Concept Byte: The Golden Ratio

Use With Lesson 7-4

ACTIVITY

In his book Elements, Euclid defined the extreme and mean ratio using a proportion formed by dividing a line segment at a particular point, as shown below. In the diagram, C divides A B ¯ eh b bar  so that the length of A C ¯ eh c bar  is the geometric mean of the lengths of A B ¯ eh b bar  and C B ¯ . c b bar , .  That is, A B A C = A C C B . fraction eh b , over eh c end fraction . equals . fraction eh c , over c b end fraction . .  The ratio A C C B fraction eh c , over c b end fraction  is known today as the golden ratio, which is about 1.618 : 1.

Rectangles in which the ratio of the length to the width is the golden ratio are golden rectangles. A golden rectangle can be divided into a square and a rectangle that is similar to the original rectangle. A pattern of golden rectangles is shown below.

Activity 1

To derive the golden ratio, consider A B ¯ eh b bar  divided by C so that A B A C = A C C B . fraction eh b , over eh c end fraction . equals . fraction eh c , over c b end fraction . .

1. Use the diagram below to write a proportion that relates the lengths of the segments. How can you rewrite the proportion as a quadratic equation?

   

2. Use the quadratic formula to solve the quadratic equation in Question 1. Why does only one solution makes sense in this situation?
3. What is the value of x to the nearest ten-thousandth? Use a calculator.

Spiral growth patterns of sunflower seeds and the spacing of plant leaves on the stem are two examples of the golden ratio and the Fibonacci sequence in nature.

Activity 2

In the Fibonacci sequence, each term after the first two terms is the sum of the preceding two terms. The first six terms of the Fibonacci sequence are 1, 1, 2, 3, 5, and 8.

4. What are the next nine terms of the Fibonacci sequence?
5. Starting with the second term, the ratios of each term to the previous term for the first six terms are 1 1 = 1 , 2 1 = 2 , 3 2 = 1 . 5 , 5 3 = 1 . 666 … , 1 over 1 , equals 1 , comma , 2 over 1 , equals 2 , comma , 3 halves , equals 1 , . 5 comma , 5 thirds , equals 1 , . 666 dot dot dot comma  and 8 5 = 1.6 . 8 fifths . equals 1.6 .  What are the next nine ratios rounded to the nearest thousandth?
6. Compare the ratios you found in Question 5. What do you notice? How is the Fibonacci sequence related to the golden ratio?

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