4.3 THE SEXIEST RECTANGLE FindingSome Scenarios Aesthetics Involving in Life, ChanceArt, and Math Th atrough Confound the Golden Our Intuition Rectangle Geometry has two great trea- sures: one is the theorem of Pythagoras; the other, the division of a line into extreme and mean ratio. Th e fi rst we may compare to a measure of gold; the second we may name a precious jewel. JOHANNES KEPLER Are you into it? On our journeys through various mathematical landscapes we have become conscious of the issue of aesthetics—in particular, the intrinsic beauty of mathematical truths. We’re discovering that mathematics is not just a collection of formulas tied together by algebra but is instead a wealth of creative ideas that allows us to investigate, explore, and dis- cover new realms. Now, however, we wonder if mathematics can be used to discover structure behind the aesthetics of art and nature. 258 Geometric Gems c04.indd Sec2:258 10/6/09 12:28:03 PM Rectangular Appeal In our discussion of Fibonacci numbers we asked the following geometri- cal question that begs to be asked again: What are the dimensions of the most attractive rectangle—the rectangle we might imagine when we close our eyes on a dark starry night and dream of the ideal rectangle? When someone says rectangle, we think of a shape. What shape is it? From the rectangles given here, choose the one you fi nd most appealing: Given these choices, a high percentage of people think that the second rectangle from the left is the most aesthetically pleasing—the one that captures the true spirit of rectangleness. That rectangle is referred to as the Golden Rectangle. It is the length of the base relative to the length of the height that makes it a Golden Rectangle. What precisely is the ratio of base to height that produces the Golden Rectangle? Recall that, in our conversations about numbers, we found a ratio that was an especially attractive number. The ratio arose in our dis- cussions of the Fibonacci numbers, and we denoted it by the Greek letter phi, ϕ. It was called the Golden Ratio because it satisfi ed the symmetrical equation of ratios: ϕ 1 ϭ . 1 ϕ Ϫ 1 Specifi cally, we found that the Golden Ratio, ϕ, is the number (1+ 52 )/ = 1.618 ... You may want to glance back at the Fibonacci discussion in Section 2.2 and revisit the relationship ϕ/1 ϭ 1/(ϕ Ϫ 1). (The Greek letter ϕ used to denote the Golden Ratio was introduced in the past century to honor the famous ancient Greek sculptor Phidias, much of whose work appears to involve the Golden Ratio.) The Golden Ratio gives us the satisfying relationship of height to width for those rectangles that many deem extremely pleasing to the eye. The precise mathematical defi nition of a Golden Rectangle is any rectangle having base b and height h such that b 15+ ==ϕ . h 2 4.3 / Th e Sexiest Rectangle 259 c04.indd Sec3:259 10/6/09 12:28:09 PM We have already discovered how the Fibonacci numbers and the Golden Ratio appear in nature’s spirals. Do the proportions of the Golden Ratio make the Golden Rectangle especially attractive and, if so, why? These questions have given rise to heated debate and much controversy. In 1876, Gustav Fechner, a German psychologist and physi- cist, conducted a study of people’s taste in rectangles—a taste test—and found that 35% of the people surveyed selected the Golden Rectangle. So, although the Golden Rectangle seems likely to win an election, we would not expect the outcome to be a landslide. The Golden Rectangle in Greece The Greeks appear to have been captivated by the proportions of the Golden Rectangle as evidenced by its frequent occurrence in their architecture and art. As a classic illustration, consider the magnifi cent Parthenon in Athens, built in the 5th century bce. The Parthenon today is pretty run-down—in fact, it’s in ruins. How- ever, perhaps you’re a step ahead of us, guessing that the big rect- angle contained in the Parthenon is a Golden Rectangle. Actually, if we measure the sides and do the division, we will see that the rect- angle is not a Golden Rectangle! So what’s the point? Well, when the Parthenon was built, it was much fancier—in particular, it had a roof. Imagine now that the roof is in place. If we form the rectangle from the tip of the rooftop to the steps, we will see a nearly perfect Golden Rectangle. Another example of the Golden Rectangle in Greek sculpture is the Grecian eye cup. The one pictured is inscribed inside a perfect Golden Rectangle. 260 Geometric Gems c04.indd Sec3:260 10/6/09 12:28:13 PM 1 1 _1 _______1 5 1_ 2 2 2 (Bohams, London, UK/The Bridgeman Art Library International) It remains an unanswered question whether Greek artists and design- ers intentionally used the Golden Rectangle in their work or chose those dimensions solely based on aesthetic tastes. In fact, we are not even certain that such artists were consciously aware of the Golden Rectangle. Although we will likely never know the truth, it is romantic to hypoth- esize that the Greeks were not conscious of the Golden Rectangle, because this then shows how aesthetically appealing its dimensions are and that we are naturally attracted to such shapes. Some people, however, believe that the occurrence of Golden Rectangle proportions is simply coincidental and random. While some believe that ancient Greek works defi nitely contain Golden Rectangles, others believe that it is nearly impossible to measure such works or ruins accurately; thus, there is plenty of room for error. In the preceding pictures, all the superimposed rectangles are perfect Golden Rectangles. Was their presence random or deliberate? Are Golden Rectangles really there? What do you think? The Golden Rectangle in the Renaissance It appears that mathematicians in the Middle Ages and the Renaissance were fascinated by the Golden Rectangle, but there is much question as to whether this enthusiasm was shared by artists of the time. Leonardo da Vinci was a math enthusiast, but did he know about the Golden Rect- angle? Did he deliberately use it in his work? While historians debate such issues, let’s take a look at Leonardo’s unfi nished portrait of St. Jerome from 1483. In the reproduction on page 262, we have superim- posed a perfect Golden Rectangle around the great scholar’s body. Intentional or otherwise, Leonardo selected proportions that were aes- thetically appealing, and such dimensions resemble those of the Golden Rectangle. Although we are not certain whether Leonardo intentionally used the Golden Rectangle, we do know that 26 years later he was aware of its existence. In 1509, Leonardo was the illustrator for Luca Pacioli’s text on the Golden Ratio titled De Divina Proportione. It was famous mainly for the reproductions of 60 geometrical drawings illustrating the Golden Ratio. 4.3 / Th e Sexiest Rectangle 261 c04.indd Sec3:261 10/6/09 12:28:17 PM Leonardo da Vinci’s illustration for Luca Pacioli’s De Divina Proportione (The Vitruvian Man, 1492, Accademia, Venice, Italy. Scala/Art Resource, NY) St. Jerome by Leonardo da Vinci (1480, Pinacoteca, Vatican Museums, Vatican The Divine Proportion is a synonym for the Golden Ratio. In fact, State, Scala/Art Resource, NY) many people, including Johannes Kepler, referred to the Golden Ratio as the Divine Proportion, or as the Mean and Extreme Ratio. Sometimes imaginations ran a bit too wild. Pacioli claimed that one’s belly button divides one’s body into the Divine Proportion. If you’re not ticklish, you can easily check that this is not necessarily true. Note the Fibonacci-like pattern in Le Corbusier’s 1946 Modulor Proportional System: 6 ϩ 9 ϭ 15, 9 ϩ 15 ϭ 24, and so on. [Le Corbusier Modular Man. © 2004 Artists Rights Society (ARS), New York.] 262 Geometric Gems c04.indd Sec3:262 10/6/09 12:28:21 PM The Golden Rectangle and Impressionism Let’s now leap ahead about 300 years to the creative age of French Impressionism. Painter Georges Seurat was captivated by the aesthetic appeal of the Golden Ratio and the Golden Rectangle. In his painting La Parade from 1888, he carefully planted numerous occurrences of the Golden Ratio through the positions of the people and the delineation of the colors. The use of the Golden Ratio in works of art is now known as the technique of dynamic symmetry. GHI B C F J E K A D Seurat’s La Parade (1888) (The Metropolitan ABCD, FGHJ, EBIK are all golden rectangles; we also GE EA Museum of Art) note that ϭϭϕ. EA FE The Golden Rectangle in the 20th Century In the 20th century, artists were still fascinated with the beautiful propor- tions of the Golden Rectangle. French architect Le Corbusier believed that people are comforted by mathematics. In this spirit, he deliberately designed this villa (below right) to conform with the Golden Rectangle. Le Corbusier, Villa (© 2009 Artists Rights Society, New York) Le Corbusier was one of the architects involved in the design of the United Nations Headquarters in New York City. Here we again see the infl uence of the Golden Rect- angle in this monolithic structure (right). Finally, we note that the Golden Rect- angle appears often in other art forms, United Nations including musical works. As an illustration, 4.3 / Th e Sexiest Rectangle 263 c04.indd Sec3:263 10/6/09 12:28:27 PM consider the work of French composer Claude Debussy.