What do the arrangements of the petals of a sunflower, the Parthenon, the Mona Lisa, and the shape of the galaxy have in common? The answer is this: the golden ratio.
The golden or divine ratio is a system of proportions that structures a composition perfectly. In particular, the concept is based on dividing a line into two parts, so the ratio of the largest and smallest parts should be equal to the ratio of the entire length and the largest part.
It is said that the golden ratio was first introduced and calculated by Pythagoras (585 – 500 BC), while Euclid (c. 325–c. 265 BC) was the one who first defined it in writing in his work Elements. Specifically, he called it “extreme and average speech.”
The most famous depiction of the golden ratio is rendered by the relationship between a square and a rectangle. The design logic is initially based on the creation of a rectangle, which is then divided internally into a square and a smaller rectangle. Each new rectangle is separated again in the same way, and the same relationship arises continuously but on smaller scales. Joining the corners of the squares with circular arcs then marks a curve called a “logarithmic spiral”. However, apart from the creation of the well-known rectangle, the golden ratio is also applied to other shapes such as the pentagon, the isosceles or right triangle, the star pentagon and the polyhedron.
The calculation of the golden ratio is based on the proportional ratio of 8:13 and the use of the number φ=1.618.
But how did the use of the letter φ as a symbol of the golden ratio come about?
The use of the letter φ as a symbol of the golden ratio was made by the mathematician Mark Barr. He proposed to use the initial letter of the name of the sculptor and architect Phidias (490–430 BC), as he is considered to have been one of the first to adopt this analogue relationship in his works.
The golden ratio in art
Indeed, in the Parthenon, designed by the architects Iktinos and Kallikratis, and whose sculptural decoration was designed by Phidias, the use of the number φ is impressive. In particular, the width of the pillar towards its length, the diameter of the columns towards the wheelbase (1,905m:4,296m), the height of the temple towards its width (13,72m:30,88=4:9) and the width of the main temple towards its length confirm the adoption of the golden rule. Moreover, the façade of the Parthenon was made using two large root side rectangles, five and four smaller, while the ratio of the length of the building to the height of the façade is φ, the golden ratio.
However, the first application of the golden ratio in architecture seems to date back to 3000 BC, since several scholars believe that the Egyptians applied it when building the great pyramids of Giza. The length of each side of the base is 756 feet, and the height is 481 feet. Thus, we can see that the ratio of base to height is 756/481 = 1.5717.
“There is no art without mathematics.”
This phrase belongs to Luca Paccioli, a collaborator of Leonardo da Vinci, who dealt in his work “Divine Proportion” with the application of the golden ratio in art.
The golden ratio is also reflected in the Mona Lisa – one of the most famous paintings – and Leonardo da Vinci’s Last Supper. The golden proportion has been used in the lines of the Mona Lisa’s face, in the piece that starts from the neck to the beginning of the hands and from the neckline to the lower hands, while in The Last Supper, the dimensions of the room, the table and even the decorations were also
designed with it in mind.
Similarly, Salvador Dalí’s Sacrament of the Last Supper is painted inside a golden rectangle, within which a huge dodecahedron is formed, with the ends appearing in golden proportion to each other, above and behind Jesus.
1. Hemenway, Priya (2005). Divine Proportion: Phi In Art, Nature, and Science. New York: Sterling. Sna. 20–21. ISBN 1-4027-3522-7.
2. Patrice Foutakis, Did the Greeks Build According to the Golden Ratio?, Cambridge Archaeological Journal, vol. 24, n° 1, February 2014, p. 71-86.
3. Eli Maor (2000). Trigonometric Delights, Princeton Univ. Press.
4. Akhtaruzzaman, M., & Shafie, A. A. (2011). Geometrical substantiation of Phi, the golden ratio and the baroque of nature, architecture, design and engineering. International Journal of Arts, 1(1), 1-22.
1. Golden ratio representation, Source: https://pixabay.com/vectors/fibonacci-spiral-golden-ratio-7225635/
2. Algebraic representation of the golden ratio, Source: https://www.mathsisfun.com/numbers/golden-ratio.html
3. Athens, Parthenon, Source: https://gr.pinterest.com/pin/302374562453891702/?lp=true
4. The Great Pyramid of Giza, Source: https://illustrarch.com/articles/13472-golden-ratio-in-architecture.html
5. Da Vinci, The Last Supper, Source: https://www.goldennumber.net/art-composition-design/
6. Salvador Dali, The Sacrament of the Last Supper, Source: http://dummyhum.blogspot.com/2016/02/the-sacrament-of-last-supper-by.html