Applied 40S June 3, 2009

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Properties and applications of fractals.

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Applied 40S June 3, 2009

  1. 1. Fractals are everywhere in the world around us, even in our bodies ... A fractal fern leaf by flickr user Paulo Henrique Zioli
  2. 2. TED Talks Ron Eglash: African fractals, in buildings and braids http://www.ted.com/index.php/talks/view/id/198
  3. 3. Question: Will the fractal ever be too large for this page? Explain.
  4. 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. 5. The Rectangle ... (a) Find the total shaded area. (b) Find the total unshaded area.
  6. 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. 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. 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?

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