286 CHAPTER 5 Number Theory and the Real Number System found abundantly in his work. The proportions of the golden rectangle can be found in the work of many artists, from the old masters to the moderns. For example, the golden rectangle can be seen in the painting Circus Sideshow (La Parade de Cirque), 1887, by Georges Seurat, a French neoimpressionist artist. In addition to using the golden rectangle in art, several artists have used Fibonacci numbers in art. One contemporary example is the 1995 work by Caryl Bryer Fallert called Fibonacci’s Garden , which is shown in the margin. This artwork is a quilt constructed from two separate fabrics that are put together in a pattern based on the Fibonacci sequence. Fibonacci numbers are also found in another form of art, namely, music. Perhaps the most obvious link between Fibonacci numbers and music can be found on the piano keyboard. An octave (Fig. 5.15) on a keyboard has 13 keys: 8 white keys and 5 black keys (the 5 black keys are in one group of 2 and one group of 3). 5 black 18 white 13 total C D E F G A B C Figure 5.15 Did You Know? Fibonacci and the Male Bee’s Ancestors An excellent example of the Fibonacci sequence in nature comes from the breeding practices of bees. Female or worker bees are produced when the queen bee mates with a male bee. Male bees are produced from the queen’s unfertilized eggs. In essence, then, female bees have two parents, whereas male bees only have one parent. The family tree of a male bee would look like this: F M M M F F F F F M M F M F F M F F M F From this tree, we can see that the 1 male bee (circled) has 1 parent, 2 grandparents, 3 great-grandparents, 5 great-great-grandparents, 8 greatgreat-great-grandparents, and so on. We see the Fibonacci sequence as we move back through the male bees’ generations. In Western music, the most complete scale, the chromatic scale, consists of 13 notes (from C to the next higher C). Its predecessor, the diatonic scale, contains 8 notes (an octave). The diatonic scale was preceded by a 5-note pentatonic scale ( penta is Greek for “five”). Each number is a Fibonacci number. In popular music, the song “Lateralas” by the band Tool uses the Fibonacci sequence in both the time signature and the lyric arrangement. The song also contains several references to the golden ratio and to the logarithmic spiral. The visual arts deal with what is pleasing to the eye, and musical composition deals with what is pleasing to the ear. Whereas art achieves some of its goals by using division of planes and area, music achieves some of its goals by a similar division of time, using notes of various duration and spacing. The musical intervals considered by many to be the most pleasing to the ear are the major sixth and minor sixth. A major sixth, for example, consists of the note C, vibrating at about 264 vibrations per second, and note A, vibrating at about 440 vibrations per second. The ratio of 440 to 264 reduces to 5 to 3, or , 5 3 a ratio of two consecutive Fibonacci numbers. An example of a minor sixth is E (about 330 vibrations per second) and C (about 528 vibrations per second). The ratio 528 to 330 reduces to 8 to 5, or , 8 5 the next ratio of two consecutive Fibonacci numbers. The vibrations of any sixth interval reduce to a similar ratio. Patterns that can be expressed mathematically in terms of Fibonacci relationships have been found in Gregorian chants and works of many composers, including Bach, Beethoven, and Bartók. A number of twentieth-century musical works, including Ernst Krenek’s Fibonacci Mobile , were deliberately structured by using Fibonacci proportions. A number of studies have tried to explain why the Fibonacci sequence and the golden ratio are linked to so many real-life situations. It appears that the Fibonacci numbers are a part of a natural harmony that is pleasing to both the eye and the ear. In the nineteenth century, German physicist and psychologist Gustav Fechner tried to determine which dimensions were most pleasing to the eye. Fechner, along with psychologist Wilhelm Wundt, determined that most people do unconsciously favor golden dimensions when purchasing greeting cards, mirrors, and other rectangular objects. This discovery has been widely used by commercial manufacturers in their packaging and labeling designs, by retailers in their store displays, and in other areas of business and advertising. m Fibonacci’s Garden by Caryl Bryer Fallert Caryl Bryer Fallert-Gentry
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