Fractals

Waclaw Franciszek Sierpiński, was born on March 14, 1882 in Warsaw and died on October 21, 1969 in Warsaw. He was a Polish mathematician, known for outstanding contributions to set theory (research on the axiom of choice and the continuum hypothesis), number theory, theory of functions and topology. Two well-known fractals are named after him (the Sierpinski triangle and the Sierpinski carpet), as are Sierpinski numbers and the associated Sierpinski problem.  A fractal is a geometric figure that is created using iteration.  Many objects in nature aren't formed of squares or triangles, but of more complicated geometric figures. Many natural objects - ferns, coastlines, etc. - are shaped like fractals.

When my students read what is above in a textbook, they still do not fully comprehend what a fractal is.  So I make them do this modeling activity that is in our textbook.  After doing that they understand the real meaning of doing the same thing over and over again.  The activity is listed below.

Instructions

1. Provide the students with blank typing paper, ruler, compass and crayons.
2. Draw an equilateral triangle 8 inches long.
3. Connect the midpoints of each side to form another triangle.
4. Shade the center triangle.  This is stage 1.
5. Repeat the process using the non-shaded triangles.
6. It would be a good idea to demonstrate in front of them for the 1st two stages while they are reading the instructions.
7. Walk around and provide help as necessary.
8. Students will not be able to finish the entire exercise in class.  They can finish stage 3 and 4 for homework. (This will make beautiful bulletin board displays.)
9. Explain what they have started and will finish up at home is called iteration, the process of repeating the same procedure over and over again.
10. If your students do not know how to construct an equilateral triangle, you will have to show them how before starting this activity.

Once I got into EMAT 6690, I learned that you can do this in GSP.  So I plan on doing this with my students next year as well.  Click below in order to see how to make Sierpinski's Triangle in GSP.

Sierpinski's Triangle in GSP

References

Back to my EMAT 6690 Page