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From the Publisher
Introduction from Gary B. Meisner
What makes a single number so captivating that it has persisted in our imaginations for more than two thousand years? So universal that it is found in the writings of an ancient Greek mathematician, the musings of a revolutionary cosmological scientist, the designs of a twentieth-century architect, and the intrigue of a novel-turned-movie blockbuster? So pervasive that it appears simultaneously in the greatest architectural monument of the ancient world, the paintings of history’s most eminent Renaissance artist, and the atomic arrangement of recently discovered quasi-crystalline minerals? And so controversial that it engenders confusing and polarizing claims about its appearances and applications? The golden ratio is not a new topic. The answers to the question may surprise you.
This portion of Leonardo da Vinci's John the Baptist (c. 1516) shows some compelling proportions. Could da Vinci have been intentionally reflecting the golden ratio in this painting?
Contents from The Golden Ratio: The Divine Beauty of Mathematics
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Golden GeometryEuclid gave us a wonderful foundation for understanding the many appearances of the golden ratio in geometry. But we can make this even simpler. Let’s look at some of the other simple geometric constructions that can be used to create a golden ratio, starting with the line, and then proceeding to the three-sided triangle, four-sided square, and five-sided pentagon. Unlike David Letterman’s 'Top 10' Lists, I’m going to start with one that is perhaps the most amazing, by virtue of its sheer simplicity. If Euclid had seen this elegant little construction, history probably would have recorded him rather than Archimedes as the one running naked through the streets, shouting, 'Eureka!' The golden ratio cut of line AC is point B. |
Phi And FibonacciThe work of al-Khwarizmi, particularly his discussion of Hindu-Arabic numerals, later caught the attention of a young Italian boy during a visit to an Algerian port city with his father, a wealthy merchant from Pisa. The boy, Leonardo Fibonacci (c. 1175–c. 1250), would later become one of history’s most famous mathematicians after the publication in 1202 of his book Liber Abaci, which promoted the Hindu-Arabic numbering system throughout Europe. This page from Fibonacci’s revolutionary 1202 work Liber Abaci, which introduced Hindu-Arabic numerals to the West, shows the association between Roman numerals and different quantities. |
The Divine ProportionNow we will examine a variety of applications of the golden ratio in Renaissance art and beyond. In doing so, we step from the world of the absolute precision and verifiable proofs of mathematics and geometry into the more subjective world of beauty and aesthetics. Thus venturing into a domain where our heart tells us what logic cannot, we’ll also be stepping into a world of controversy, full of conflicting and polarizing claims that lead to much misinformation and confusion about the golden ratio. Did the Renaissance masters truly and intentionally incorporate the golden proportion into some of their most revered works? The Creation of Adam. |
Contents from The Golden Ratio: The Divine Beauty of Mathematics
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Golden Architecture & DesignIf the ancient Egyptians’ knowledge and application of the golden ratio were limited to this one exceptionally accurate appearance in the Great Pyramid, it could still be argued that it was due to chance. However, we now have additional evidence that suggests that the golden ratio also appears in the positions and relative sizes of the pyramids at the Giza site. These recent findings make for a much more compelling case. The larger rectangle has perfect golden proportions, whereas the second has proportions within 0.08 of phi. The Tedder grid shows the two golden rectangles formed in the distances between the apexes of Menkaure (left) and Khafre (center), and Khufu (right). Note: The top of the diagram faces west. |
Golden LifeSo, what about the human face? Do golden ratios exist there, too? The basic structure of all of our faces is fundamentally the same. There’s a wide variety within that basic human structure though, so how do we pick a face that would be representative of all humankind? These are visual representations of the mathematically averaged proportions of fifty male and fifty female faces, based on 189 facial markers, providing a very good statistically valid benchmark for assessing the appearance of the golden ratio in various features. These are visual representations of the mathematically averaged proportions of fifty female faces, based on 189 facial markers, providing a very good statistically valid benchmark for assessing the appearance of the golden ratio in various features. |
A Golden Universe?The frequent appearances of the golden ratio in living organisms is intriguing, but there are still other instances that are far more unexpected, even astounding. One that amazes me the most involves the relative sizes of the Earth and Moon. As we observed in chapter 4, the Kepler triangle represents the proportions of the Great Pyramid with a variance of less than 0.2 percent. This same triangle defines a rather amazing relationship between the radii of the Earth and Moon. The radii of the Earth and Moon. |
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