Finding Golden Rectangles

Finding Golden Rectangles

In today's pop culture people like to find and make patterns, patterns in nature, patterns in crimes, patterns in architecture, etc. One such pattern is the golden ratio or the divine proportion. The golden ratio was believed to be prevalent in nature, and thus, must be from the workings of God, leading people to actively seek out this specific proportion. This golden ratio has also seemingly found its way into modern art through the golden rectangle. A golden rectangle is a rectangle with sides in the golden ratio, 1: 1.618, or, the greek alphabet, phi. It is said that golden rectangles are also the most aesthetically pleasing type of rectangles. In paintings by Piet Mondrian, golden rectangles appear occasionally, however, just how many appear, and did Mondrian intentionally make these golden rectangles?

In this painting by Mondrian, there is a single individual rectangle that meets the standard of a golden rectangle by having a ratio of 1: 1.645, within the 2% accepted range as used by George Markowsky, between 1.586 and 1.650. However, when all other rectangles are taken into account by combining rectangles to form bigger ones, three rectangles were found to have the golden ratio. Counting all rectangles in the painting, 33 rectangles were found. In the 33, three were found to be of having divine proportions. One of eleven rectangles in the painting is found to be of golden ratio. Having a 9.1% chance of obtaining a golden rectangle seems to be oddly high, as the precise measurements need to be obtained. However, as said before, the golden rectangle is said to be the most aesthetically pleasing rectangle, and therefore, Mondrian may have unconsciously included more of these rectangles purely because of aesthetical reasons.

But just how likely would Mondrian paint three golden rectangles in his painting if he were unconscious of the golden rectangle phenomena? To figure this out, we will need to generate one of these "rectangle paintings" by random. I first drew a 15cmx15cm square. Then I used Excel to create six different random numbers between 0 and 1. Three of these numbers would be used to draw the lines of the horizontal plane while the remaining three would be used to draw the vertical plane. The numbers between 0 and 1 would be the percentage of the total; 15cm. The six numbers that were randomly generated were 0.847, 0.158, 0.027, 0.096, 0.210 and 0.916. These divisions will lead me to create a 4x4 grid. A 4x4 grid contains a total of 100 combined rectangles. Out of these 100 combine rectangles, I only found one that was close to the golden ratio of 1.618, being 1:1.64. The equation "number of golden rectangles/number of rectangles" would be used to give a rough estimate of the probability of obtaining a golden rectangle. So we have 1 golden rectangle/100 rectangles, giving us a probability of 0.01 or 1%.

Figure 2. Rectangles generated from random percentages of 15cm. Red colored rectangle is in golden ratio.

According to this estimation, there shouldn't even be a single golden rectangle in Mondrian's painting (0.01 * 33 = 0.33). This is considerably lower than the percentage of golden rectangles in Mondrian's painting of 9%, and considerably lowers than the number of golden rectangles actually found in his painting. This may suggest that Mondrian was conscious of the golden rectangle concept, and may have incorporated these into his paintings. But was Mondrian really aware of the golden rectangle concept? Maybe he just thought these rectangles in the golden ratio to be aesthetically pleasing, and so, included them into his work. To find out, we'll take a look at his background.

Being paramount in modern architectural design and having a fascination in geometrical abstractions, I'd say that Piet Mondrian was very likely trained in math. Mondrian was also very fascinated by a philosophical movement known as Theosophy. Theosophists are people that try to seek knowledge of "presumed mysteries of being and nature, particularly concerning the nature of divinity"1, 10. Later, in 1909, Mondrian even joined the Dutch branch of the Theosophical Society. Through this, Mondrian drove his search for the aesthetics even further, and, as a result, his paintings became increasingly abstract, experimenting with Cubism, triangles and rectangles, and finally, breaking entirely off from representational painting. Mondrian's artistic view moved to creating paintings entirely of lines perpendicular to one another with basic colors. Mondrian believed that there were hidden truths and concepts in the geometric shapes of triangles and rectangles that people may perceive with his paintings. If Mondrian were well aware about fact of golden rectangles, it would not be surprising at all, as Mondrian was fascinated with these sorts of concepts. The concept of the golden ratio started spreading around the time of German Romanticism, dating back to the late 18th century3. Being this old, Mondrian may have been made aware of the golden rectangle theory as he painted his works of art. There is not direct evidence of Mondrian applying or even knowing of the golden rectangle, but with his interests and hobbies in Theosophy, it would not be one bit surprising to find out he was quite familiar with the concept. In 2005, Dr. Vladimir Konecni of University of California, San Diego published a research article detailing the accuracy of Mondrian's golden rectangles. He found that, among all the other artists he studied, Mondrian had the highest accuracy of golden rectangles. This may also suggest his familiarity with the golden rectangle phenomena9.

There is also information supporting the fact that things in the golden ratio are the most aesthetically pleasing. This may be the reason why Mondrian included any rectangles that resembled a golden ratio in the first place; purely based on aesthetic means. A US academic, Adrian Bejan, believes that things in the golden