Activity 2
The Golden Ratio

The Golden Ratio is a number that occurs in both mathematics and in nature. In this activity you will examine how this ratio occurs aesthetically, geometrically, and mathematically.

Your Favorite Rectangle

Examine the five rectangles drawn below from several different views. Choose the one rectangle that is most appealing to you and place an X on it. Use a metric ruler to measure both sides of each rectangle to the nearest millimeter. For each rectangle, divide the length of the longest side by the length of the shortest side and write this ratio on the rectangle.

Which rectangle was the most liked in the entire class?_____ What is its ratio?_____

Beauty of the Body

The Golden Ratio can also be found in the dimensions of the human body. Measure your height to the nearest centimeter and record it below. Measure the distance of your navel from the floor to the nearest centimeter and record that also. Divide your height by your navel height to find the height/navel ratio rounded to two decimal places and record that below.

Height = ________ Navel Height = ________ Ratio = ________

What is the class average of these ratios rounded to two decimal places? _______

The Pentagram

The Golden Ratio also appears in the measurements of many common geometric figures. The common pentagram on the next page has the golden ratio hidden in its structure. This star is composed of a regular pentagon with an isosceles triangle on each side. Use a metric ruler to measure the lengths of the segments AB, AC, AD, and DC to the nearest millimeter. Divide to find the ratios of AB to AC; AC to AD; and AD to DC rounded to two decimal places.

AB to AC = ________ AC to AD = ________ AD to DC = ________

The Fibonacci Sequence

The Fibonacci Sequence is a mathematical sequence of numbers constructed so that each number in the sequence is the sum of the two previous numbers. The first few numbers in the sequence are given below.
1 2 3 5 8 13 21 34 55 ...
Let Fi represent the ith number in the Fibonacci Sequence. For each value of i from 2 to 10, find the ratio of Fi / Fi-1 rounded to two decimal places and write these ratios in the table below. The Fibonacci Number is the limit of the sequence of these ratios as i increases without bound. This number is known as the Golden Ratio.


Value of i 2 3 4 5 6 7 8 9 10
Ratio Fi / Fi-1 ____ ____ ____ ____ ____ ____ ____ ____ ____
Can you guess the value of the Fibonacci Number? ______

The Golden Rectangle

The Golden Rectangle also shows the Golden Ratio. In the diagram below ACDF is a golden rectangle, ABEF is a square, and BCDE is a golden rectangle. The ratio of the length of the longest side to the length of the shortest side of a, golden rectangle is the golden ratio. The golden ratio can be mathematically derived from this relationship by the proportion shown at the right of the diagram.

The Golden Spiral

The Golden Ratio may also be seen in a spiral. In fact, a spiral can be drawn from the diagram above by dividing BCDE into a square and another golden rectangle, and by continuing to separate the golden rectangles into squares and new golden rectangles. The first part of the spiral is drawn with a compass point at B and an arc from A to E; next, the compass point at G and an arc from E to H. Continue in this way until the spiral is completed to your satisfaction. The spiral appears in nature in the nautilus shell, flower petals, and pine cones.

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