The Fibonacci sequence has many applications in the real world. Not only are there applications in other mathematics, but different forms of the Fibonacci sequence and Golden Ratio can be found in nature, art and architecture, fashion, the human body, and so much more. The calculations behind how a golden section is formed has previously been discussed. A golden section is used to make several different things, including Golden Rectangles and Golden Triangles. The two unequal parts that form the golden section make up the side lengths of Golden Rectangles (Watson, p. 12). Another way to think about constructing a Golden Rectangle is by creating squares with side lengths equivalent to the Fibonacci numbers. For example, two squares with side lengths equal to one could be constructed where they share one side. Another square could be added to that with a side length of two and connect with the previous two squares where their total side lengths equals two. This pattern continues to create rectangles that can be considered Golden Rectangles.
Use the applet below to investigate the relationship between golden sections and Golden Rectangles.
Directions: Use the slider for b to change the size of the golden rectangle. Notice that the relationship given below the rectangle always holds true.
To view this applet in a different tab, please use the following link: Golden Rectangles Applet
There are several interesting applications of Golden Rectangles in art, architecture, and other forms of design. The most notable pieces of art are "Leonardo da Vinci's Last Supper, Mary Cassat's The Boating Party and Georges Seurat's Bathers at Asnieres are just a few paintings composed using golden rectangles" (Nature’s MATHterpiece). There are several building designs that also include the Golden Ratio and Golden Rectangles. For example, "the front side of Parthenon building in Greece can be easily framed with golden rectangles" (Omotehinwa & Ramon, p. 636). Another famous building that historians and mathematicians have discovered they used some form of the Golden Ratio to construct was the Great Pyramid of Giza (Omotehinwa & Ramon, p. 637).
Several aspects of fashion design and construction use the Golden Ratio and the Fibonacci numbers. "The design and pattern making of elements, which are created with applied directly or by geometrical figures the Golden ratio and Fibonacci numbers proportions, are: different types of seams in the bodice and sleeves; contours like necklines and armholes; 3D elements..." (Kazlacheva). It also mentions that if the Golden Ratio and Fibonacci numbers are not directly used in the design then they are used in the frames of the pieces (Kazlacheva). There are several examples shown of how different designs and parts of patterns use the Fibonacci numbers and Golden Ratio to create "beautiful and harmonic forms directly or with the help of geometrical figures" (Kazlacheva).
The Golden Spiral is another common form of seeing the Fibonacci sequence. It is constructed "By drawing arcs through opposite corners of connected golden rectangles..." (Nature’s MATHterpiece). The Golden Spiral can be found in many things in nature, including pinecones, pineapples, the layout of seeds in sunflowers, "the unfurling of a growing fern", etc. (Nature’s MATHterpiece). In pinecones, there are 8 spirals in the clockwise direction and 13 in the counterclockwise direction (Omotehinwa & Ramon, p. 632). Pineapples have three different spirals on their exterior, one with 5 scales, another with 8 scales, and the final one with 13 scales of the pineapple in it (Omotehinwa & Ramon, p. 632). Golden Spirals are found in the seed patterns in the heads of sunflowers with 21 spirals in the clockwise direction and 34 spirals in the counterclockwise direction (Watson, p. 16-17). Not only do these spirals show up in plants, but they also show up in other places in nature including "in Nautilus shells..., a spiral galaxy's arms, a hurricane, an ocean wave..." (Bortner & Peterson).
Use the applet below by clicking the play button to show how a Fibonacci Spiral can be created using squares with side lengths of the Fibonacci numbers.
To view this applet in a different tab, please use the following link: Fibonacci Spiral Applet
A final lesser known application of the Fibonacci sequence is in stock market analysis. "Many investors use what is called the Fibonacci Retracement Technique to estimate the action that the price of a particular stock will take, based on certain ratios found within the Fibonacci numbers" (Bortner & Peterson). This is an interesting and unexpected application of the Fibonacci numbers.
Thus, the Fibonacci sequence and Golden Ratio have a significant role in several diverse fields of study. Understanding the background and history of the Fibonacci sequence and Golden Ratio help clarify the mathematics used in finding different applications of these principles. Our walk through the forest ends with us finding that many things in nature and the world connect to the Golden Ratio and Fibonacci sequence.