Visual Arts16 Jan 2008 10:04 am

The Mathematics of the Parthenon

Admirers of the Parthenon often mention the fact that its proportions often approximate the “golden ratio.” What exactly does this mean?

Wikipedia has a concise and accurate definition of the golden ratio: “two quantities are in the golden ratio if the ratio between the sum of those quantities and the larger one is the same as the ratio between the larger one and the smaller.” This is probably a bit confusing, so an example might help.

Let’s consider two values, a and b, with a being the greater of the two values. The values form a golden ratio if the value of a divided by b are equally to the value of a+b (the sum of the values) divided by a (the larger of the two):

If you do the math, the proportion between a and b comes out to about 1.618 to 1.

A fascinating instance of the golden ratio is a golden rectangle, that is, a rectangle with sides that have the relationship 1:1.618. If you take a rectangle of this size and remove the square formed by the shorter side, you’re left with another golden rectangle. In the image of a golden rectangle below, for instance, the sides a and b have the relationship 1:1.618. Remove the square formed by the shorter side (the one in blue) and the rectangle that’s left (the one in pink) is itself a golden rectangle. You can repeat the process to infinity.

So what does all of this have to do with the Parthenon? It turns out that the facade of the Parthenon is a golden rectangle, and important elements of the structure, like the sculptures in the frieze, are deliberately placed in areas where new golden rectangles are formed by the process described above. It seems that the Parthenon’s reputation for exactitude and perfection is well earned.