Perfect Proportion? The Golden Ratio?

Golden Ratio in Photography

Photographic Composition & Proportion

Golden Ratio Misconceptions: Putting The Golden Ratio To The Test

In my sustained quest to learn more about photographic composition and preferred proportions, I’ve encountered the apparent and perhaps misleading use of the Golden Ratio on the internet,  time and time again.  And it’s annoyed me enough to initiate this prelude to a forthcoming article I’m musing over.

Woman’s Face

And this is what grates at me: strategically placed lines, Golden Rectangles and Golden Spirals on artwork, humans, animals and anything man-made to portray that the Golden Ratio is indeed embedded within them and it is that which gives those things beauty.

Taj Mahal, Agra

I’m dubious about these claims and these seemingly arbitrarily-placed rectangles.

Violin

Could there actually be hidden proportions in imagery, nature and buildings that the majority prefer over any other?

Let’s actually put the Golden Ratio to the test!

Silva Lucimara

Butt for that, I will need your help. Please take a few moments to study the rectangles in the grid below and vote for your favorite one. The one that most appeals to you. The one you find the most attractive. The one you find aesthetically pleasing. The one you believe to have the best proportions.

Only vote if you see a black square here. A black rectangle means that your desktop size is not configured to display aspect ratios correctly.
And submit your vote before reading the remainder of this post.

All the rectangles in the grid below are unique in size and proportion. It is not a trick. I just want to measure what people actually prefer.

Vote for the Rectangle that Pleases You the Most

And if you want to help further, please re-blog this article  and share it so that as many people as possible get to see it and vote. The empirical data that are collected in this exercise are crucial for my next article.

There are plenty of comprehensive references to the Golden Ratio on the internet already, so here is just a quick summary of what it’s all about.

The Golden Ratio

For centuries, scholars have tried to identify hidden patterns in art and nature; patterns that we find attractive and pleasing to look at. And it is said, that one proportion is known to please us more than any other. It is known as Phi (φ) or the Golden Ratio.

It’s existence has been known for millennia but the first written record of it dates back to the time of the Greek mathematician, Euclid in  300 B.C., although Euclid referred to it as the extreme and mean ratio in his work, Elements.

The idea is quite simple.

Take a line and divide it into two pieces such that the ratio of the whole line to the big piece is equal to the ratio of the big piece to the small piece. Shown to fourteen decimal places, this ratio turns out to be:

φ = 1.61803398874989 ...

The Golden Ratio | (a + b) / a = a/b = 1.618…

The Golden Ratio is actually an irrational number. However, ratios are rational, which means that they can be expressed by the quotient of two whole numbers.

Any number that can be obtained by dividing one whole number by another whole number is a rational number.

Yet this number is irrational. It cannot be expressed as a ratio of two integers because of its decimal expansion. The decimal places continue forever without settling into any sort of repeating pattern. The Golden Ratio is not an actual ratio of whole numbers. It is a ratio of lengths.

A rectangle made with sides in these proportions is said to be the most pleasing to look at of any rectangle. This is the Golden Rectangle, and during the Renaissance it had a resurgence in popularity.

Golden Rectangle: Formed with the Phi Ratio of 1 : 1.618…

In 1509, the Franciscan friar and Italian mathematician, Fra Luca Bartolomeo de Pacioli published a manuscript entitled Divina Proportione. The subject: mathematics and the artistic proportion. The Golden Ratio was re-branded with the title of that work: Divine Ratio, and with such an auspicious new title associated with God, interest for it rallied.

Mona Lisa, Leonardo da Vinci

It is claimed that this divine proportion can be detected in the works of artists from that period, such as Leonardo da Vinci and Michelangelo.

Centuries earlier, another Italian mathematician, Leonardo Fibonacci found that the Golden Ratio had a connection to a seemingly unconnected problem in mathematics. In his book, Liber Abaci, Fibonacci tackled a difficult problem about rabbits:

Take a pair of rabbits that take one month to become mature. And then they start to have pairs of baby rabbits, which also take one month to reach maturity before they can have pairs of babies of their own. After, say 10 months, how many rabbits would there be?

What Fibonacci noticed was that with each month, the new number of rabbit pairs would be the sum of the previous two. This became known as the Fibonacci Sequence.

Fibonacci Sequence – Rabbit Pairs

Fibonacci Sequence: 1, 1, 2, 3, 5, 8, 13, 21, 34, 55, 89, 144, ....

Now if you take two consecutive numbers from the Fibonacci Sequence and divide the larger number by the smaller one, the result is a close approximation to Phi(φ). And the further up the Fibonacci Sequence you go, the number becomes closer and closer to the Golden Ratio.

1/1 =1.000
2/1 - 2.000
3/2 = 1.500
5/3 = 1.667
8/5 = 1.600
13/8 = 1.625
21/13 = 1.615
34/21 = 1.619
55/34 = 1.618

So there appears to be a profound connection between Phi(φ) and the Fibonacci Sequence. And the breeding pattern of rabbits is not the only natural phenomenon that can be described with this sequence of numbers.

The Fibonacci Sequence certainly appears often in nature, but not all the time.

More importantly though,  it does appear that a number of misconceptions abound. This interesting paper, written by George Markowsky uses science, logic, empirical data and evidence to highlight a lot of the misconceptions surrounding the Golden Ratio. He demonstrates:

• That the name  “Golden Ratio” was NOT used in antiquity
• The Great pyramid was NOT designed to conform to the Golden Ratio φ
• The Greeks did NOT use φ in the Parthenon
• Many painters, including Leonardo da Vinci, did NOT use φ
• The UN building does NOT embody the Golden Ratio φ
• That the Golden Rectangle is NOT the most esthetically pleasing rectangle
• The human body does NOT exhibit φ

It certainly appears that a formidable cottage industry has evolved  from this mystical number; a force that continues to repeat misleading and false information; repeated so many times that it starts to become folklore unless challenged.

For me, the magical properties and real beauty of the Golden Ratio remain with the mathematics and the tendency for it to appear in mother nature.

And as for seeking that illusive pattern in photographic images that we would all find pleasing…

… Isn’t beauty in the eye of the beholder anyway?