Applied 40S June 2, 2009


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Introduction to fractals.

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Applied 40S June 2, 2009

  1. 1. Properties of Fractals (What is a fractal anyway?) Infinity by flickr user azarius
  2. 2. What is a Fractal? Fractal Zoom on YouTube
  3. 3. The Bouncing Ball A ball is dropped from one metre, and the height is recorded after each bounce. A 'Super Bouncer' sold locally is guaranteed to bounce to 90 percent of its drop height if it is dropped onto concrete from a height of less than two metres. 1. How high does the ball bounce on its eighth bounce? 2. How many times does the ball bounce before it rises to less than half of its original drop height? 3. How many times does the ball bounce before it stops bouncing? 4. How far has the ball travelled as it reaches the top of its 4th bounce. 5. Construct a graph that shows the bounce height versus bounce number.
  4. 4. The Sierpinski Triangle Waclaw Sierpinski, a Polish mathematician, developed another fractal known as the Sierpinski Triangle. This fractal also starts with an equilateral triangle. To draw the fractal, you find the midpoint of each side of the original triangle, and then draw three segments joining the midpoints. There are now four triangles inside the original triangle. The middle triangle is not shaded, and the process is continued with the other three shaded triangles, as shown in the diagram below.
  5. 5. The Koch Snowflake This fractal -- the Koch Snowflake -- was developed by in 1904 by Helge von Koch, a Swedish mathematician. The fractal is started by drawing an equilateral triangle. Each side of the triangle is trisected, and the middle section forms the base of a new equilateral triangle outside the original one. The process is then continued. The diagram below shows three generations of the Koch Snowflake. 1st iteration 2nd iteration
  6. 6. A Fractal: The Koch Snowflake All about the Koch Snowflake on wikipedia
  7. 7. Draw a Fractal Use pencil and paper (metric graph paper if possible) to draw the fractal described below. • Draw a square with 8-cm sides in the middle of the paper. • Position the paper horizontally (in landscape format). Extend the diagram to the left and right by drawing a square on each side of the original square -- touching the original square. The sides of the new squares should be half as long as the side lengths of the original square. • Repeat the previous step three times. Your fractal should now have five generations, including the original square.
  8. 8. Question: Will the fractal ever be too large for this page? Explain.
  9. 9. The Rectangle ... HOMEWORK Draw a rectangle that measures 12 cm by 8 cm, and shade the inside of the rectangle. Construct the midpoints of each side of the rectangle, and then draw a quadrilateral by joining these points. Shade the quadrilateral white. Now continue the process by finding the midpoints of the quadrilateral, drawing the rectangle, and shading it the same colour as the first rectangle. Draw six generations. (The initial rectangle is the first generation.)
  10. 10. The Square ... HOMEWORK Create a fractal that begins with a large square 20 cm on each side. Each pattern requires that the square be divided into four equally sized squares, that the bottom-left square be shaded, and the process continues in the upper-right square. Repeat the process four times.