Teacher Information: Koch and Sierpinski Fractals

There are two fractal constructions in this virtual manipulative, the Koch Snowflake and the Sierpinsky Carpet. As is typical of fractal constructions, each portion is repeated on a smaller and smaller scale, so that if we were able to blow up each portion sufficiently, each little piece would be identical to any corresponding piece at any other stage. (See the Mandelbrot and Julia Sets manipulative.) Koch's curve is the boundary of the Koch Snowflake obtained by beginning with an equilateral triangle, removing the middle third of each side and replacing it with two sides of a smaller triangle, and continuing the process forever.

The Sierpinski Carpet is normally constructed by starting with a square, removing the middle ninth, removing the middle ninth of each remaining square, and continuing. The construction in this virtual manipulative accomplishes the same result differently, but the result is the same. As with the Snowflake, we can show only a few steps, but in this case we place a pattern of rectangles in a square region, and then repeat on a smaller scale, then again and again. When we reach the limit of what is visible on our computer monitors, we have an excellent approximation of the fractal Carpet.

Many references to Sierpinski in the context of fractals speak of the "Sierpinski Triangle," which is constructed much like the Sierpinski Carpet. Starting with an equilateral triangle, remove a middle equilateral triangle (a fourth of the triangle rather than a ninth of the square), and then repeat with each of the remaining small triangles, and continue indefinitely. The Sierpinski Triangle, built in an entirely different way from this manipulative, can be constructed in the virtual manipulative Polygon Fractals, setting the right-hand slider at 3 and using a probability of about .50.

For each construction here, the user can control the Play, Pause, and Stop (clearing the screen) buttons, and can choose colors (even during the construction), but the interactivity is limited.

For use in class discussions, take advantage many other web discussions and constructions of the Koch Snowflake, the Sierpinski Triangle, and related fractal topics.