### There’s the golden rule, the golden mile, and of course the ubiquitous golden arch. For artists, though, there is the golden rectangle, also known as the golden or divine mean, ratio, or proportion. A GOLDEN IDEA The golden proportion has long served as a foundation for pictorial composition. No one is quite certain when it was discovered, but there is evidence of its use going back at least 5,000 years. Humans, throughout their history, seem to have gravitated toward rectangles that conform to the golden proportion. We like them and find them comfortable. The golden proportion is often described thus: “The smaller is to the larger as the larger is to the whole.” In simpler terms, it means that if you have a length of rope and cut it into two sections, the proportion of the shorter section to the longer piece is the same as the proportion of the longer segment to the length of the two sections combined. For example, if the short piece is 1" long and the long segment is 1.618", the ratio is 1.0 to 1.618. Should you mend the rope, you would learn that the combined length of both ropes is about 2.618", or 1.618 times the length of the long segment. By now it should be apparent that the golden ratio is 1:1.618 in numerical terms. The golden proportion is an established numerical ratio of 1.0 to 1.618 (also 0.618 to 1.0). Thus, if the shorter of two ropes was 1" long (A), the longer piece would be 1.618" long (B) as indicated by the figures at the top of the illustration. Further, if the two pieces of rope were tied together, their combined length would be 1.618 times that of the longer section. In the illustration, the longer section measures 1.618" (B). When multiplied by the golden proportion of 1.618, the result is 2.618" (C), the combined length of the two pieces of rope joined. The golden proportion has some interesting properties that have fascinated artists and geometers for millennia. Geometers are taken by the mathematical beauty inherent in the golden mean, and artists see in it an aesthetic purity. One of the most engaging qualities of the golden rectangle is its potential for the infinite. If you begin with a single small golden rectangle, and attach to it a square with its sides equal in length to the long side of the rectangle, you get a new rectangle with the same proportions as the first, albeit larger. You can then build another square onto this to arrive at a third and yet larger golden rectangle. It is possible to continue adding squares to make ever larger golden rectangles without end, theoretically expanding beyond the limits of the universe. Each of the shapes above is a golden rectangle. All have the same proportions, with one side being 1.618 times as long as the other. Golden rectangle can be built upon golden rectangle to potentially infinite extent. Not only can a golden rectangle be erected that extends past the boundary of the universe, it has also been found that many naturally occurring structures within the universe conform to the golden rectangle’s proportions. This includes objects as immense as our galaxy, and as small as a human eye. Due to the potentially infinite nature of the golden mean, and the fact of nature often coinciding with the golden mean, ancient geometers have ascribed mystical qualities to it and artists have come to see the golden rectangle as an ideal construct for their creations. Examples of nature coinciding with the golden proportion can be found virtually everywhere, such as (from left to right) spiral galaxies like our Milky Way, the human eye, a nautilus shell, the hand, and the wing of an eagle. Megalithic architects frequently designed their structures around the golden proportion, either by intent or through accident. The layout of the first stage of Stonehenge, erected about 4000 years ago, conforms exactly to the golden proportion. The central circle of uprights has a diameter that is 0.618 times the length of each of the rectangles. The rectangles themselves overlap one another. A boulder stands at each of the outer corners of combined rectangles. One of the oldest such monuments is Stonehenge in England. It was constructed in stages over a period of hundreds of years, between roughly 2000 and 1500 B.C. Archaeo-astronomer Gerald S. Hawkins has noted “that the first and third stages of the monuments’ (sic) design are laid out in strict accordance with the golden mean.” [Curtis, 99] Researchers have also seen a correlation between the golden proportion and the slopping sides of the Great Pyramid to its base. Built 4600 years ago, it is even older than Stonehenge. The dimensions of the Great Pyramid are highly suggestive of the golden proportion. The base of the pyramid measures 755.8 cubits, half that being 377.9 cubits. When divided by the length of one of its inclined sides, which are 612 cubits, the result is 1.619, nearly the same as the divine ratio. Ictinus and Callicrates, designers of the Parthenon at Athens, were clearly affected by the geometry set down by Pythagoras in the 6th century B.C. Among his many impressive discoveries, Pythagoras expounded upon the primacy of the golden mean. The proportions of the golden rectangle are repeated throughout the mid-5th century Greek temple. The golden rectangle is repeated over and over throughout the design of the Parthenon, which dates from the first century A.D. Romans carried on with the Greek idea of designing buildings around the framework of the golden mean. Virtruvius, the first century A.D. architect, wrote about the golden proportion and its mystical implications in his Ten Books on Architecture. We can see the manifestation of his theories in such structures as the Arch of Constantine in Rome, built around 325 A.D. The Arch of Constantine was erected in Rome about 325 A.D. Proportions of its major elements are clearly those of the golden rectangle, favored by Roman architects. Interiors of places of worship during the Middle Ages were often laid out according to the golden mean. Among them, the 6th century Hagia Sophia in Istanbul and the 12th century St. Mark’s Cathedral in Venice are notable. The interior of the 6th century Hagia Sophia (left) in Istanbul has as its design framework a series of golden rectangles. The same is true of the Venetian Cathedral of St. Marks (right), completed during the 12th century. Careful examination of the illustrations will reveal many more golden rectangles than have been highlighted. Painters of the Renaissance, who were interested in classical ideals, used the golden rectangle to devise many of their compositions. Leonardo da Vinci’s Vitruvian Man is one such composition. Completed around 1490, it was intended as an illustration for Fra Luca Pacioli’s Sacra Proportione. Among the proportions that Pacioli wrote about was the golden rectangle, which Leonardo used in plotting out his Vitruvian Man. Leonardo da Vinci’s Vitruvian Man is a study in proportions based in part on the golden ratio. Michelangelo, Leonardo’s rival, places the divide between the smaller and larger sectors of a golden mean at the divine intersection of the Creation of Adam, the moment when Adam’s fingertip meets that of his life-giving god. In his La Grande Odalesque, the 19th century neoclassical artist Jean Auguste Dominique Ingres used a similar method for establishing the placement of the woman’s hand. Both Michelangelo (left) and Ingres (right) used the golden mean to establish the locations of key elements in their compositions. Salvador Dali’s 1956 Quick Still-life is virtually a treatise on the golden ratio. Only major golden rectangles have been diagrammed in the reproduction, but the painting contains many others as well. While the picture as a whole has been designed around the relationships amongst the several rectangles, the artist didn’t stop there. Dali has treated each section of each rectangle as an individual picture, giving careful thought to their compositions. The purple rectangle at left, for instance, is bisected by one of the large white rectangles. The result is that the purple rectangle has been reduced to four sections. In the upper right segment we see a bird placed so as to create a dynamic and interesting interplay between positive and negative shapes. Beneath it the placement of bottle and stream of water form an organic diagonal that effectively unifies the space and activates all its parts. You will see similar phenomena occurring with the yellow outlined rectangle in the upper right portion of the painting. As you let your eyes wander throughout the picture, examining each golden rectangle and its sub-sections, you will discover the same sort of well considered treatment everywhere, along with numerous mini-paintings. Quick Still-life, a 1956 painting by Salvador Dali is a study of golden rectangles. Artists have long relied on constructs like the golden rectangle to serve as skeletons around which to fashion their pictures. The golden rectangle has particular appeal due to its pleasing proportions, its history, and its array of symbolic associations. It also has the benefit of breaking up the canvas into unequal but complementary parts. The astute artist pays close attention to the interplay of these shapes and frequently, as is demonstrated by the Dali painting, considers the compositions of each of the many subdivisions which the picture as a whole has been reduced. Art material manufacturers are well aware of artists’ affinity for the golden proportion, and produce stock frames, canvases, and papers accordingly. The golden ratio of 1 to 1.618 is very close to that of 5 to 8, a proportion that artists are well familiar with: 5" X 8", 10" X 16", 16" X 20", and so on are common frame and canvas sizes. Some artists oppose using schemes like the golden rectangle to form their compositions, believing that they are mechanical and lacking in creative innovation. Most, however, find such pre-determined formats liberating. With basic compositional issues already resolved, devices like the golden rectangle free the artist to invest his or her energies and time in developing original ideas and compelling techniques. For them the golden rectangle is indeed golden, its radiance illuminating a pathway toward sound pictorial structures. BIBLIOGRAPHYDrawing From Observation, Brian Curtis, New York, McGraw-Hill 2002

... golden proportion can be found virtually everywhere, such as (from left to right) spiral galaxies like our Milky Way, the human eye, a nautilus shell, ...