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Angela Perkins History of Mathematics Essay 1: the Golden Ratio In Angela Perkins History of Mathematics Essay 1: The Golden Ratio in Ancient Greece The Golden Ratio, though not named this until the nineteenth century, has been used and studied for thousands of years. Though the exact origin of discovery is not known, the Golden Ratio is thought to be discovered in ancient Greece. Some researcher have said that the Pythagoreans, particularly Hippasus of Metapontum, discovered the Golden Ratio. The first known written definition of the Golden Ratio was by Euclid, a Greek philosopher and mathematician that lived around 300 B.C, in his collection of mathematic works called Elements. Another prominent Greek philosopher that also was known to have an interest in the Golden Ratio was Plato. Though Plato didn’t make much mathematical contributions like Euclid, his interest influenced many important people. The Golden Ratio goes by many names, including; Golden Number, Golden Ratio, Golden Section, Divine Proportion, phi (φ), and extreme and mean ratio (in Elements). The definition that Euclid gives in Elements is, “A straight line is said to have been cut in extreme and mean ratio when, as the whole line is to the greater segment, so is the greater to the lesser.” Though the words might be hard to follow on first read, the concept is fairly simple. In Figure 1, the line segment A is to B as B is to C. The Golden Ratio is (1 + √5)/2 and can be reduced to 1.61803398… It is irrational because it never ends. Figure 1. Golden Ratio as a proportion of a line Though the concept of irrational numbers is known and accepted readily today, irrational numbers were not known about or accepted in ancient times. Whole numbers and fractions of whole number were, for a long time, the only known numbers to exist. Hippasus of Metapontum, who was a follower of Pythagoras in fifth century B.C. Greece, discovered irrational numbers. The Pythagoreans strongly believed that the universe was ruled and could be understood by whole numbers and fractions of whole numbers. Hippasus’s discovery made them question one of their most fundamental and important beliefs. Because this discovery discredited their beliefs, it was kept a secret. The Pythagoreans were interested in pentagrams and pentagons. The pentagram was the symbol of their brotherhood. As shown in Figure 2, a pentagon is formed at the center of the pentagram. A smaller pentagram can be formed inside this pentagon. This pentagram forms an even smaller pentagon, and so on and so on. Each pentagon is exactly to the previous, by the Golden Ratio. The pentagrams also follow this trend. Because of the Pythagoreans’ interest in pentagrams and pentagons, some researchers believe that they discovered the Golden Ratio. Particularly, it is thought that the Pythagorean that discovered the Golden ratio was Hippasus. Figure 2. Golden Ratio and the pentagram Another Greek philosopher that is known to be connected to the Golden Ratio is Plato (427 B.C. - 347 B.C.). Plato is considered one of the greatest minds of ancient Greece. Though he studied mathematics with Theodorus of Cyrene, he did not contribute greatly to mathematics directly. His influence of other mathematicians is what was important. Some of the best mathematicians of ancient Greece studied at his institution called the Academy. In Plato’s Timaeus, he describes the Golden Ratio by saying, ”For whenever in any three numbers, whether cube or square, there is a mean, which is to the last term what the first term is to it; and again, when the mean is to the first term as the last term is to the mean-then the mean becoming first and last, and the first and last both becoming means, they will all of them of necessity come to be the same, and having become the same with one another will be all one.”. Plato looked at mathematics as something divine and beautiful. Plato attempted to explain the cosmos and also the structure of matter. He used five regular solids. These five solids have been studied previously by the Pythagoreans and by Theaetetus. These solids are called the five Platonic solids. They are the tetrahedron, cube, octahedron, dodecahedron, and the icosahedron, as shown in Figure 3. His theory was that the solids represented different fundamental particles. The Platonic solids have a connection with the Golden Ratio, in particular the dodecahedron and icosahedron. The total surface area of the dodecahedron, with an edge length of 1, is 15φ/√3 − 휑 and the volume is represented by 5φ3/(6-2φ). The volume of the icosahedron, with a side length of 1 is 5φ5/6. (φ representing the Golden Ratio). Also, another connection between Platonic solids and the Golden Ratio is it appears if an icosahedron is mapped into a dodecahedron. The ratio of the solid’s edge length can be expressed as φ2/√5. The math of the solids were not known to this extent to Plato and his followers, but they still appreciated the beauty that they held. Figure 3. Platonic Solids It is believed that Euclid was educated by Plato’s pupils. He wrote many books about things such as music and optics. Euclid also wrote one of the best known books about mathematics in history called Elements. Elements is a collection of thirteen volumes about geometry and number theory. Elements was used for thousands of years for the study of mathematics. There are multiple locations in this work that the Golden Ratio is mentioned. In Elements, the Golden Ratio is referred to as extreme and mean ratio. Book II gives the definition of the Golden Ratio in relation to areas. Book VI provides another definition related to proportion as shown in Figure 1. In Book IV, the Golden Ratio was used to construct a pentagon, and in Book XIII, to construct icosahedrons and dodecahedrons. The Golden Ratio has been of human interest for thousands of years. The mathematicians and philosophers of ancient Greece have been recorded of having knowledge and studying it. To Hippasus of Metapontum and the Pythagoreans, the Golden Ratio, being an irrational number, was detrimental to their fundamental beliefs of the universe. For Plato, it was about geometry, beauty, and seeking understanding of Platonic solids and the cosmos. Finally, a recorded definition and explanation was given by Euclid in Elements. Though the history of the Golden Ratio spans much father in time than this, these three people (and groups of people) were an important beginning. Works Cited Dunlap, Richard, A. The Golden Ratio and Fibonacci Numbers. World Scientific, 1997. Euclid, Heiberg, J L, and Richard Fitzpatrick. Euclid's Elements of Geometry: The Greek Text of J.l. Heiberg (1883-1885) : from Euclidis Elementa, Edidit Et Latine Interpretatus Est I.l. Heiberg, in Aedibus B.g. Teubneri, 1883-1885, 2008. Huntley, H.E. The Divine Proportion. Dover Publications, 1970 Livio, Mario. The Golden Ratio. Broadway Books, 2002. Plato, , and Benjamin Jowett. Timaeus. Champaign, Ill: Project Gutenberg, 1990. Internet resource. Walser, Hans. The Golden Section. The mathematical Association of America, 1996. Figures: Figure 1. “Golden Ratio as a proportion of a line.”. http://blog.minitab.com/blog/statistics-and- quality-data-analysis/six-sigma-sweet-spot-the-search-for-a-divine-proportion Figure 2. “Golden Ratio and the pentagram”. http://www.freemasonry.bcy.ca/symbolism/golden_ratio/index.html Figure 3. “Platonic Solids”. http://www.maths.gla.ac.uk/~ajb/3H-WP/Platonic%20solids.html .
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