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In nature's portfolio of architectural works, the magnificent shell of the chambered nautilus holds a place of special distinction. The spiral-shaped shell, with its revolving interior stairwell of ever-growing chambers, is more than a splendid piece of natural architecture – it is also a work of remarkable mathematical creativity. Humans have imitated nature's wondrous spiral designs in their own architecture for centuries, all the while trying to understand how the magic works. What are the physical laws that govern spiral growth in nature? Why do so many different and unusual mathematical concepts come into play? How do the mathematical concepts and physical laws mesh together? What is the source of the intrinsic beauty of nature's spirals? Trying to answer these questions is the goal of this chapter. DEF: Fibonacci numbers F1 = 1, F2 = 1, FN = FN-1 + FN-2, for any N > 2. Therefore, after the first two numbers (both of them being equal to 1), each subsequent number in the Fibonacci list is the sum of the two numbers before. 1, 1, 2, 3, 5, 8, 13, 21, 34, 55, 89, 144, 233, 377, 610, 987, … OBS: The list of Fibonacci numbers is an infinite sequence and is called Fibonacci sequence. The previous definition is a recursive one, i.e. each number is defined in terms of the other (earlier) numbers. Binet’s formula for Fibonacci numbers N N 1 5 1 5 2 2 , for any N a whole number. FN 5 OBS: In spite of its rather nasty appearance, this formula has one advantage over a recursive definition: it gives us an explicit rule for calculating any Fibonacci number without having to first calculate the preceding Fibonacci numbers. For this reason, Binet's formula is called an explicit definition of the Fibonacci numbers. Fibonacci Numbers in Nature: Our interest in Fibonacci numbers stems from their frequent occurrence in nature. Take flowers, for example. Consistently, the number of petals in a daisy is a Fibonacci number, which depends on the variety: 13 for Blue daisies; 21 for English daisies; 34 for Oxeye daisies; 55 for African daisies, and so on. What's true for daisies is also true for many other types of flowers (geraniums, chrysanthemums, lilies, etc.). Fibonacci numbers also appear consistently in conifers and seeds. The bracts in a pine cone spiral in two different directions in 8 and 13 rows; the scales in a pineapple spiral in three different directions in 8, 13, and 21 rows; the seeds in the center of a sunflower spiral in 55 and 89 rows. Exactly why and how this unusual connection between a purely mathematical concept (Fibonacci numbers) and natural objects (daisies, sunflowers, pineapples, etc.) occurs is still not fully understood. But this connection certainly has something to do with the spiraling way in which these objects grow. Reminder: The quadratic equation a x2 + b x + c = 0 has two solutions given by the quadratic formula b b 2 4ac b b 2 4ac , x2 . x1 2a 2a Therefore, the solutions of the equation x2 = x + 1 are 1 5 1.6180339887... 1.618 2 1 5 0.6180339887... 0.618 2 DEF: The Golden Ratio 1 5 1.618 2 OBS: Ф2 = Ф + 1 Ф3 = 2Ф + 1 [since Ф3=Ф*Ф2=Ф(Ф+1)=Ф2+Ф=(Ф+1)+Ф=2Ф+1] Ф4 = 3Ф + 2 [since Ф4=Ф*Ф3=Ф(2Ф+1)=2Ф2+Ф=2(Ф+1)+Ф=3Ф+2] Ф5 = 5Ф + 3 [since Ф5=Ф*Ф4=Ф(3Ф+2)=3Ф2+2Ф=3(Ф+1)+2Ф=5Ф+3] Ф6 = 8Ф + 5 [since Ф6=Ф*Ф5=Ф(5Ф+3)=5Ф2+3Ф=5(Ф+1)+3Ф=8Ф+5] Ф7 = 13Ф + 8 [since Ф7=Ф*Ф6=Ф(8Ф+5)=8Ф2+5Ф=8(Ф+1)+5Ф=13Ф+5] and so on. Hence: ФN = FNФ + FN-1 Example (page 322): Ratios of consecutive Fibonacci numbers The magic number that the ratios of two consecutive Fibonacci numbers approach is Ф = (1 + V5)/2. Essentially this means that, except for the first few, each Fibonacci number is approximately equal to Ф times the preceding one, and the approximation gets better as the numbers get larger. The Golden Ration in Art, Architecture, and Design: As a special number, the golden ratio Ф is considered, right along with π (the ratio between the circumference and the diameter of a circle) among the great mathematical discoveries of antiquity. The golden ratio was known to the ancient Greeks, who ascribed to it mystical and religious meaning and called it the divine proportion. The Greeks used the golden ratio as a benchmark for proportion and scale in art and architecture. The famous Greek sculptor Pheidias consistently used the golden ratio in the proportions for his sculptures, as well as in his design of the Parthenon, perhaps the best-known building of ancient Greece. Many others example where the Golden Ratio appear can be found in worksheets and in the textbook.