Fractals

In colloquial usage, a fractal is "a rough or fragmented geometric shape that can be subdivided in parts, each of which is (at least approximately) a reduced/size copy of the whole

Because they appear similar at all levels of magnification, fractals are often considered to be infinitely complex

Look at the triangle you made in Step One Look at the triangle you made in Step One. What fraction of the triangle did you NOT shade? What fraction of the triangle in Step Two is NOT shaded? What fraction did you NOT shade in the Step Three triangle? Do you see a pattern here? Use the pattern to predict the fraction of the triangle you would NOT shade in the Step Four Triangle. Confirm your prediction and explain. Find another interesting pattern in the fractal called the Sierpinski Triangle. Write a paragraph descibing this pattern.

                      Step One                        Step Two                         Step Three                        Step Four                        Step Five

Cynthia Lanius' Lesson Koch Snowflake Fractal, Using JAVA

KOCH Questions What is the perimeter of the snowflake at stage 1? At stage2? Work out the perimeter of the snowflake at each stage. What will the perimeter be after n stages?   What about the area of the snowflake - what is the area at each stage? Hint: work with fractions, not decimals

Now think of doing this many, many times. The perimeter gets huge! But does the area? We say the area is bounded by a circle surrounding the original triangle. If you continued the process oh, let's say, infinitely many times, the figure would have an infinite perimeter, but its area would still be bounded by that circle.

An infinite perimeter encloses a finite area... Now that's amazing!!