By Francis D. Hauser Phd
Euclid’s masterpiece textbook, The Elements, used to be written twenty-three hundred years in the past. it really is essentially approximately geometry and includes dozens of figures. 5 of those are developed utilizing a line that “is reduce in severe and suggest ratio.” at the present time this can be referred to as the golden ratio and is usually observed by way of the logo Φ.
Many myths have grown up round this ratio. This publication was once written to benefit approximately them. They come up from the pyramids, the Pythagorean Brotherhood, the platonic solids, the Fibonacci numbers, sea shells, and others. there's a universal thread between those myths. Φ is an irrational quantity (a quantity whose digits after the decimal element move on ceaselessly and not shape a repeating pattern). Φ can be utilized to attract wonderful figures. yet its numerical worth can't be written down utilizing integers and fractions, that have been the single numbers utilized in Euclid’s time.
Mathematicians ahead of Euclid knew that irrational numbers existed. yet to many folks, a bunch that can’t be written down was once absurd. for hundreds of years, many scientists and engineers believed that Φ used to be godlike.
This booklet discusses the myths from an engineering standpoint. The final bankruptcy of the e-book indicates how Euclid dealt with irrational numbers; how Euclid did algebra utilizing geometry; and a straightforward visible evidence of why there are just 5 platonic solids.
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Euclid’s masterpiece textbook, the weather, used to be written twenty-three hundred years in the past. it really is essentially approximately geometry and comprises dozens of figures. 5 of those are developed utilizing a line that “is minimize in severe and suggest ratio. ” this present day this is often known as the golden ratio and is frequently pointed out by way of the emblem Φ.
Additional resources for The Golden Ratio: The Facts and the Myths
It is one of the five Platonic solids. • Regular Dodecahedron. This three-dimensional figure is bounded by twelve equal faces inscribed in a sphere. Each face is a regular pentagon, and Φ is used to build its internal scaffolding. It is one of the Platonic solids. Figure 2: Figures Constructed Using the Golden Ratio Euclid never used the name the golden ratio, and he never used its symbol or its value. The mythological name was not used until the Renaissance, and that was eighteen hundred years after Euclid.
Mineola, NY: Dover Publications, 1998. Huntley, H. E. The Divine Proportion. New York: Dover Publications, 1970. Livio, Mario. The Golden Ratio. New York: Broadway Books, 2002. O’Daffer, Phares, and S. Clemens. Geometry, an Investigative Approach. New York: Addison-Wesley, 1976. Olsen, Scott. The Golden Section, Nature’s Greatest Secret. New York: Walker & Company, 2006. Posamentier, Alfred, and I. Lehmann. The Glorious Golden Ratio. Amherst, NY: Prometheus Books, 2012. _____. The (Fabulous) Fibonacci Numbers.
Euclid himself says nothing about myths. They arose from very famous historical figures, in times when there were very close connections between religion, science, and math. There is a common thread among the myths. Φ is an irrational number. In Euclid’s time, mathematicians couldn’t write down its value using integers and fractions. They had to describe it using words and figures. To many people, the concept of an “unknowable” number was absurd. For centuries, even great scientists and engineers believed that Φ was godlike.