Golden Ratio

The Golden Ratio is the ratio of two adjacent sides of a rectangle that the ancient Greeks held to be the most aesthetically pleasing when applied to building. The Classical Greek Parthenon, near Athens, embodies the golden ratio in much of its construction.

One definition of the golden ratio is that "A rectangle conforming to the golden ratio when cut to leave leave a square will also leave another rectangle conforming to the golden ratio". Such a rectangle is often called a "golden rectangle".

The image below shows a rectangle conforming to the golden ratio. The rectangle conforming to the golden ration is shown in the colour . When a square is cut from this golden rectangle, it leaves a rectangle shown in the colour . By moving the cut rectangle and then drawing a diagonal line from the corner to corner of the original golden rectangle (shown in ), it is clear to see that the cut rectangle (shown in ) has the same ratio as the original rectangle.

Calculating the Golden Ratio

Calculating the Golden Ratio is very easy. Using the diagram on the right, the golden ratio (φ) is defined as a/b as this is a golden rectangle. Now, we know that if a square is cut from the golden rectangle, a golden rectangle remains. So, if c=b, which is the square to be cut off, the dimensions of the remaining rectangle is (a-b) by b. As we know this to be a golden rectangle, then b/(a-b) = φ which equals a/b. Putting this in mathematical notation:

φ = b/(a-b) = a/b

or by rearranging

1 = a(a-b)/b²

and by expanding

1 = (a²-ab)/b²

on simplifying

1 = (a/b)² - a/b

or by rearranging

0 = (a/b)² - a/b -1

and as φ = a/b,

0 = φ² - φ -1

which is a familiar quadratic equation the solution which is:

which gives φ = 1.618034

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