October 7, 2009

Geometry in Nature and Art

A few days ago on my walk, I noticed that a lot of pine cones were littering the ground, so picked a couple of them up and carried them home. Looking at the spiral form of the cone took me back to graduate school, where many of us were neo-classicists in inclination, so learned about and used the Golden Section, or Mean, or Ratio, in composing our paintings. This ratio is expressed in different ways, one of which is: in a divided line, the proportion of the longer segment to the shorter is the same as the longer to the whole line. Expressed as a number, it is approximately 1.618. This mean was supposed to allow for perfect proportion, and some ancient buildings, possibly the Parthenon, were designed using it. The amazing thing about all this is that things in the natural world conform to this number, such as the spiral of the pine cone, or the arrangement of leaves and sprouts on a brussels sprouts plant.

The golden mean is also expressed as numbers in a Fibonacci sequence. In the image below, we can see how a spiral is created starting with two squares of 1, then to a square with a side of 2, then adding to that a square of 2 + 1, giving us 3, and so on for the sequence of 0, 1, 1, 2, 3, 5, 8, 13, 21, 34. It's remarkable how orderly nature is.

The following illustration shows how to draw a Golden Rectangle, which is supposed to be an ideal shape. Finding the center of a square, draw a diagonal to the corner and use that line as a radius to draw a circular arc; where it hits the continuation of the square is the golden rectangle.

For many artists, using these proportions was a way to achieve pleasing compositions. Many years ago I painted figure compositions, inspired by painters such as the fifteenth century Italian painter Piero della Francesca, and used the golden section for main compositional elements. In The Flagellation of Christ, Piero seems to have composed this rather remarkable painting (in that the main subject of the picture is in the background) as a series of golden rectangles. It has great power in its geometry and emotional solemnity.

Although I'm no longer interested in using a geometric ideal to compose my paintings, I find it fascinating that this ratio, intrinsic to the natural world, has engaged mathematicians, architects, and artists for centuries.


  1. What's really interesting in a history lesson such as this is the relevance/ connection to the present - that the simple act of picking up a pine cone brings it all to the fore again.

  2. Altoon, you have so much to share and I am fortunate to have a found this way to learn with you. You give a new meaning to Golden Mean to me on this morning and the pine cone photo is lovely.