5.8 The Fibonacci Sequence and the Golden Ratio 285 From the golden proportion, the golden rectangle can be formed, as shown in Fig. 5.13. C B a a a A b Figure 5.13 a b a a b Length Width 5 1 2 = + = = + Note that when a square is cut off one end of a golden rectangle, as in Fig. 5.13, the remaining rectangle has the same properties as the original golden rectangle (creating “like from like”) and is therefore itself a golden rectangle. Interestingly, the curve derived from a succession of diminishing golden rectangles, as shown in Fig. 5.14, is the same as the spiral curve of the chambered nautilus. The same curve appears on the horns of rams and some other animals. It is the same curve that is observed in the plant structures mentioned earlier: sunflowers, other flower heads, pinecones, and pineapples. The curve shown in Fig. 5.14 closely approximates what mathematicians call a logarithmic spiral. m The Parthenon Ancient Greek civilization used the golden rectangle in art and architecture. The main measurements of many buildings of antiquity, including the Parthenon in Athens, are governed by golden ratios and rectangles. It is for Phidias, considered the greatest of Greek sculptors, that the golden ratio was named “phi.” The proportions can be m Circus Sideshow (La Parade de Cirque), 1887, by Georges Seurat Figure 5.14 Learning Catalytics Keyword: Angel-SOM-5.8 (See Preface for additional details.) Olga Drabovich/Shutterstock Danny Iacob/Shutterstock AKG London/SuperStock Instructor Resources for Section 5.8 in MyLab Math • Objective-Level Videos 5.8 • PowerPoint Lecture Slides 5.8 • MyLab Exercises and Assignments 5.8 • Chapter 5 Projects
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