The Golden Ratio
Body, art, music, architecture, nature – all connected by a simple irrational number – the Golden Ratio. According to Posamentier & Lehmann in their work The (Fabulous) Fibonacci Numbers, there is reason to believe that the letter φ (phi) was used because it is the first letter of the name of the celebrated Greek sculptor Phidias (490-430 BCE). He produced the famous statue of Zeus in the Temple of Olympia and supervised the construction of the Parthenon in Athens Greece (Posamentier & Lehmann, 2007). In constructing this masterpiece building, Phidias used the Golden Ratio to create a masterpiece of work. Figure 1: This is a model of Zeus in the Temple of Olympia. The red lines show the use of the Golden Ratio. (www.scarletcanvas.com/) Phidias brought about the beginning of the one of the most universally recognized form of proportion and style used throughout history (Posamentier & Lehmann, 2007). The irrational number Phi, also known as the Golden Ratio, has had tremendous importance. To properly understand this mathematical concept, it is important to explore the definition, history, and the relations to architecture, art, music and the Fibonacci sequence. Figure 2: This model shows the line segments in the Golden Ratio. (Wikipedia.org)
As is with any new topic, it is essential to have a foundational understanding. According to Wikipedia, the definition of the Golden Ratio: In mathematics and the arts, two quantities are in the Golden Ratio if the ratio of the sum of the quantities to the larger quantity is equal to the ratio of the larger quantity to the smaller one. The figure on the left illustrates the geometric relationship (Wikipedia.org). In a documentary from the Discovery Channel titled, “Assignment Discovery: The Golden Ratio”, the narrator explains the Golden Ratio as; “the ratio of the whole segment to the larger piece is equal to the ratio of the larger piece to the smaller piece (see figure 2)”(DiscoveryFigure 3: This models the Golden Ratio in equation form (Wikipedia.org).
Multiple definitions and diagrams bring great clarity to this magnificent mathematical concept, as well as the different names it has been called. In the 16th Century, Luca Pacioli – geometer and friend of the great Renaissance painters rediscovered the “golden secret” (Jovanovic, 2003). Fra Luca Bartholomeo de Pacioli defines the golden ratio as the "divine proportion" in his published book Divina Proportione in 1509 (Pickover, 2012). Though the name Golden Ratio has collected many names, they all refer to phi (φ). Figure 4: Golden Ratio Model of phi. (Wolframalpha.com).
As noted previously, Phidias used this ratio to construct the Temple at Olympus Posamentier & Lehmann, 2007, p.110-1). This was just the beginning. Euclid (325-265 B.C.) in his work Elements, “Euclid ca. 300 BC gave an equivalent definition of by defining it in terms of the so-called ‘extreme and mean ratios’ on a line segment” (Wolfram Alpha, 2012). Figure 4 shows the relation of the sides to the whole of the golden ratio. The point C is called a golden cut of the line segment AB (Kalajdzievski, 2008). This cut creates the ratio of the sides to one another. The Golden Ratio can also be illustrated with the golden rectangle. The whole rectangle’s area is equal to the length multiplied by width. This rectangle has side length 1 and φ. The following steps will show the relationship between the sides refer to Figure 5. 1-φ is to φ as φ is to 1
This is solved by multiplying both sides by φ, to get
The Quadratic Formula applies here with a=1, b=1, c=-1, and yields the answer
Now that there has been a foundational basis to this topic, let us shift gears and look at the deep mathematical concepts that make this ratio “golden”. As stated before, the Golden Figure 6: This model represents the golden rectangle with side length 1. (Wikipedia.org )...
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