Composition:Fibonacci’s Divine Ratio

 

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This is the  Fibonacci spiral with some of the numbers of the Fibonacci sequence: 1,1,2,3,5,8.  Notice that the side length of each square corresponds to the numbers of the Fibonacci Sequence.

I first encountered the Fibonacci sequence while I was in 6th grade and we watched the filmstrip “Donald in Mathmagic Land” by Disney.  It was quite a treat to watch a film in class and it was even more enjoyable to see a cartoon about math.  The narrator of the film introduced Donald to the wonders of the Fibonacci spiral and how it is found throughout nature.  Much later, when I was teaching 6th grade math, the Fibonacci sequence was one of many sequences used to help students recognize patterns in numbers.  Look at the following sequence to see if you can figure out the pattern: 0,1,1,2,3,5,8,13,21,34….  If you begin with zero and add 1, the sum is 1. If you continue with the same pattern, each the previous two numbers will have a sum of the following number.  1 + 1 = 2, 1+2 = 3, 2 + 3 = 5 and so on.  The successive numbers in the sequence form ratios referred to as the Golden Mean, the Golden Ratio, or the Divine Proportion. Fibonacci, who lived around 1200 A.D., did not “discover” the Golden Ratio but his mathematical work with patterns and spirals made the idea more prominent. Many scholars believe that people had been subconsciously using the idea of the Golden Ratio in art and architecture long before the mathematician Euclid first defined it around 300 B.C..

The bigger the pair of Fibonacci numbers, the closer the approximation to the value of the golden ratio: 1.618….

A            B             B/A

2            3              3/2 = 1.5

3            5               5/3 = 1.66………..

…          ….

144        233          233/144 = 1.618055556…..

There is a lot of complicated math attributed to Fibonacci, but the basics of his sequence and spiral are woven throughout many aspects of our daily lives. If you want to know more about the math involved click here.

Famous historical architecture, such as the Parthenon, used the mathematics of the golden ratio in its design. Leonardo DaVinci employed the mathematics of the golden ratio while painting the Mona Lisa.  The Fibonacci spiral and Fibonacci numbers are found throughout the natural world from the numbers of petals on a flower to the number of spirals on a pine cone, to the pattern of a nautilus shell.  Scientists have even proven that the human faces we find most attractive are those whose features are arranged closest to the Golden Ratio. Here is a good website that gives more information about the Fibonacci and nature  click here.

So what does this have to do with photography? Well, if nature has been arranged according to the Divine Proportion since the beginning of time then it is hard to argue with the results.  We are surrounded by so much natural beauty that represents the mathematics of the Divine Proportion and it seems like it would be difficult to improve on the idea.  Our eyes are naturally drawn to faces, flowers, architecture, and landscapes that fit into these proportions.  The Fibonacci spiral and Golden Ratios are two guidelines that help create a well-balanced and pleasing photo.

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Here you can see that the center of the spiral frames the sun. The sun is the brightest part of the picture and is the first thing to catch your eye.  Your eye will naturally roam around the image after noticing the focal point, often following the path of the spiral.  Here it leads the eye under Turret arch.
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Here, the gray rock is in the middle of the spiral.  The rock is a different color than it’s surroundings, which is why I chose to make it the focal point of the photo.  Even though the arch is a more interesting element, the eye is drawn to the location of the gray rock because of its placement in the frame. Following the path of the spiral, they eye is then led over the top of Broken Arch.
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There are many elements of interest in this photo, but the anchor is the red rock formation in the center of the spiral.  The red rock formation on the left is situated on the far left of the continuing spiral,  keeping the image balanced and drawing the eye through the frame.

You can read more about composition guidelines in these previous posts: The Rule of Thirds and Leading Lines and 3 Ways to Improve Your Photography.

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