Judge The Golden Ratio by its cover. With this book, you will have done so rightly, for the book’s interior perfectly delivers what the stunning exterior promises: extraordinarily beautiful illustrations, interesting analysis, and entertaining instruction. Six chapters contain clear and entertaining explanation for why the Golden Ratio has captured the attention of philosophers, scientists, and artists for thousands of years.
The first chapter, “Golden Geometry,” takes the reader from the discoveries of the properties of the Golden Ratio by ancient Greece’s Pythagoras and Euclid to the work of German mathematician Johannes Kepler. Meisner thoroughly covers Kepler’s work identifying the properties of the unique triangle which was later named for him, and the laws of harmonious planetary spheres, depicted in models of the universe as a series of nesting Platonic solids. Meisner ends the chapter with precise descriptions of the properties of Platonic solid shapes in Penrose tiles accompanied by attractive two-dimensional figures.
In “Phi and Fibonacci,” the second chapter, Meisner introduces Leonardo Pisano (whom he calls Leonardo Fibonacci) and explains how he was influenced by scholars of the Islamic Golden Age, such as Muhammad ibn Musa al-Khwarizmi and Abu Kamil Shuja ibn Aslam. With gorgeous full-page color images, Meisner educates the reader on the magnificent contributions of Fibonacci through his book Liber Abaci and the Fibonacci Sequence. Blaise Pascal’s work is shared with the reader, specifically his triangular algebraic expansion of binomial coefficients (p 44). The chapter includes information about the Fibonacci spiral, volutes, and triangles and rectangles produced using numbers in the Fibonacci series. A very explicit instruction on how to calculate Phi ends the chapter.
“The Divine Proportion” (chapter 3) begins with an introduction to the Italian genius Luca Pacioli followed by an examination of how the Golden Ratio has been applied throughout history by countless artists, architects, and engineers. Dozens of historical references, anecdotes, and literary sketches (as well as exquisite images) offer verifiable proof of mathematics’ importance in the conscious creation of things with enduring aesthetic quality. Meisner includes exceptionally detailed discussion of individual works of many famous artists, such as Botticelli, Raphael, and Michelangelo, and showcases Divine Proportion overlays upon astounding photos of dozens of iconic works to demonstrate assertions that artists purposefully used this beautiful math to enhance their projects. For fans of Leonardo da Vinci, there are nine full pages showing and explaining with interesting observation how his most beloved masterpieces exhibit elements of the Divine Proportion.
“Golden Architecture & Design,” the fourth chapter, shows us how everything “can be described mathematically and geometrically” and many things can be described using Phi as well as Pi. Meisner devotes eleven pages to analyzing how the pyramids of Giza and the Great Sphinx embody the geometry of the Golden Ratio. The dimensions of the Parthenon, several cathedrals, and the Taj Mahal are also examined thoroughly. The chapter includes exploration of the works of Seurat and Le Corbusier, and ends with a look at the Rule of Thirds and applications of the Golden Ratio in fashion, logo, and product designs.
In the fifth chapter, “The Golden Life,” Meisner ponders the possible significance to biology, chemistry, and physiology of this mathematic proportion that appears ubiquitous throughout the world in Nature, our living bodies, and perhaps even the universe. He covers the history and relevance of phyllotaxis, five-fold symmetry, and equiangular spirals in the Plant and Animal Kingdoms; he also discusses theoretical applications of growth structures using golden shapes, such as in the modern-day creation of fractals.
The section covering spirals in the Animal Kingdom is absolutely delightful. When he interprets the evidence for Golden Human Proportions, he shows how the Golden Ratio is present even in our DNA and shares scientific corroboration for theories about the impact of Golden Ratio proportions on perceptions of human beauty.
I find it significant that the title of the last chapter is a question: “A Golden Universe?” returns to the cosmic relationships studied by Kepler, Plato, and others before touching upon research by modern scientists. The sizes and shapes of structures in our vast universe appear to be related; moreover, the pathways (orbits) of planets and moons, and the pulsations of stars, also suggest patterns of Golden numbers. The chapter journeys from observing vast space to microscopic molecular structures, contemplating how atoms, crystals, and carbon molecules reflect Golden Ratio symmetry. Meisner ends the chapter by explaining quantum frequencies. He then discusses the modern applications of Fibonacci numbers in gambling and investing, which follows phi-based patterns of economic cycles.
Over two hundred high-resolution color images illustrate the concepts, principles, and ideas presented in this exquisite 224 page book. Araujo’s astounding Leonardesque drawings are featured, but all of the other images in the book enhance the text, also. Numerous drawings, diagrams, and charts accompany Meisner’s clear explanations. Examples include numerous diagrams of Euclid’s propositions (pp 22-26) and the color-delineated lines transposed upon the image of a Lady Blunt Stradivarius to show how Antonio Stradivarius constructed his beautiful violins using the Golden Ratio (p 136). In short, every page is filled with literary gold and visual gems.
The Golden Ratio is perfect for educational purpose as well as for gift-giving and/or display. It is Meisner’s hope—and my expectation—that readers of this book will develop an admiration for the divine beauty of mathematics. Of Fibonacci books, this one is my favorite.
Gary B. Meisner’s lifelong love of math, science, and nature led him to pursue a career in finance and technology. After earning BS and MBA degrees from two top business schools, passing the CPA exam, and working in public accounting, he spent most of his career in CFO/CIO roles with operating units of five Fortune 1000 public companies. He is now an independent financial and business systems consultant, which allows more time to pursue his research, software development, and writing on the golden ratio.
Having created websites since 1997, Gary created “Phi 1.618 – The Golden Number” in 2001, a leading website dedicated to the mathematics, prevalence, and design applications of the golden ratio (www.goldennumber.net). In 2004, he developed PhiMatrix Golden Ratio Design and Analysis software, which is used by thousands of artists, architects, designers, and photographers in over seventy countries, as well as in cosmetic, medical, and stock market analysis applications. www.phimatrix.com
Rafael Araujo has been hand drawing intricate, architectural illustrations of nature for more than forty years. Rather than computer programs, Rafael uses pencil, ink, a compass, a ruler, and a protractor to portray the marvelous mathematical brilliance of the natural world. Each illustration takes him upwards of 100 hours, for the architect and illustrator from Caracas, Venezuela, meticulously applies the Golden Ratio to geometric formulas while constructing precise frameworks from which his three-dimensional creatures appear to fly.
Araujo’s work has been featured in publications, including Wired Magazine, and exhibited at Stanford University as well as many other prestigious galleries. www.rafael-araujo.com
Meisner, Gary B. The Golden Ratio: The Divine Beauty of Mathematics. Race Point Publishing, 2018.
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Paperback: 128 pages
Author: Shelley Allen, M.A.Ed.
Publisher: Fibonacci Inc.; 1st edition (2019)