The Golden Mean
To see how the Fibonacci sequence relates to visual space, consider the following representation. We stack squares based on the numbers in the sequence. The first two numbers are one, so the first two squares have sides equal to one unit. The next number is two, so this square lines up nicely with the first two. As the arrangement becomes larger, it begins to fall into a recognizable, repeating pattern.
Creating an arc that traces the edges of each block leads us toward the center in what is called the Fibonacci spiral. Starting with just a few blocks, the spiral is not yet very accurate, but we can extrapolate what the spiral should look like given an infinite number of iterations. Artists call the center of the spiral the cradle and mathematicians call it the pole.
Consider how Phi applies within a composition. The ratio between lines and masses will correspond to the golden ratio. In the previous illustration, the ratio of line B to line A is Phi, 1.618. For example if line B is 16.18 inches long, line A is 10 inches, 16.18/10 = 1.618. The borders of the outside rectangle follow the same proportion. The area in pink is a square. The remaining rectangle is divided in the same way, by creating a square whose proportion to the overall rectangle is Phi. Each subsequent rectangle can be subdivided, and so on, to infinity.
The preceding diagrams are a form of armature, or a framework on which a photograph can be composed. The armature can be flipped horizontally, vertically, or rotated 180 degrees and it would still be in accordance with the golden mean. Additional golden mean armatures can be applied to subsections of the image. One way to use the golden mean in a composition is to place the design elements along the lines indicating the golden rectangles and Fibonacci spiral. Also, the golden mean should appear in the subjects. This means showing body proportions and other compositional elements that correspond to the golden ratio when compared to one another. For example, if you photograph a model laying on a rock, you might choose a rock that is 1.6 times her length rather than one that is twice her length. A group of rocks with a visual mass totaling this ratio would also suffice, especially if the subdivisions of the rocks also conform to the golden ratio.
Some photographers follow the golden mean religiously and sometimes sacrifice everything else in their photos in the process. Others reject the golden ratio outright, deeming it an over-practiced fad and actively avoid it. I prefer to produce images that I find interesting and innovative, rather than photograph according to a formula-but I also apply lessons learned from the classic methods of composition.
Also see: Rule of Thirds
Also see: Comparing Golden Mean and Thirds (soon to be published)