Thursday, April 7, 2011

Parthenon and the Golden Ratio

Today I visited the Parthenon which is one of my favourite buildings and the preeminent landmark of Athens and located in the Acropolis. And yet, many people are unaware of the beautiful mathematical design it holds in plain view. Built into the Parthenon’s design you will often discover that many porportions are designed to the golden ratio, also known as the divine proportion, the ratio that many Greek’s believed to be the source of proportional mathematical beauty.

Shown above: Parthenon

Many believed that this ratio occurred often in nature and was the basis for beautiful proportions (for example it appears in Leonardo Da Vinci’s Vitruvian Man as the ratio between the length of the arms and legs).  Another occurance is with Fibonacci numbers as the ratio between them is an approximation of this same ratio (Fibonacci numbers are numbers in a sequence in which the next number is the sum of the previous two numbers). The approximation becomes more accurate the further out you go. Note: It's only an approximation because the first two numbers are one and one.

The golden ratio is mathematically defined as follows:

(A + B) / A = A / B

Or in layman’s terms, the proportion between two measures is the same as the proportion of their combined length and the larger length. Note that this definition can be recursive and applied over multiple lengths repeatedly.

Also note, that the proof for the actual ratio is self is quite elegant:

(A + B) / A = A / B (Multiplying both sides by A x B we get)

AB + B^2 = A^2

0 = A^2 – AB – B^2

= A^2 – AB + 1/4 B^2 – 1/4 B^2 – B^2

= (A – 1/2 B)^2 – 5/4 B^2

Now assume that B is 1.

= (A – 1/2)^2 – 5/4

5/4 = (A – 1/2)^2

Sqrt(5) / 2 = A – 1/2

Therefore, the golden ratio, A = [1 + sqrt (5)] / 2

Note that this makes A approximately 1.62. (it is an irrational mathematical constant)

Intricate in its beauty, elegant in its simplicity.

Near the Parthenon was once the statue of Athena overlooking the city.

Shown above: Athena's perch from which she offered Victory to the Athenians, now long vacant.

Previously, this building was home to statue of the grey-eyed goddess who watched over her namesake and in her raised hand held Nike, the winged goddess of victory. While the statue is long gone, perhaps it is time for the Greeks to look after her and her fellow Olympians as they recover and protect artifacts of the past.

1 comment:

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