Assignment: Fractals Generated With Positive Exponents

In the beginning of this module, we presented the Sierpinski gasket, made from triangles:

Initial, black equilateral triangle is completely filled in. Step 1, the triangle has been divided into three black and one white equilateral triangles, with the white triangle in the center. In step 2, each black triangle has been further divided into into three black and one white equilateral triangles with the white triangle in the center. In step 3, each black triangle has once again been divided into three black and one white equilateral triangles with the white one in the center.

Create this object with paper and pencil by drawing a triangle, then finding the midpoints, and creating a new triangle by connecting the midpoints you found. The black triangles represent those that “stay” and the white represent those that “go”. Draw a few more steps, you will want to refer to this drawing throughout the exercise.

In this activity, we will discover more interesting features of this well-known fractal.

Goals:

  1. Develop a mathematical expression that represents the number of triangles in a Sierpinski gasket that “stay” at any step in the generating process. (see above for context for the meaning of “stay”)
  2. Develop a mathematical expression that represents the perimeter of a Sierpinski gasket at any step in the generating process.
  3. Develop a mathematical expression that represents the area of a Sierpinski gasket at any step in the generating process.

You can probably find the answers to these goals by searching the internet. It will be your loss if you do, this is an interesting activity that will help you understand how math can be used to express the behavior of fractal generation in a concise and elegant way. So go ahead, miss out on the opportunity for self-discovery.  Or, read on…

To achieve goal #1, do the following:

  • Answer this question – how many midpoints can you find on a triangle? Keep this in mind as you develop your answer to #1. Go ahead, Google midpoint if you need a reminder.
  • For each stage, count the number of black triangles. It may help to put your data in a table like this:
Step Number of Black Triangles
 0 (Initial)  1
 1
2
3
4
  • 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 3 1 3
1
2
3
4
  • Using a mathematical expression, define the perimeter of the Sierpinski gasket at any  step n.

Now answer the following questions:

  1. How does the length of each side of the initiator triangle change at each step?
  2. How does the number of triangles at each step effect the perimeter of the gasket?
  3. 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.
  • Use a table to track the area at each step for a few more steps
n (step) Area
0 1
1
2
3
4
  • Define a mathematical expression for the area of the Sierpinski gasket at any step using the variable n to define the step.

Now answer the following questions:

  1. How does the number of triangles in each step effect the area remaining?
  2. Once you have generated an expression for the area of the gasket at any step, try it using these values for n: 20, 50, 100.
  3. Describe how the area of the gasket changes as the number of steps increases.