Instructions: A fractal is a never-ending pattern. Fractals are infinitely complex patterns that are self-similar across different scales. They are created by repeating a simple process over and over in an ongoing loop. An example of a fractal using the mid-segments of an equilateral triangle is shown below. Now, you will create a fractal pattern using RIGHT triangles. Instructions: 1) 1st Iteration: Using unlined paper and a ruler, draw a LARGE RIGHT triangle on your paper. For the right angle you may use the corner of a piece of paper. (You may want to make the two legs have whole number lengths for easier computation later). a. What are the lengths of the two legs of this triangle? (don’t forget units) (6 points) _, _ b. Find the area of this triangle. (don’t forget units) _ (3 points) 2) 2nd Iteration: Find the midpoints of each side; you may fold the paper or you may use a ruler for this. Connect each midpoint with a mid-segment. a. What are the lengths of the two legs of the small inner triangle? (don’t forget units) (6 pts) _, _ b. Find the area of this inner triangle. (don’t forget units) _ (3 points) c. Color this inner triangle a solid color.
3rd Iteration: There should be three un-colored triangles surrounding the colored one. For each of these white triangles, find the midpoints of each of the sides, and draw the mid-segments connecting them. a. What are the lengths of the two legs of each small inner triangle? (don’t forget units) (6 pts) _, _ b. Find the area of each inner triangle. (don’t forget units) _ (3 points) c. Color each inner triangle a solid color. 4) 4th Iteration: There should now be 9 small white triangles in the picture. Repeat the steps above to draw the mid-segments of each of these triangles. a. What are the lengths of the two legs of each small inner triangle? (don’t forget units) (6 pts) _, _ b. Find the area of this inner triangle. (don’t forget units) _ (3 points) c. Color each inner triangle a solid color. d. How many small white triangles are now in the picture? Look for a pattern for finding the number of white triangles that will be in the next iteration of the fractal. Describe the pattern. (9 points) _, _ _ e. How many small white triangles would there be in the 5th iteration? (3 points) _ 5) Look back and compare the areas of the innermost triangles for each of the 4 iterations. a. Write the areas from largest to smallest (3 pt)_,_,_,_ b. Look for a pattern for finding the area of the next iteration. Describe the pattern. (6 points) _ _ c. What would be the area of the innermost triangle of the 5th iteration? (3 points) _