25 Apr Precalculus – Project 3 – Fractals

Purpose: My purpose for this project is for you to apply your knowledge of sequences and

 

series to another branch of mathematical study. Your purpose for completing this project is to

 

communicate to me your understanding of geometric sequences as they apply to fractals. The

 

project may be handwritten but all work must be included and shown in an organized manner.

 

All work should be done on another sheet of paper with answers clearly marked. Answers

 

should be given in exact form. (Exact form means NO decimals.)

 

Fractal: A fractal is a geometric shape that can be split in parts that are reduced scale patterns

 

of the original. This pattern is repeated an infinite number of times. If you zoom in on any part

 

of it you will see it looks the same.

 

One such fractal is the Sierpinski Triangle. This fractal is created by connecting the midpoints of

 

the three sides of an equilateral triangle. This creates a new equilateral triangle which is then

 

“removed” from the original. The remaining three triangles are smaller versions of our original.

 

This process can then be repeated to continue to create other iterations of the figure.

 

The first three stages are shown here:

 

Stage 1 Stage 2 Stage 3

 

Part I:

 

a) Write a sequence to represent how many shaded triangles there are at each stage.

 

b) Based on this sequence, write a formula for the nth term of the sequence.

 

c) Using at least two sentences, describe what is represented by each term in the sequence.

 

Part II: Assume the original triangle in stage 1 had a side length of one.

 

a) Write a formula that would give the length of a side of one of the triangles at the nth stage.

 

If we look at the stage 1 triangle, we can find its area by multiplying one-half times the length of

 

its base times the length of its height. Assume again that it has a side length of one.

 

b) Find the height of the triangle in stage 1.

 

c) Find the area of the triangle in stage 1.

 

Part III: The stage 2 triangle has side length one-half.

 

a) Find the height of one of the triangles in stage 2.

 

b) Find the area of one of the triangles in stage 2. Be sure to simplify.

 

Part IV:

 

a) Use your knowledge of geometric sequences and your answers from (c) in Part II and (b) in

 

Part III to write a formula for the area of one of the triangles at the nth stage.

 

b) Use your answers from (b) in Part I and (a) in Part IV to write a formula for the total area of

 

all the shaded triangles at the nth stage. Be sure to simplify.

 

c) Using at least three complete sentences:

 

a. Describe what is happening to the triangle as n gets infinitely large.

 

b. Discuss what value you believe the total shaded area will approach and why.