Properties and applications of fractals.

1. Fractals are
everywhere in the
world around us,
even in our bodies ...
A fractal fern leaf by flickr user Paulo Henrique Zioli

2. TED Talks Ron Eglash: African fractals, in buildings and braids
http://www.ted.com/index.php/talks/view/id/198

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

4. The Rectangle ...
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.) (a) Find the total
shaded area.
(b) Find the total
unshaded area.

5. The Rectangle ...
(a) Find the total
shaded area.
(b) Find the total
unshaded area.

6. 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.
(a) Find the total
shaded area.
(b) Find the total
unshaded area.

7. The fractal shown below, The Circle-Square, consists of a
square inscribed in a circle, and then a circle inscribed in a
square, et cetera. Calculate the total area of the shaded parts of
the fractal if there are eight circles and eight squares. The
diameter of the original circle is 16 cm.
HOMEWORK

8. People on Mars HOMEWORK
A group of 100 astronauts is sent to Mars to colonize the planet. NASA
scientists have predicted that the population will increase by 12 percent
every 20 years. Find the terms of the sequence for the first 100 years.
(a) Write a recursive formula for this sequence.
(b) Draw a graph of the sequence of populations over 100 years.
Describe the shape of the graph.
(c) Draw a graph of the sequence of populations over 300 years.
Does the graph still look the same as it did before?