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Fibonacci sequence is a sequence type or a set of numbers where a number is determined by summing up the two previous numbers. It starts with zero or one, one as the following number, and then the following subsequent terms are determined by the summation of the two previous terms. The sequence is written as:

The mathematical rule or expression for this sequence is:

Where, A n is the n th term of the sequence. The seed values of these sequence are:

or

Therefore,

The golden ratio is the ratio whose value is approximately equal to 1.618. A ratio is said to be a golden ratio if the ratio of the sum of two quantities to the larger quantity is equal to the ratio of the larger quantity to the smaller quantity. The quadratic expression of the golden ratio is:

Given, quantities x and y and y > x, then the golden ratio can be presented as:

The golden ratio has a unique relationship the Fibonacci sequence such that the ratio of the summation of its two terms following each other to the larger term is approximately equal to the value of the golden ratio. Furthermore, the ratio of the two terms following each other, that is, the ratio of the larger value to the smaller value also is approximately equal to the golden ratio value. Mathematically, it can be expressed as the limit as n approaches infinity of the Fibonacci consecutive numbers ratio is equal to the golden ratio value.

Given the Fibonacci sequence of terms:

In other words, the ratio of the consecutive terms of Fibonacci converges at the golden ratio.

The mathematical concept was used by many artist and architectures to create or design their works. Sandro Botticelli, in his painting, The Birth of Venus, used the golden ratio in the navel of the middle lady’s height as well as the height of the painting itself. The canvas dimensions were 172.5 cm by 278.5 cm. The width to height ratio was approximately equal to the golden ratio value 1.618. The frames of the painting, 171.5 cm by 277.5 cm, were also dimensioned by the concept of golden ratio. Red line shows a golden ratio from the lady’s hair top to her bottom lower foot. The green line represents the golden ratio from the lady’s hairline at her forehead top to her upper foot bottom. The blue line represents the golden ratio of the lady’s height from the feet middle to head top at her hair back part (Gary, 2014).

Figure 1: The Birth of Venus by Botticelli Sandro (Lauren, 2015)

In Da Vinci’s painting The Annunciation, the golden ratio proportion appears. He painted courtyards brick wall in golden ratio proportion to the painting’s top and bottom. Furthermore, the positioning of the emblems fine details on the table was based on the golden ratio of the table’s width (Gary, 2014).

Figure 2: The Annunciation by Leonardo Da Vinci (Gary, 2014)

Michelangelo Buonarroti painted The Creation of Adam, by the concept of golden ratio. Looking at the picture, God’s finger touches that of Adam’s at a precise golden ratio point of the height and width of their surrounding area (Gary, 2016).

Figure 3: The Creation of Adam by Michelangelo (Gary, 2016)

Application of Golden ratio and Fibonacci in nature, art, and music

Music

Design of violins. Fibonacci and golden ratio proportion is used in the design of violins. The horizontal length of the scroll tip to the bottom tip of the neck of the violin falls in the golden ratio of the lower bout tip to the upper bout tip. From the F-Holes to the lower tip of the fingerboard is also in the golden ratio of the lower tip of the fingerboard and the lower tip of the neck.

Figure 4: Violin design by the golden ratio (Gary, 2012)

Musical Scale. The music scale lies on the fundamentals of the Fibonacci sequence. The 5th note is the dominant note on the major scale that is also the 8th note in 13 notes in the octave. It forms Fibonacci sequence whose ratio gives the golden ratio 5/8 ≈ 8/13 ≈ 1.618. More interesting is that, a three-chord song in the A key is formed by A, E, and D, which are Fibonacci partners.

Figure 5: Fibonacci sequence and golden ratio in the piano board (Gary, 2012)

Art

Stained glass art made by the golden ratio concept. The below-stained glass was designed and made by the golden rectangle concept. The triangle ADE is a golden triangle. The ratio of either of the longest sides AE or AD to the shortest side ED gives the value of the golden ratio 1.618.

Figure 6: Application of golden triangle in the design of the tainted glass window (Veena, 2015)

Visually, the Eiffel Tower that was designed by a French engineer, Alexandre Gustave Eiffel, was by golden ratio (Go Geometry, 2014). The top part of the structure is about 2/3 while the bottom part is 1/3.

Figure 7: The Eiffel Tower in Paris (Go Geometry, 2014)

Nature

The American Giant Millipede. Naturally, the American millipede coils to form a spiral Fibonacci, which constitutes of golden triangles.

Figure 8: The American Giant Millipede (Keiren, 2016)

The golden triangle ABC is formed where AB or AC divided by BC gives the golden ratio 1.618.

The Aloe Plant. This plant naturally grows to assume the Fibonacci sequence and produce series of golden triangles whose ratio of the sides gives a golden ratio.

Figure 9: The Aloe Plant (Keiren, 2016)

Currently, Fibonacci and the golden ratio concept is used in the technical analysis and finance and design buildings. In financial analysis, it is used in the Fibonacci arcs to predict the stock exchange by the Meta Trader. The arcs are divided into three curved lines at 38.2 %, 50 %, and 61.8 % (Kuepper, 2018). In architectural design, the Fibonacci sequence and golden ratio are used to produce buildings with very high aesthetic values to the surrounding and, at the same time, meet the engineering needs.

Figure 10: Financial analysis by use of golden ratio concept (Kuepper, 2018)

Figure 11: Building designed by the concept of golden ratio. (MEDIATOP, 2018)

References

Gary, M. (2012, May 4). Music and the Fibonacci Sequence and Phi. Retrieved from Golden Number: https://www.goldennumber.net/music/

Gary, M. (2014, March 7). Botticelli, The Birth of Venus and the Golden Ratio in Art Composition. Retrieved from The Golden Number: https://www.goldennumber.net/botticelli-birth-venus-golden-ratio-art/

Geometry. (2014, July 22). The Golden Rectangle and the Eiffel Tower in Paris. Retrieved from Geometry.com: http://gogeometry.com/wonder_world/golden_rectangle_eiffel_tower_paris.html

Keiren. (2016). The Fibonacci Sequence in Nature. Retrieved from Insteading.com: https://insteading.com/blog/fibonacci-sequence-in-nature/

Kuepper, J. (2018, August 5). Fibonacci and the Golden Ratio. Retrieved from Investopidia: https://www.investopedia.com/articles/technical/04/033104.asp

Lauren, P. (2015, October 2). See How Artists Discover Simplicity as an Art Form in Works Which Reflect the Golden Ratio. Retrieved from Artnet News: https://news.artnet.com/art-world/golden-ratio-in-art-328435

MEDIATOP. (2018). Math meets Art: Fibonacci, The Golden Ratio, Video & Film. Retrieved from MediaTop: https://www.mediapop.ca/fibonacci-golden-ratio-video-film/

Sally. (2018, June 21). A Crash Course in the Golden Ratio. Retrieved from Already Pretty: https://www.alreadypretty.com/a-crash-course-in-the-golden-ratio/

Veena, R. (2015, November 20). Fibonacci Storybooks and Art projects for kids. Retrieved from Artsy Craftsy Mom: https://artsycraftsymom.com/fibonacci-storybooks-and-art-projects-for-kids/

September 18, 2023

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