Difference between revisions of "Golden Ratio Exploration"

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[[category:Similarity and Fractals Explorations]]

Revision as of 08:14, 24 January 2009


Objective: Explore properties of the Golden Ratio.

Suppose we start by drawing one small square, and attach another square to it on the right, making a 2x1 rectangle. Now draw another square along the long edge of the 2x1 rectangle, making a 3x2 rectangle. Continue this process, spiraling outward, until you're out of room on the page.


We are in effect creating larger and larger rectangles by attaching squares to them. After attaching several squares, we get the following figure:


  1. If we make a table showing the side lengths of each rectangle along the way, then we find the flowing values:
    Dimension of rectangle Ratio of the two sides
    1 x 2 2/1 = 2
    2 x 3 3/2 = 1.5
    3 x 5 5/3 = 1.666666666
    5 x 8 8/5 = 1.6

    a. Find the dimensions of the next two rectangles.

    b. Look at the list of the dimensions of the rectangles. How is the biggest number related to the two number in the row above it? Knowing this, continue the following sequence of numbers: 2, 3, 5, 8, 13,...
  2. The sequence of numbers we have found above is the famous Fibonacci sequence. If you do a little research on the Fibonacci sequence, you will find for instance that on some plants the number of petals is a Fibonacci number.
  3. We have calculated the ratio of the long side to the short side for the first four big rectangles. Find the ratio for the remaining rectangles. What number does this value seem to approach?
  4. Note that the rectangles changed shape less and less as the process continued (the ratios of long to short sides didn't change much). Suppose you want a rectangle that stays exactly the exactly the same shape when a square is attached. Goldenratio3.jpg

    This means the ratio <math>\frac{x}{1} = \frac{1+x}{x}</math>

    Solve this equation for x. (Cross multiply and use the quadratic formula!)
  5. Calculate the ratio of your height to the height of your navel. Do the same for four friends. How does this number compare to the ratios you have found before?
  6. This triangle is called the Golden triangle. Compute the ratio of the length of the side to the length of the base. Goldenpentagon.jpg
  7. The Acropolis in Athens, Greece is built according to the golden ratio. Measure the width and the height, and check that they form a golden rectangle.
  8. How many Golden Rectangles can you find in Mondrian’s Broadway Boogie Woogie?
  9. Construction.png This section is unfinished.