The Golden Ratio
What is it about the 35mm rectangle format that attracts so many photographers to its dimensions of 36x24mm – a proportion of 1.5 to 1 – that gives such a pleasing sense of harmony and balance, which is so appealing to the eye? (fig.2)
The proportion of 1.5 to 1 comes very close to 1.6180… to 1, the proportions of the golden rectangle (fig. 1) whose side lengths are a ration of 1.6180 to 1, a ratio so particularly interesting that it has fascinated more brilliant minds than pi or e put together. The reverence shown to this ratio is indicative of the names attributed to it: the transcendental ratio, divine proportion, divine number, golden section, golden number, golden mean, are all common names for what is known as the golden ratio.
It is based on the number phi (φ) = 1.61803398874… discovered by the Italian mathematician, Leonardo Pisano (1170-1250, who is better known today as Fibonacci), being the ratio between the number sequence 1, 1, 2, 3, 5, 8, 13, 21 etc where the next number in the sequence is derived by adding together the two previous numbers. So, 1+1 = 2, and 1+2 = 3, and 2+3 = 5… the Fibonacci sequence.
As one progresses through the series eg, by continuously sub-dividing a golden rectangle by the ratio, the ratio between each successive pair gets closer and closer to phi and the spiral produced exactly matches the growth of the nautilus shell or the arms of a spiral galaxy (fig. 1). Is this not peculiar or what?
Once one is aware of it, the golden ratio seems to be everywhere: the elegant spirals of shells, in the solar system such as the Milky Way, the branching of trees, the curves of waves, the seed head of a sunflower, architectural masterpieces such as the Greek Parthenon, the pyramids, Notre Dame and works by Corbusier.
Not only does the golden ratio seemingly crop up everywhere, it also seems that the human eye is very attracted to the results of the ratio. It is a variation of the golden ratio that artists over the centuries have used to give structure, order and harmony to their creations – the rule of thirds.
The Rule Of Thirds
To better understand the rule of thirds divide your 36x24mm frame into three horizontal parts and three vertical parts, creating nine separate zones (fig. 2). Many digital cameras have the facility to overlay this grid pattern in the viewfinder eg, my Nikon D90, and with some film cameras the screen can be changed, eg Olympus OM, Nikon F, F2 etc – check your manual.
It seems that the human eye is not too happy resting on objects in the centre of a photograph. Studies show that it is more natural for our eyes to rest on one of the intersecting lines on the rule of thirds grid. Placing your main objects of interest in one of these positions allows your viewer’s eye to rest comfortably on this spot instead of fighting against an element placed at the centre (figs. 3 & 4).
In landscape photography experiment by placing the horizon in the picture one third of the way up or one third of the way down. Putting the horizon in the middle, thereby splitting the picture in half, is less appealing and natural to look at: it creates a tension that the eye and brain finds difficult to resolve satisfactorily.
When composing your picture you really have to ask yourself one simple question: which is more interesting, the sky or the ground? If it is the sky place the horizon two-thirds down from the top, if it is the ground place the horizon a third down from the top. Of course, rules are not set in stone, but the rule of thirds is a good compositional starting point, being one artists have used for centuries to good advantage.
The world of math and the world of fine art are linked and related, and math is undeniably artistic, evidenced by the elegant mathematical solutions to enhance our love of form and composition, which echo those found in nature.