Using a mathematical expression, describe the number of triangles that “stay” at any step in the generating process, use the letter n to represent the step. To achieve goal #2 do the following: First, define the perimeter of the initiator triangle. Let each side of the triangle have a length of 1. Next, use the definition of a midpoint to define the lengths of each side of the new triangles that are generated in step 1. Now write an expression for the perimeter of one of the three triangles in step 1. Continue this process for a couple more steps, keep track of your data using the following table: Side length of Perimeter of one n (step) Number of triangles Perimeter at step n one triangle triangle 0 1 2 3 4 Using a mathematical expression, define the perimeter of the Sierpinski gasket at any step n. Now answer the following questions: How does the length of each side of the initiator triangle change at each step? How does the number of triangles at each step effect the perimeter of the gasket? Once you have generated an expression for the perimeter of the gasket at any step, try it using these values for n: 20, 50, 100. Describe how the perimeter changes as the number of steps increases. To achieve Goal #3 do the following: First, define the area of the initiator triangle as 1. This will make life easier. The area of the Sierpinski gasket at step 0 is 1. Next, think about how much area was removed at step 1. Hint: there are 4 triangles in step 1, and three “stay”. Write down the remaining area for step 1. In step 2, we remove the same fraction of the remaining area. Write down an expression for the area at step 2.
Using a mathematical expression, describe the number of triangles that “stay” at any step in the generating process, use the letter n to represent the step. To achieve goal #2 do the following: First, define the perimeter of the initiator triangle. Let each side of the triangle have a length of 1. Next, use the definition of a midpoint to define the lengths of each side of the new triangles that are generated in step 1. Now write an expression for the perimeter of one of the three triangles in step 1. Continue this process for a couple more steps, keep track of your data using the following table: n (step) Side length of one triangle Perimeter of one triangle Number of triangles Perimeter at step n 0 1 2 3 4 Using a mathematical expression, define the perimeter of the Sierpinski gasket at any step n. Now answer the following questions: How does the length of each side of the initiator triangle change at each step? How does the number of triangles at each step effect the perimeter of the gasket? Once you have generated an expression for the perimeter of the gasket at any step, try it using these values for n: 20, 50, 100. Describe how the perimeter changes as the number of steps increases. To achieve Goal #3 do the following: First, define the area of the initiator triangle as 1. This will make life easier. The area of the Sierpinski gasket at step 0 is 1. Next, think about how much area was removed at step 1. Hint: there are 4 triangles in step 1, and three “stay”. Write down the remaining area for step 1. In step 2, we remove the same fraction of the remaining area. Write down an expression for the area at step 2.