Cynthia Lanius

The Sierpinski Triangle

To make a fractal: Take a familiar geometric figure (a triangle or line segment, for example) and operate on it so that the new figure is more "complicated" in a special way. Then operate on that figure in the same way and get an even more complicated figure. Then do it again and again...and again.

Let's make a famous fractal called the Sierpinski Triangle. You will need triangular gridpaper.

Step One
Draw an equilateral triangle with sides of 2 triangle units each.
Connect the midpoints of each side.

How many equilateral triangles do you now have?
Cut out the triangle in the center.

Step Two
Draw an equilateral triangle with sides of 4 tiangle units each. Connect the midpoints of the sides and cut out the triangle in the center as before.

Notice the three small triangles that also need to be cut out in each of the three triangles on each corner.

Step Three
Draw an equilateral triangle with sides of 8 triangle units each. Follow the same procedure as before, making sure to follow the cutting pattern.

Step Four
For this one, you'll need a larger paper, or cut smaller triangles. Follow the above pattern and complete the fourth stage of the Sierpinski Triangle. Use your artistic creativity and shade the triangles in interesting color patterns. Does your figure look like this one? Then you are correct!

Teachers' Notes: Combine the students' triangles to form a larger Sierpinski Triangle. This makes a really nice wall hanging for the classroom.
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