Phi in Art
Many artists believe that mathematics plays a role in what we perceive as aesthetically pleasing and beautiful (Burger & Starbird). The Golden Rectangle, a rectangle whose ratio of length to with is the Golden Ratio, has been noted to have an aesthetic property that makes it pleasing to the human eye. A professor at Duke University, Adrian Bejan argues that there is an evolutionary reason for this. According to Bejan, the human eye is able to interpret images that feature the Golden Ratio faster than others. His argument is that animals, including humans, have vision that is horizontal in orientation. In the wild, danger tends to approach from behind or from the side view of an individual. This explains why our eyes gather information more efficiently when scanning side to side rather than up and down. Bejan also argues that our natural inclination of viewing the world explains why artists, whether knowingly or unknowingly tend to draw and make things that conform to Golden Rectangles (McVeigh, 2009). For this reason many artists have used Golden Rectangles to frame entire works of art or as a basis for structuring the creation so that a sense of balance and harmony of proportion are achieved (Hemenway, 2005). However, there is a difficulty in determining whether some instances of art conforming to the Golden Rectangle are intentional or not. In situations where the artist left no indication of his purpose, it is impossible to know his intent. For example, although the Parthenon fits inside of a Golden Rectangle, one cannot prove that this was the intent of the architect because there is no documentation. In cases like this, all one can do is wonder and marvel at how frequently the Golden Ratio seems to appear in things that we create.
Although there are many artists for whom one cannot be sure if the Divine Proportion was intentionally used, there are some whose use of the Golden Ratio is unmistakable. Many scholars believe that Leonardo Da Vinci who would have had extensive time working with the Golden Ratio. As he was the illustrator for Pacioli’s book, The Divine Proportion, he would have had ample opportunity to be heavily influenced by this ratio. (Meisner, 2013). His “Mona Lisa” and “The Last Supper” contain many instances of Golden Rectangles. (Hemenway, 2005). Michelangelo has blessed the world with his painting “The Creation of Adam” captured on the ceiling of the Sistine Chapel. In this painting Adam’s finger touches God’s finger at the exact point of the Golden Ratio of the height and width of the sections that contain them (Meisner, 2013). Other artists whose works prominently features the Divine Proportion are Albrecht Durer and Georges Seurat. (Hemenway, 2005).
Below are some examples of the Golden Rectangle in art. Click on the images for more information.
Although there are many artists for whom one cannot be sure if the Divine Proportion was intentionally used, there are some whose use of the Golden Ratio is unmistakable. Many scholars believe that Leonardo Da Vinci who would have had extensive time working with the Golden Ratio. As he was the illustrator for Pacioli’s book, The Divine Proportion, he would have had ample opportunity to be heavily influenced by this ratio. (Meisner, 2013). His “Mona Lisa” and “The Last Supper” contain many instances of Golden Rectangles. (Hemenway, 2005). Michelangelo has blessed the world with his painting “The Creation of Adam” captured on the ceiling of the Sistine Chapel. In this painting Adam’s finger touches God’s finger at the exact point of the Golden Ratio of the height and width of the sections that contain them (Meisner, 2013). Other artists whose works prominently features the Divine Proportion are Albrecht Durer and Georges Seurat. (Hemenway, 2005).
Below are some examples of the Golden Rectangle in art. Click on the images for more information.
Phi in Plants
Of all the places in which Phi appears, none may be as awe inspiring as those which occur in nature. The easiest way to see Phi in action is to notice how plants grow. Phyllotaxis is the name given for how plants distribute leaves on a stem as well as the mechanisms that govern the process. This mechanism is so prevalent in plants that mathematicians and botanists have come to use the term to describe the repetitive arrangement of petals, seeds, florets, and in some cases, branches (Adam, 2003). Upon examining many flowers, one would notice that most types have a Fibonacci number for the number of petals. Buttercups have 5 petals, marigolds have 13 petals, and daisies have 34, 55, or 89 petals. (Adam, 2003). This is true for many flowers, but there are exceptions. However, generally speaking most flowering plants use Fibonacci numbers for the number of petals.
Plants also use phyllotaxis to place their leaves. Leaf arrangement is important due to plants needing to obtain the optimal amount of sunlight and water. As the plant grows upward, it grows leaves at regular intervals that originate from the stem. If these intervals consisted of regular whole number divisions of 360 (the circular path around the stem is 360 degrees), then the leaves would overlap each other and block the sunlight and some moisture from the leaves below it. (Adam, 2003). Phi’s designation as the most irrational number helps in this case (Livio, 2002). Dividing 360 degrees by Phi gives an angle that provides the optimal distance between consecutive leaves; 225.5 degrees (or 135.5 degrees, depending on which way the rotation is measured). The 135.5 measurement is usually what is used since it is less than 180 degrees and is called the Golden Angle. When plants use this angle as the interval between consecutive leaves, then the spacing is optimized for maximum efficiency in gathering light and moisture (Adam, 2003). This same phyllotactic process is found on the arrangement of seeds in a flower’s head. It is the most efficient way to pack seeds in the head without wasting space. A sunflower gives an easy to see example of this phenomenon.
Plants also use phyllotaxis to place their leaves. Leaf arrangement is important due to plants needing to obtain the optimal amount of sunlight and water. As the plant grows upward, it grows leaves at regular intervals that originate from the stem. If these intervals consisted of regular whole number divisions of 360 (the circular path around the stem is 360 degrees), then the leaves would overlap each other and block the sunlight and some moisture from the leaves below it. (Adam, 2003). Phi’s designation as the most irrational number helps in this case (Livio, 2002). Dividing 360 degrees by Phi gives an angle that provides the optimal distance between consecutive leaves; 225.5 degrees (or 135.5 degrees, depending on which way the rotation is measured). The 135.5 measurement is usually what is used since it is less than 180 degrees and is called the Golden Angle. When plants use this angle as the interval between consecutive leaves, then the spacing is optimized for maximum efficiency in gathering light and moisture (Adam, 2003). This same phyllotactic process is found on the arrangement of seeds in a flower’s head. It is the most efficient way to pack seeds in the head without wasting space. A sunflower gives an easy to see example of this phenomenon.