The Golden Ratio

So, once you’ve decided on where to paint, and what format to choose, it’s time to start thinking about how to place the elements of the picture on the canvas. And one of the methods artists use to divide up the canvas is geometry. As a former math geek, I have to admit that I really enjoy this topic…

Geometry has been important in art since ancient times; it appeared in ancient Egyptian, Indian, Greek, and Roman art. During the Renaissance, math was seen as the rational key to God’s perfect creation. So geometry was the artist’s way of translating creation into a painting.

So, for example, in della Francesca’s The Baptism of Christ (at right, from Composition, by Sarah Kent) Heaven is represented by a circle, and earth by a rectangle. The spot where they meet – the centre of the circle – is marked by the dove, representing the Holy Spirit. Jesus is placed on the vertical median, and the tree and John are on the verticals that divide the rectangle into thirds.

Because it symbolized the Christian Trinity (God, Jesus, and the Holy Spirit), the triangle (especially the equilateral triangle) was often used to compose religious paintings. It was particularly commonly used in paintings of the Madonna and Child (as in Raphael’s Madonna of the Meadow, at left), and of the Crucifixion.

The most famous mathematical compositional tool, though, is the Golden Ratio, also known as the Golden Mean or the Golden Section. It is based on the number Φ (phi), which is approximately 1.618. Phi is an incredibly cool number (well, to math geeks, at least).

Essentially, the Golden Ratio is the point at which a line can be divided so that the ratio of the long segment to the whole is the same as that of the short segment to the long.  So:

1 : 1.618 = .618 : 1

And did you notice that the short segment’s length is Φ-1?  Even better, Φ² works out to 2.618, or Φ+1.  It has other fascinating properties, too, like being the number toward which the Fibonacci series converges.

A rectangle with sides of length 1 and Φ, called the Golden Rectangle. is shown at right. If you divide off a 1 x 1 square at one end, you’re left with a small rectangle with sides 0.618 and 1 – another Golden Rectangle! And if you divide off a square at the end of that rectangle, you’re left with an even smaller Golden rectangle. And so on. If you then draw quarter circles in all those squares, you get a spiral that duplicates many found in nature:

Totally awesome. No wonder in the Middle Ages it was known as the Divine Proportion, embodying God’s perfect logic!

Ahem. Perhaps I’m not such a former math geek, after all!

In the next post, I’ll look at using the Golden Mean in composition.