Phidias and Fibonacci Recall that a proportion is a statement that two ratios are equal. Proportions have a special quality that helps us solve for a missing value. We usually remember it by saying that the “cross-products” are equal. a c If = , then ad = bc b d A geometric mean is a value between two others, chosen so that the square of that value equals the product of the other two. For example, 4 is the geometric mean between 2 and 8, since 42 = 16 and 2⋅ 8 = 16. Here are a few proportions; the last one is from the Golden Rectangle that you constructed. 2 = 4 1 = 3 5 = 20 1.61803 ≈ 1 4 8 3 9 20 80 1 0.61803 The geometric mean would also fill-in a missing term in a geometric sequence, as shown below. 1, 2, ______, 8, 16… 1, _____, 9, 27… 5, _____ , 80, 400,… 0.61803, _______, 1.61803, 2.61803,… The proportions above all show a geometric mean, but only the last one also shows the relationship: ab+ = a. This is so special that ab ab+ == a φ is called the Golden Ratio or the Golden Mean, Phi. ab The Golden Ratio, which we found after we constructed it, equals: φ = 1.6180339887...51+ ≈ 2 Phi is named for the ancient Greek sculptor, Phidias, who was both a sculptor and a mathematician. He created the statue of Zeus at Olympia, which was considered one of the seven wonders of the ancient world, and directed the construction of the Parthenon and sculpted its statues. When he sculpted the gods, he used the Golden Ratio; when he sculpted mortals, he did not. Though Phidias lived in the 5th century B.C., the golden ratio has been found further back in time in the Great Pyramids of ancient Egypt. Leonardo of Pisa, called Fibonacci (son of Bonacci), was a 12th century Italian mathematician, best known to the modern world for a number sequence named after him, commonly called the Fibonacci Numbers. He did not discover this sequence of numbers but used them as an example in his most important work, Liber Abaci. The sequence begins with two ones, and the next term is always found by adding the two previous terms. 1, 1, 2, 3, 5, 8, 13, 21, 34, 55, 89, 144, … The numbers in this sequence have special mathematical properties as well. Ratio Value If you take these numbers two at a time, in order, and compute their ratios, you approach the Golden Ratio. Note that each of these ratios is of the 1/1 1.00000 form: ab+ . (That makes sense, doesn’t it?) a 2/1 If we look at the sequence geometrically, by constructing squares whose 3/2 side lengths are Fibonacci Numbers, we approach a Golden Spiral. 5/3 8/5 13/8 21/13 34/21 55/34 The Fibonacci numbers can be seen in art and nature, too. 89/55 http://factoidz.com/interesting-facts-about-the-golden-ratio-in- nature-art-math-and-architecture/ .