Introduction to fractals.

1. Properties of Fractals
(What is a fractal anyway?)
Infinity by flickr user azarius

2. What is a Fractal?
Fractal Zoom on YouTube

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. 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. 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.
http://www.youtube.com/watch?v=JdMgvSWSKZI
1st iteration 2nd iteration

6. A Fractal: The Koch Snowflake
All about the Koch Snowflake on wikipedia

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. Question: Will the fractal ever be too large for this page? Explain.

10. 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.)

11. 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.