# Assignment: Fractals Generated With Positive Exponents

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

- 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")
- Develop a mathematical expression that represents the perimeter of a Sierpinski gasket at any step in the generating process.
- Develop a mathematical expression that represents the area of a Sierpinski gasket at any step in the generating process.

- 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.

- 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.

- 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.

- 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.

- How does the number of triangles in each step effect the area remaining?
- 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.
- Describe how the area of the gasket changes as the number of steps increases.

Download the assignment from one of the links below (.docx or .rtf):

Fractals Generated With Positive Exponents: Word Document

Fractals Generated With Positive Exponents: Rich Text Format