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F : Fanatic Fractal Fever BROWN FORGER INSLEY MAHONEY An exploration of patterns for Middle and High School students 3 F3: Table Of Contents A series of activities that explore fractals with a progression from Middle School to High School. Middle School: Rep-Tiles - Students use pattern blocks to create a sequence of growing similar figures. They analyze patterns created and predict results for even larger figures. Sierpinski Triangle - Students create the beginning stages of the Sierpinski Triangle by building-up triangular regions and exploring patterns in perimeter and area. Sierpinski Carpet - Students create the beginning stages of the Sierpinski Carpet by building-up square regions and exploring patterns in perimeter and area. Self-Similarity - Students use their knowledge of similarity to explore the meaning of self-similarity in basic shapes. Connections are made to the rep-tile activity. Self-Similarity in the Sierpinski Triangle - Students extend their knowledge of self- similarity as it applies to their activity on the Sierpinski Triangle. Dimension - Students explore the meaning of dimension, from a point to a three- dimensional space and then relate that to the concept of fractal dimension. Sierpinski Tetrahedron - Students extend their understanding of fractals to three dimensions by creating the first stages of the Sierpinski Tetrahedron. They explore patterns in surface area and volume with challenges for even the brightest students. Pascal’s Triangle - Students look for patterns in Pascal’s Triangle, specifically with the placement of odd and even integers. Furthermore, through shading, students discover a connection to the Sierpinski Triangle. High School: Cutting Away - Students cut paper into thirds and repeat the process to create stages. This exploration leads to the discovery of an exponential function and its connection to recursive formulas. This introductory activity begins student thinking of repetitive processes and pattern recognition. Sierpinski Triangle - Students create the beginning stages of the Sierpinski Triangle by removing triangular regions and exploring patterns in perimeter and area. They also connect these patterns to functions through graphing. Sierpinski Carpet - Students create the beginning stages of the Sierpinski Carpet by removing square regions and exploring patterns in perimeter and area. Graphs and functions are also explored, as well as connections to the Sierpinski Triangle. Dimension - Students explore the meaning of dimension, from a point to a three- dimensional space and then relate that to the concept of fractal dimension. Sierpinski Tetrahedron - Students extend their understanding of fractals to three dimensions by creating the first stages of the Sierpinski Tetrahedron. They explore patterns in surface area and volume with challenges for even the brightest students. Pascal’s Triangle - Students look for patterns in Pascal’s Triangle, specifically with the placement of odd and even integers. Furthermore, through shading, students discover a connection to the Sierpinski Triangle. For more information, or copies of the activities, contact: Nathan Brown ncbrown76@gmail.com Lauren Forger lforger@hotmail.com Dave Insley David.Insley@comcast.net Kelly Mahoney k_a_mahoney@yahoo.com Name: __________________ Class: ___________ Date: ________ Rep-Tiles Begin with an equilateral triangle pattern block, like the one shown: 1. Now, tile as few of these shapes together as possible to make an enlarged copy of the original shape that contains no holes or missing pieces. Draw the arrangement below: 2. Is your new triangle similar to the original triangle? How do you know? 3. Repeat Questions 1–2 for the next bigger enlargement you can make with the original shape. Draw and describe your figure below. 4. Complete the following table to show how the side length, perimeter, and number of pieces change as the figure grows larger: Stage 1 2 3 4 5 Side Length 1 2 Perimeter 3 Number of unit 1 triangles 5. Looking at the patterns in the table above, what would be the perimeter for the Stage 7 triangle? the Stage 15 triangle? (draw the figures if it helps) Name: __________________ Class: ___________ Date: ________ 6. How does the perimeter change from one figure to the next? Describe the pattern in words or symbols. 7. Looking at the patterns in the table above, what would be the number of unit triangles in the Stage 7 triangle? the Stage 15 triangle? 8. How does the number of unit triangles change from one figure to the next? Describe the pattern in words or symbols. 9. Name at least two other shapes you can use to create a rep-tile. Draw them below. Do you have any conjectures? Name: __________________ Class: ___________ Date: ________ The Sierpinski Triangle We are going to create a famous fractal called the Sierpinski Triangle. Using a sheet of isometric dot paper, draw an equilateral triangle with side length of 2 units. Now connect the midpoints of each of the three sides of your triangle. Does this look familiar? Shade in the three triangles that are pointing up. This is Stage 1 of the Sierpinski Triangle! You may notice that stacking three equilateral triangles can create the figure above leaving a hole in the middle. If we call one triangle Stage 0 of the Sierpinski Triangle, then the next stage can be made from three copies of the previous stage. 1. Draw Stage 2 below. Stage 0 Stage 1 Stage 2 2. What fraction of the triangle in Stage 1 is shaded in? 3. What fraction of the triangle in Stage 2 is shaded in? 4. Use this information to predict the fraction of the triangle in Stage 3 that will be shaded in. Confirm your answer by drawing Stage 3. Name: __________________ Class: ___________ Date: ________ 5. Can you predict the fraction of the triangle in Stage 10 that will be shaded in? Describe the pattern in words or symbols. 6. Let’s explore some patterns in the Sierpinski Triangle. Remember that the white areas in the stages are holes, so the perimeter includes both the outside of the triangle as well as the lengths around the holes. Stage Number 0 1 2 3 4 5 Side Length 1 3 Perimeter 3 9 Number of Shaded 1 3 Triangles Fraction of Triangle 1 3 that is Shaded 4 Fraction of Triangle 0 1 that is not Shaded 4 7. How could you find the number of shaded triangles for any stage number? Describe the pattern in words or symbols. 8. Write the fractions from the last column in order from the least to the greatest. Write a statement about how their order connects to the shading in process. 9. How does the fraction that is shaded change as the stage number gets larger and larger? 10. Find another interesting pattern in the Sierpinski Triangle. Write a paragraph describing your pattern. Name: __________________ Class: ___________ Date: ________ The Sierpinski Carpet Now that you have explored the basic concepts of a fractal and the idea of self- similarity, let’s take a look at another fractal, called the Sierpinski Carpet. The Sierpinski Carpet is similar to the Sierpinski Triangle in that both are fractals based on basic shapes and have various patterns that can be seen and explored. To create a Sierpinski Carpet begin with a square of any size. We will call this square Stage 0. Stage 0 We can create a new stage by arranging 8 elements of the previous stage in a square pattern with a “hole” in the center the size of one of the elements from the previous stage. 1. Draw Stage 2 below. Stage 0 Stage 1 Stage 2 2. What fraction of the square in Stage 1 is shaded in? 3. What fraction of the square in Stage 2 is shaded in? Name: __________________ Class: ___________ Date: ________ 4. Use this information to predict the fraction of the square in Stage 3 that will be shaded in. Confirm your answer by drawing Stage 3. 5. Can you predict the fraction of the square in Stage 10 that will be shaded in? Describe the pattern in words or symbols. 6. Let’s explore some patterns in the Sierpinski Carpet. Remember that the white areas in the stages are holes, so the perimeter includes both the outside of the squares as well as the lengths around the holes. Stage Number 0 1 2 3 4 5 Side Length 1 3 Perimeter 4 16 Number of Shaded 1 8 Unit Squares Fraction of Square 1 8 that is Shaded 9 Fraction of Square 0 1 that is not Shaded 9 7. How could you find the number of shaded squares for any stage number? Describe the pattern in words or symbols. 8. Write the fractions from the last column in order from the least to the greatest. Write a statement about how their order connects to the shading in process. Name: __________________ Class: ___________ Date: ________ 9. How does the fraction that is shaded change as the stage number gets larger and larger? 10. Do you see any relationship between the generalizations for the Sierpinski Triangle and the Sierpinski Carpet? What role does the number of sides of the unit shape in each fractal play in the patterns you see? Name: __________________ Class: ___________ Date: ________ Self-Similarity We have used the term ‘similar’ many times. If you and a friend are similar, then you are alike in some ways. In Geometry, similarity has a special meaning. Geometric figures are similar if they have the same shape. For example, the following squares are similar, in fact all squares are similar: However, not all rectangles are similar. For example: are not similar rectangles. To be similar, the corresponding sides need to be proportional and the angles need to be congruent (have the same measure). 1. Give an example of two Geometric figures that are similar. Draw your similar figures below: 2. Self-similarity is a slightly different concept. We can go back and look at our reptiles for an example of self-similarity. We have a bunch of copies of a shape making up a larger shape. Draw a rep-tile using trapezoids that shows an example of self-similarity: Adapted from http://math.rice.edu/~lanius/fractals/self.html Name: __________________ Class: ___________ Date: ________ Self-Similarity in the Sierpinski Triangle Above is a picture of the Sierpinski Triangle that we made in this unit. Notice that the outline of the figure is an equilateral triangle. Now look inside at all the equilateral triangles. Remember that there are infinitely many smaller and smaller triangles inside. How many different sized triangles can you find? All of these are similar to each other and to the original triangle - self-similarity. See all the copies of the original triangle inside? How many copies do you see where the ratio of the outer triangle's sides to the inner ones is 2:1? 4:1? 8:1? I think we have a pattern here. Can you find it? Let’s look at some questions about self-similarity 1. If figure 1 is the original figure, how many similar copies of it are contained in figure 2? Figure 1 Figure 2 2. Are squares self-similar? (Can you form bigger squares out of smaller ones?) Are hexagons? (Can you form larger hexagons out of smaller ones?) Draw examples to justify your answer. 3. Are circles similar? Are they self-similar?(Can you form larger circles out of smaller ones? Draw examples to justify your answer. 4. Experiment with designing another self-similar figure. Do you have any conjectures? Adapted from http://math.rice.edu/~lanius/fractals/self.html Name: __________________ Class: ___________ Date: ________ Dimension A point is a single place in space that has no length, width, or height. We say that a point has zero dimension. We can attempt to draw a point, but anything we draw will be huge in comparison to an actual point, so we’ll have to trust that a point can be represented and still has zero dimension. ●P A line has only one dimension, length. It has no width, but it has infinite length. You will have to imagine a pencil line so thin that it cannot be measured. This won’t ever happen, but there are still lines that have no width and infinite length. l A plane has two dimensions, length and width, but it has no depth. Think of a very thin desktop. So thin that you cannot measure it, but it stretches an infinite distance in all directions. Space has three dimensions, length, width, and depth. Think of the universe as a space or a large empty box. It stretches an infinite distance in all directions. Fractals, like the Sierpinski Triangle, could have fractional dimension. Let’s explore how this can be. Think of a self-similar object like a line segment, and double its length: Doubling the length gives two copies of the original object. Let’s look at a square. If we double the length of the side of a square we will get four copies of the object: Finally, let’s look at a cube. If we double the length of a side of a cube we will get eight copies of the object: Adapted from http://math.rice.edu/~lanius/fractals/dim.html Name: __________________ Class: ___________ Date: ________ Let’s organize our data in the following table and see if we can see a pattern: Figure Dimension Number of Copies Line Segment 1 2 21 Square 2 4 22 Cube 3 8 23 1. What do you notice about the relationship between the Dimension and the Number of Copies? 2. How many copies would we get if we doubled an object of dimension d? We can use the answer to the above question to determine the dimension of the Siepinski Triangle! 3. If we double the length of our Siepinski Triangle, how many copies did we get? Doubling the side length gives us 3 copies, so we can use the equation 3 2d to determine the dimension. Go ahead and experiment using a calculator to see if you can find a value of d that works in the above equation. Remember: 21 2 and 22 4 4. So, our exponent has to be between 1 and 2. What do you think the exponent will be? Explain your answer. 5. Use a calculator and try different possibilities for d. What is the approximate dimension of the Sierpinski Triangle? Adapted from http://math.rice.edu/~lanius/fractals/dim.html Name: __________________ Class: ___________ Date: ________ Sierpinski Tetrahedron We are going to create the Sierpinski Tetrahedron, a three-dimensional version of the Sierpinski Triangle. Cut out the attached tetrahedron templates. Each template will fold along the dashed lines to form a tetrahedron. Glue or tape each tab to form three edges of the tetrahedron. The first step in making the Sierpinski tetrahedron is to arrange three of your tetrahedra so that they look like this from above: Glue these three tetrahedra together at the vertices where they touch. To make the new figure as strong as possible, you may want to insert a toothpick into the vertex of each tetrahedron and glue it in place. Complete the new figure by gluing a fourth tetrahedron above these three, attaching each of the vertices of its base to one of the three vertices that are pointing upward: Your final shape should be a larger tetrahedron with a hole in the middle. This is Stage 1 of the Sierpinski tetrahedron! We can consider one of the original tetrahedra to be Stage 0. 1. How many vertices, edges, and faces are there in the original tetrahedron, or Stage 0? 2. How many vertices, edges, and faces are there in the Stage 1 tetrahedron? Follow this rule for building the Sierpinski tetrahedron: The next stage can be created by taking 4 of the previous stage and gluing these figures together to form a larger tetrahedron. Name: __________________ Class: ___________ Date: ________ 3. Use this rule to build the Sierpinski tetrahedron: complete the following table. You don’t have to build more than Stage 2, unless it helps you figure out the patterns. Stage 0 1 2 3 4 5 6 Side Length 1 2 Number of Vertices Number of Edges Number of Faces 4. How could you find the number of vertices for any stage number? Describe the pattern in words or symbols. 5. How could you find the number of edges for any stage number? Describe the pattern in words or symbols. 6. The pattern for determining the number of faces for any stage number is similar to the one of the patterns in the last two problems. To which is it similar? Explain why the patterns are similar, and why the other pattern is different. 7. GEOMETRY. The four Stage 0 tetrahedra form an empty space within the Stage 1 tetrahedron. Describe the shape of this hole by naming it and counting its number of vertices, edges, and faces. Name: __________________ Class: ___________ Date: ________ 8. CHALLENGE. In Stage 1, what fraction of space is taken up by the hole compared with the entire figure? What about in Stage 2? Explain your reasoning, and describe the pattern as the stage number increases. 9. CHALLENGE. Imagine a solid tetrahedron the same size as the Stage 1 tetrahedron. How does the surface area of Stage 1 compare to the surface area of this solid? Answer the same question for Stage 2. Explain your reasoning, and describe the pattern as the stage number increases. 10. CHALLENGE. Combine your Sierpinski Tetrahedron with your classmates’, and make as large a Sierpinski tetrahedron as you can! Can you make one large enough to get inside? Name: __________________ Class: ___________ Date: ________ Pascal’s Triangle Pascal’s Triangle is a triangular array of numbers. As the triangle grows, more rows are added through addition and symmetry. Look at the beginning of Pascal’s Triangle on the next page and see if you can discover how each row is constructed. 1. Fill in the missing numbers of Pascal’s Triangle shown on the next page. What procedure did you use to fill in the triangle? 2. What are some patterns that you see in Pascal’s Triangle? 3. How do you know if a number is going to be even or odd? 4. What do you notice about the location of the even and odd numbers? Why do you think their locations have these patterns? 5. Shade in the odd numbers. What do you notice? Why do you think this pattern happens? 6. Combine your triangle with those of your classmates. What did you discover? Triangle image from Connors et al (2004) Name: __________________ Class: ___________ Date: ________ Triangle image from Connors et al (2004) Name: __________________ Class: ___________ Date: ________ Cutting Away 1. Start with a piece of 8 ½ X 11 paper. Cut the paper into three pieces. This represents stage 1. Next, cut each of these pieces of paper into three pieces. This represents stage 2. Repeat this procedure a few times. As you are working, complete the table below. Can you find a function that represents the number of pieces of paper you will have after n times of repeating this process? Try to look for patterns to fill in the table. Stage Number Number of Pieces of Paper 0 1 2 3 4 … n 2. The process of repeating the same procedure based on a prior action again and again is called iteration. Explain how cutting the paper represents an example of iteration. 3. What type of function is being represented here? Explain your reasoning. 4. Represent this iteration in a recursive formula. A recursive formula has a defined starting value and represents an example of iteration. In other words, you are using the output of the previous function calculation as the input for the next one. 5. Another example: Given the following set of data: G(0) = 4, G(1) = 12, G(2) = 36, G(3) = 108 a.) What type of function is represented? What is the function? Explain what each part of the equation represents. b.) What is the recursive formula for this data? Explain your reasoning. Adapted from McDougal, Littell, and Company’s Algebra 2 textbook Name: __________________ Class: ___________ Date: ________ The Sierpinski Triangle Fractals are geometric shapes that are self-similar. If one “zooms in” on a section of a fractal, it looks exactly the same as the original image. The Sierpinski Triangle is one of the most famous examples of a fractal. It contains a wide variety of patterns and applications to other areas of mathematics. To create a Sierpinski Triangle begin with a triangle of any size. We will call this triangle Stage 0. Stage 0 Locate the midpoints of each side and connect them forming four triangles similar to each other and the original. Remove the middle triangle, leaving three smaller triangles at the corners of the larger. This is Stage 1. Stage 1 Name: __________________ Class: ___________ Date: ________ Repeat this process on the three shaded triangles remaining in Stage 1 by locating the midpoints of each side of the shaded regions, connecting them to form four similar triangles, and “removing” the center triangle each time. This is Stage 2. Try creating Stage 2 based on the given midpoints. Stage 2 Following the same procedure, complete Stage 3 and Stage 4 on the triangles shown below. Stage 3 Stage 4 How many Stage 1 triangles are in Stage 2? How many Stage 2 triangles are in Stage 3? How many Stage 3 triangles are in Stage 4? How many Stage 1 triangles are in Stage 3? How many Stage 1 triangles are in Stage 4? What patterns do you see here? Explain. Name: __________________ Class: ___________ Date: ________ The process of repeating the same procedure based on a prior action again and again is called iteration. How do your drawings and answers from above represent an example of iteration? If the process described above is repeated an infinite number of times, the outcome will be the Sierpinski Triangle (shown in the pictures on the previous page). 1. Patterns in the Sierpinski Triangle In a discussion with your group members, what are some patterns that you see developing in the Sierpinski Triangle? For example, take a look at the number of triangles, both shaded and unshaded, edge lengths, and fractional parts. 2. Exploring The Perimeter of the Sierpinski Triangle Let’s take a closer look at the perimeter of the Sierpinski Triangle by analyzing the triangle at each stage and looking for patterns. Assume that the measure of each side of the stage 0 triangle has a length of one. Fill in the table below for stages 0 – 5. Please use fractions in your answers. (NOTE: Number of triangles refers to the number of small triangles left after the removal of the middle triangles.) Stage 0 1 2 3 4 5 … n Number of Triangles 1 Length of 1 Each Side Perimeter of 3 Each Triangle Total Perimeter 3 Ratio of Total Perimeter to Previous a. Looking at the pattern of the number of triangles, how many triangles do you think will be in the Stage 7 triangle? the Stage 15 triangle? the Stage n triangle? Describe the pattern in words and symbols. b. Looking at the pattern of the length of each side, how long would the side of a triangle be in the Stage 7 triangle? the Stage 15 triangle? the Stage n triangle? Describe the pattern in words and symbols. Name: __________________ Class: ___________ Date: ________ c. Looking at the pattern of the perimeter of each triangle, what is the perimeter of each triangle in the Stage 7 triangle? the Stage 15 triangle? the Stage n triangle? Describe the pattern in words and symbols. d. Looking at the pattern of the total perimeter, what is the total perimeter in the Stage 7 triangle? the Stage 15 triangle? the Stage n triangle? Describe the pattern in words and symbols. e. What do you notice about the ratio of the total perimeter to the perimeter of the previous stage? Why does this pattern occur? f. What types of functions do your Stage n triangles represent? Why? Draw a sketch of the graphs of each of the functions represented in parts a – d. Name: __________________ Class: ___________ Date: ________ g. Do you see any connections between the shape of the graphs and your generalizations of the perimeter discoveries in parts a – d? h. Using your equations, graphs, and your answer from part g, what do you think will happen in each case above as the process of creating the Sierpinski Triangle is done an infinite number of times? Explain your findings. 3. Exploring the Area of the Sierpinski Triangle Now, let’s take a closer look at the area of the Sierpinski Triangle by analyzing the triangle at each stage and looking for patterns. Assume that the area of the stage 0 triangle is 1. Fill in the table below for stages 0 – 5. Please use fractions in your answers. Stage 0 1 2 3 4 5 … n Number of Triangles 1 Area of Each Triangle 1 Total Shaded Area Ratio of Total Shaded Area to Previous Total Unshaded Area Ratio of Unshaded Area to Shaded Area a. Looking at the pattern of the area of each triangle, what is the area of each triangle in the Stage 7 triangle? the Stage 15 triangle? the Stage n triangle? Describe the pattern in words and symbols. b. Looking at the pattern of the total shaded area, what is the total shaded area in the Stage 7 triangle? the Stage 15 triangle? the Stage n triangle? Describe the pattern in words and symbols. Name: __________________ Class: ___________ Date: ________ c. Looking at the pattern of the total unshaded area, what is the total unshaded area in the Stage 7 triangle? the Stage 15 triangle? the Stage n triangle? Describe the pattern in words and symbols. d. What do you notice about the ratio of the total area to the previous area? Why does this pattern occur? e. What do you notice about the ratio of the total unshaded area to the total shaded area? Why does this pattern occur? f. What types of functions do your Stage n triangles represent? Why? Draw a quick sketch of the graphs of each of the functions represented in parts a – c. g. Name: __________________ Class: ___________ Date: ________ g. Do you see any connections between the shape of the graphs and your generalizations of the area discoveries in parts a – c? h. Using your equations, graphs, and your answer from part g, what do you think will happen in each case above as the process of creating the Sierpinski Triangle is done an infinite number of times? Explain your findings. Name: __________________ Class: ___________ Date: ________ The Sierpinski Carpet Now that you have explored the basic concepts of a fractal and the idea of self- similarity, let’s take a look at another fractal, called the Sierpinski Carpet. The Sierpinski Carpet is similar to the Sierpinski Triangle in that both are fractals based on basic shapes and have various patterns that can be seen and explored. To create a Sierpinski Carpet begin with a square of any size. We will call this square Stage 0. Stage 0 Divide the square into nine smaller squares of equal size. Remove the middle square, leaving eight smaller shaded squares on the outside edges of the larger. This is Stage 1. Stage 1 Name: __________________ Class: ___________ Date: ________ Repeat this process on the eight squares remaining in Stage 1 by dividing each square into nine smaller squares of equal size and removing the square in the middle (while shading the remaining squares). This is Stage 2. Try creating Stage 2 based on the given picture. Stage 2 Following the same procedure, complete Stage 3 and Stage 4 on the squares shown below. Stage 3 Stage 4 How many Stage 1 squares are in Stage 2? How many Stage 2 squares are in Stage 3? How many Stage 3 squares are in Stage 4? How many Stage 1 squares are in Stage 3? How many Stage 1 squares are in Stage 4? Do you see a pattern here? Explain. Name: __________________ Class: ___________ Date: ________ How do your drawings of the Sierpinski Carpet and answers from above represent another example of iteration? If the process described above is repeated an infinite number of times, the outcome will be the Sierpinski Carpet (shown in the pictures on the previous page). 1. Patterns in the Sierpinski Carpet In a discussion with your group members, what are some patterns that you see developing in the Sierpinski Carpet? For example, take a look at the number of squares, both shaded and unshaded, edge lengths, and fractional parts. 2. Exploring The Perimeter of the Sierpinski Carpet Let’s take a closer look at the perimeter of the Sierpinski Carpet by analyzing the square at each stage and looking for patterns. Assume that the measure of each side of the stage 0 square has a length of one, and we will define perimeter as any interior or exterior exposed edge. Fill in the table below for stages 0 – 5. Please use fractions in your answers. (NOTE: Number of squares refers to the number of small squares left after the removal of the middle squares.) Stage 0 1 2 3 4 5 … n Number of Squares 1 Length of 1 Each Side Perimeter of 4 Each Square Total Perimeter 4 Ratio of Total Perimeter to Previous a. Looking at the pattern of the number of squares, how many squares do you think will be in the Stage 7 square? the Stage 15 square? the Stage n square? Describe the pattern in words and symbols. b. Looking at the pattern of the length of each side, how long would the side of a square be in the Stage 7 square? the Stage 15 square? the Stage n square? Describe the pattern in words and symbols. Name: __________________ Class: ___________ Date: ________ c. Looking at the pattern of the perimeter of each square, what is the perimeter of each square in the Stage 7 square? the Stage 15 square? the Stage n square? Describe the pattern in words and symbols. d. Looking at the pattern of the total perimeter, what is the total perimeter in the Stage 7 square? the Stage 15 square? the Stage n square? Describe the pattern in words and symbols. e. What do you notice about the ratio of the total perimeter to the previous perimeter? Why does this pattern occur? f. What types of functions do your Stage n squares represent? Why? Draw a sketch of the graphs of each of the functions represented in parts a – d. Name: __________________ Class: ___________ Date: ________ g. Do you see any connections between the shape of the graphs and your generalizations of the perimeter discoveries in parts a – d? h. Using your equations, graphs, and your answer from part g, what do you think will happen in each case above as the process of creating the Sierpinski Carpet is done an infinite number of times? Explain your findings. 3. Exploring the Area of the Sierpinski Carpet Now, let’s take a closer look at the area of the Sierpinski Carpet by analyzing the square at each stage and looking for patterns. Assume that the area of the stage 0 square is 1. Fill in the table below for stages 0 – 5. Please use fractions in your answers. Stage 0 1 2 3 4 5 … n Number of Squares 1 Area of Each Square 1 Total Shaded Area Ratio of Total Shaded Area to Previous Total Unshaded Area Ratio of Unshaded Area to Shaded Area a. Looking at the pattern of the area of each square, what is the area of each square in the Stage 7 square? the Stage 15 square? the Stage n square? Describe the pattern in words and symbols. b. Looking at the pattern of the total shaded area, what is the total shaded area in the Stage 7 square? the Stage 15 square? the Stage n square? Describe the pattern in words and symbols. Name: __________________ Class: ___________ Date: ________ c. Looking at the pattern of the total unshaded area, what is the total unshaded area in the Stage 7 square? the Stage 15 square? the Stage n square? Describe the pattern in words and symbols. d. What do you notice about the ratio of the total area to the previous area? Why does this pattern occur? e. What do you notice about the ratio of the total unshaded area to the total shaded area? Why does this pattern occur? f. What types of functions do your Stage n squares represent? Why? Draw a quick sketch of the graphs of each of the functions represented in parts a – c. h. Name: __________________ Class: ___________ Date: ________ g. Do you see any connections between the shape of the graphs and your generalizations of the area discoveries in parts a – c? h. Using your equations, graphs, and your answer from part g, what do you think will happen in each case above as the process of creating the Sierpinski Carpet is done an infinite number of times? Explain your findings. 4. Connections Do you see any relationship between the generalizations for the Sierpinski Triangle and the Sierpinski Carpet? What role does the number of sides of the basic shape in each fractal play in the generalizations you made?