 Luca Pacioli defined the Golden Ratio as the “divine proportion”. Then in between 1550 and 1631 Michael Maestlin published the first known approximation of the inverse golden ratio as a decimal fraction which is 1.61803398875. Very soon afterwards Johannes Kepler proved that the golden ratio is the limit of the ratio of consecutive Fibonacci numbers. Then in between 1842 and 1891 Edouard Lucas named the numerical sequence the Fibonacci sequence. In 1974 Roger Penrose discovered Penrose tiling which is a pattern that is related to the golden ratio both in the ratio of areas of its two rhombic tiles and in their relative frequency within the pattern.

The golden ratio from a linear aspect is described as a number that is found by dividing a line into two parts so that the longer part divided by the smaller part is also equal to the whole length divided by the longer part. The number that is found is called phi which is symbolized by ⱷ. The value of phi is found by the following equations:

(a+b)/a=1+b/a=1+1/ⱷ

Therefore,

1+1/ⱷ=ⱷ

Multiplying by ⱷ gives

ⱷ+1=ⱷ^2

This can be rearranged to

ⱷ^2-ⱷ-1=0

Using the quadratic formula, this solution is obtained:

ⱷ=(1+√5)/2=1.6180339887…

The golden ratio described from a rectangular perspective is as follows. When there is a rectangle with the length A+B and a width of A there is a square inside of this rectangle with the dimensions AxA. If the square is removed from the rectangle, the ratio of the remaining rectangle would still be the same as the original rectangle before the square was removed. The golden rectangle can also be formed if there is a square and from a top corner of the square (point B) a curve is drawn down to a point (F) which is on the same line as segment DC and is half the distance of segment DC. If a perpendicular line is drawn from this point vertically the distance of segment AD to point G then this...

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