Project C01 - The Golden Ratio Project C01 - The Golden Ratio
Most of us tend to have the same idea of what it means for an object to be ``in proportion'' or ``out of proportion.'' In fact, human beings seem to have a talent for determining what looks ``just right.'' In this project, you will study the Golden Ratio and maybe you'll get the feeling that nature, in general, shares our talent.
  1. Choose any two natural numbers and call them a1 and a2. Now consider the sequence of numbers obtained by adding consecutive pairs. For n = 3,4,5,..., the nth term of this sequence will be an = an-1+an-2.
    1. Complete the following table.

    2. The ratios of consecutive terms of your sequence are approaching a fixed number. That number is the Golden Ratio. Use your table to approximate the golden ratio.
    3. The exact value of the golden ratio is Find an approximation that is correct to 25 decimal places.
  2. Examine some of the fonts available on your personal computer. In particular, look at the different B's. Print several large B's (of different sizes and different fonts) and place them inside rectangles that are just the right size. Do this with some hand-written B's as well. Determine the ratio of length to width for each rectangle. Discuss your findings.
  3. Alex keeps three Eastern Box Turtles as pets. The dimensions of their shells are shown below.

    Turtle Shell Length Shell Width
    Leonardo 143 mm 97 mm
    Michaelangelo 138 mm 104 mm
    Chunky 153 mm 100 mm

    1. As far as box turtles are concerned, what seems to be the ``perfect'' ratio of shell length to shell width?
    2. How close is this ratio to the golden ratio?
    3. If Leonardo's shell grew to a length of 150 mm, about how wide would it be?
  4. Obtain the following measurements from several of your friends:

    Round to the nearest half-inch or centimeter.

    1. For each person, find the ratio of total height to height from floor to navel. Determine the average ratio.
    2. The average ratio for all humans is approximately the golden ratio. How does your average value compare with the golden ratio?
    3. Suppose that while rummaging through some old clothes, you stumbled upon some pants that were 56 in from belt-loop to cuff. About how tall was the person who wore those pants?
  5. On the attached sheet, you will find the beginnings of a logarithmic or equiangular spiral. You should also notice many golden rectangles (rectangles whose ratio of length to width is equal to the golden ratio).
    1. Identify each golden rectangle on the attached sheet.
    2. Find an example of an equiangular spiral in nature.
    3. Find an unrelated example of the golden ratio appearing in nature.
    4. Find an example of the golden ratio appearing in architecture or art.

File translated from TEX by TTH, version 1.98.
On 25 May 2000, 21:44.