University School of Nashville Sixth Grade Math Self-Guided Challenge Curriculum Unit 2 Fractals This curriculum was written by Joel Bezaire for use at the University School of Nashville, funded by a grant from Quaker Hill in the Summer of 2011. We are making it available for anyone to use. You may not alter, distribute, or disseminate this document except for use in your own personal classroom or home-school/tutoring situation. You may not charge a fee for this document. Do not email this document or share it via USB drive or the Cloud if you wish to share it with someone please direct them to pre-algebra.info so that person can download it for themselves from the source website. All rights reserved by the author and USN (www.usn.org) About This Curriculum
What you hold in your hands is a special 6 th grade curriculum, designed by the math teachers at USN to make sure that the best and brightest math students in the grade are being challenged in the way they should be challenged. There are a number of Units in this curriculum. They are very different in the way the activities are arranged: Some use computers, and some don t. Some require a great deal of reading, and some don t. Some might require you to do a great deal of writing, and some won t. Because that is the case you may enjoy some Units more than others, depending on your preferred activities. Because of your particular strengths, you may be invited to do some Units and not invited to do others. What they all have in common is as follows: 1) These are all designed to be self-studies. While you may need to receive some help from a teacher from time to time, if you find yourself having to constantly receive assistance, you might be better served to be with the rest of your 6 th grade class during this unit. 2) This curriculum is designed to be difficult. Since these Units are designed to give a challenge above and beyond the regular 6 th grade curriculum, you should expect to spend some time and effort completing the tasks. Sometimes you may need to set this booklet aside and just think. Sometimes you may need to walk away and work on something else while your brain stews on a difficult problem. All of that is OK. Only when you re convinced that you are stuck should you seek help from a teacher. Of course, the work in this booklet should represent the work of the student, not a parent, sibling, or classmate. 3) This curriculum assumes that you enjoy learning math. We don t put a lot of effort into selling mathematics to you, trying to convince you that mathematics can be fun and worthwhile. We assume that you already think so, and that s why you re tackling this curriculum! 4) Since this is a self-guided curriculum, please let a teacher know if something sparks your interest! There might be a way to spend more time on a topic that you find particularly interesting. You won t know unless you ask Have fun!
Fractal geometry will make you see everything differently. There is danger in reading further. You risk the loss of your childhood vision of clouds, forests, flowers, galaxies, leaves, feathers, rocks, mountains, torrents of water, carpets, bricks, and much else besides. Never again will your interpretation of these things be quite the same. Michael F. Barnsley Introduction Have you ever noticed that the shapes in nature often look differently than the shapes that we learn about in geometry? There aren t a lot of squares, rectangles, or even circles that occur naturally. That doesn t mean it isn t important to study squares, rectangles, circles, and other geometric shapes. In fact, we ll do just that in a later Unit. In recent years, however, mathematicians have tried studying the types of shapes that are seen in nature. For example: Can we describe the shape of a cloud? Of the veins in a leaf? Of a snowflake? Enter fractals: The study of fractals is relatively new, and still developing. What mathematicians and scientists are discovering, though, is that fractals often represent shapes or patterns found in nature. You ll complete this Unit on fractals at the same time your classmates are studying fractions. Fractals. Fractions. Hmmm could they have something in common?
1. Introduction To Fractals Fractals: Hunting The Hidden Dimension Questions These questions refer to the video Fractals: Hunting the Hidden Dimension (available streaming on Amazon and other sources). Answer them as the video plays (you may have to pause the video, or even re-watch a section of the video). The answers to these questions are found sequentially throughout the video. 1) What was the name of the mathematician who wrote the book Fractals: Form, Chance and Dimension? 2) In the video, an employee of Boeing purchased the book. What problem was this employee trying to solve? 3) Explain what a fractal is and how a fractal is created. 4) What word do mathematicians use that means endless repetition? 5) Complete the quote: Think not of what you see but. (This quote is one of the keys to understanding fractal geometry) 6) Describe the concept of self-similarity.
7) What is one of the most familiar examples of self-similarity? Explain. 8) Complete the quote: The basic assumption that underlies classical mathematics is that everything is...(c)lassical mathematics is really only well suited to study. 9) In what decade did Mandelbrot introduce his new mathematics? 10) What realization did Mandelbrot have that made him interested in mathematics for the first time? 11) What company did Mandelbrot work for beginning in 1958? Why did he leave his teaching position in France? 12) What problem was Mandelbrot trying to solve for IBM when he discovered selfsimilarity?
13) Explain what the late 19 th -century mathematicians called monsters. 14) What measurement problem did the Koch Curve help to address? 15) Instead of the length of a coastline, what did Mandelbrot attempt to measure? 16) Describe what Mandelbrot meant by a fractal dimension. 17) Explain the Julia Set. How did Mandelbrot achieve a breakthrough in solving a Julia Set? 18) How is the Mandelbrot Set related to the Julia Set?
19) What was the initial reaction of most mathematicians when the Mandelbrot Set was introduced? 20) What did Nathan Cohen demonstrate about small, wide-range (or wide-band) antennas? Where are those antennas used today? 21) Explain how self-similarity is found in the way the human heart and eye works. Why might this be important? 22) How are fractals related to the way an organism s energy intake and mass are related?
2. Examining The Hilbert Curve We re going to examine some basic fractals, so we can start to understand the ideas of iterations and self-similarity. Let s start with the Hilbert Curve. It is named after German mathematician David Hilbert, who lived until 1943. Start with this figure, and assume each side to be one (1) unit in length: This is the first iteration of the Hilbert Curve. To get the next iteration of the Hilbert Curve, replace each line segment in the original iteration with this shape: Assume each segment is one-third the length of the original segment. If a segment has two endpoints that join another segment, (like the top line does) then simply replace it with this shape: So, the second iteration of the Hilbert Curve would look like this: The first two iterations of the Hilbert Curve together look like this: In the space to the right (above) of the second iteration, can you draw the third iteration of the Hilbert Curve? The answer is on the next page, but try it yourself first.
Answer: Fill in the table below to represent the length of each iteration of the Hilbert Curve. Remember to treat each side length in the first iteration as one unit, and each iteration breaks the previous unit into thirds. See if you can complete the table without drawing out subsequent iterations. Iteration Total Length 1 3 2 3 4 5 Use the space below to show any work:
a) What do you notice about the total length of the curve in each case? b) Will the length of the Hilbert Curve eventually level off? Will it ever stop getting bigger? Explain. c) Does the space that the Hilbert Curve occupies ever get any larger? In other words, does the area within the original boundary every change? d) Based on your answer to parts b) and c), is it possible for an curve of infinite length to occupy a finite space? Explain.
3. Heron s Formula Before we examine our next fractal, we need to study a formula that can be used to find the Area of a triangle. Now, you probably already know that the area of a triangle is found by the formula A =!!, where b represents the length of the base and h represents! the length of the height. Heron s Formula (found by the ancient Greek mathematician Heron of Alexandria, who lived in the first century C.E.) helps us solve for the Area of a triangle when we don t know the height when all we know is the length of the three sides. Here are the steps to finding the area of a triangle using Heron s Formula. 1) Find the perimeter of the triangle. 2) Divide the perimeter by two. This is called the semi-perimeter or s. 3) The formula for the Area, given the semi-perimeter s and the three sides a, b, and c is: A = s(s a)(s b)(s c) Problem Solving with Heron s Formula: 1) Find the area of the following triangle using Hero s Formula: In this triangle, a = 14, b = 17, c = 23 The solution is on the next page.
Solution: a) Find the semi-perimeter: s = 14 + 17 + 23 2 s = 54 2 s = 27 b) Find the Area: A = 27(27 14)(27 17)(27 23) A = (27)(13)(10)(4) A = 14 040 A 118.5 square units 2) Draw a triangle below using a ruler. Measure the sides of the triangle in centimeters. Be as accurate as you can. Label the length of each side. Find the area of the triangle using Heron s Formula.
4. The Koch Snowflake Now that we know how to find the area of a triangle using Heron s Formula, we can examine another famous fractal: The Koch Snowflake. You will likely recognize this fractal from the introduction video. It is named after Swedish mathematician Helge von Koch. Start with an equilateral triangle that has each side length of one (1) unit: This is the first iteration of the Koch Snowflake. For the second iteration, replace each line in the Koch Snowflake with this shape (assume that each segment is one-third the length of each original segment): That makes the second iteration of the Koch Snowflake look like this: In this picture, you can see the second iteration of the Koch Snowflake with the original triangle highlighted in green: Here are the first two iterations of the Koch Snowflake together: In the space to the right of the second iteration (above), draw the third iteration of the Koch Snowflake. (Remember: Divide each segment into thirds and replace the middle segment with an equilateral triangle). The answer is on the next page.
Solution: (Note: Each side length of the original triangle is still one unit long in each iteration the drawings above aren t necessarily to scale). Fill in the table below to represent the area and perimeter of each iteration below. Remember to treat each side length in the first iteration as one unit, and each iteration breaks the previous unit into thirds. See if you can complete/estimate the table without drawing out subsequent iterations. Remember to take advantage of Heron s Formula when finding the areas. Verify the solutions already in the table. Iteration Area Perimeter 1 0.433 3 2 3 4 5 Use the area below to show any work:
a) How are the patterns in the perimeters of Koch s Snowflake and the lengths of Hilbert s Curve similar? How are they different? b) Describe what is happening to the area of Koch s Snowflake. How is it similar to what is happening to the area of Hilbert s Curve? How is it different? c) Consider the billionth iteration of Koch s Snowflake. Estimate the area and perimeter of the figure.
5. A Chaos Game The short film that you watched at the beginning of this unit referenced the fact that fractals often appear chaotic, but really represent a great deal of order. Mathematicians were hesitant to accept fractals as a branch of mathematics because they look so disorderly. One of Mandelbrot s (and his allies ) main arguments was the fact that fractals do, in fact, contain a great deal of order. We are going to show this order arising from chaos by playing a chaos game. Here are the extra materials you will need for this game: 1) A regular, 6-sided die: 2) A ruler with centimeters marked. On the next page is an equilateral triangle with vertices marked A, B, and C. Place a point randomly inside the triangle. This is point P, but don t label the point. Instructions: 1) Roll your 6-sided die. On a result of 1 or 2, place a point halfway between point P and point A (use your ruler to make sure you re measuring in a straight line segment between the two points). On a result of 3 or 4, place a point halfway between point P and point B. On a result of 5 or 6, place a point halfway between point P and point C. 2) Repeat step one above, but use each newly placed point instead of point P each time. 3) Repeat about 400 times. Seriously. You don t have to do all of the repetitions you can have a friend help you (as long as you explain the game to them properly). A parent or a sibling can help you. You shouldn t do them all at once. But you need at least 400 repetitions of this game. You can do more if you want your results will be even better the more times you play this game. 4) Answer the questions on the following page once you have completed the game.
Questions: a) Describe the pattern that you see inside the equilateral triangle. b) Would you describe the game that you played (involving the dice rolling) as ordered or chaotic? Explain. c) Would you describe the pattern inside the equilateral triangle as ordered or chaotic? Explain. d) Use the equilateral triangle below to draw what you think the picture would look like after a billion plays. e) How does the triangle you created relate to the topic of fractals? (In other words, where is its self-similarity?)
6. Sierpinski s Triangle The triangle that you created in the previous chaos game is a famous fractal called Sierpinski s Triangle. It is named after Polish mathematician Waclaw Sierpinski, who passed away in 1969. Here are the first five iterations of Sierpinski s Triangle: a) Explain in words how each new iteration of Sierpinski s Triangle is obtained. b) Find the area of the shaded part of each iteration of Sierpinski s Triangle below. See if you can complete the table without drawing out subsequent iterations. Remember to take advantage of Heron s Formula when finding the areas. Iteration 1 2 3 4 5 Area of Shaded Part Show any work below and on the next page:
c) Is there a pattern in the Area column? If so, what is it? d) Based on that pattern, will the area of the shaded section ever become zero? If so, after how many iterations? If not, why not?
7. Your Own Fractal Create your own fractal. For your first iteration use either a line segment (with a length of one unit), an equilateral triangle (sides of one unit in length) or a square (sides of one unit in length). a) Explain in words how you will obtain each subsequent iteration of your fractal. b) Draw the first four iterations of your fractal: c) Find the area and perimeter for each of the first four iterations of your fractal drawn above. Iteration Area Perimeter 1 2 3 4 Show any work below and on the top of the next page:
d) Are there any patterns in the area or perimeter of your fractal? If so, what are they? e) Can you predict the dimensions (area, perimeter) of your fractal after billions of iterations? Can you draw what you think your fractal will look like after billions of iterations? f) Name your fractal. Choose a name that describes how it looks (like Snowflake, Curve, Triangle, etc.) The (Your Name) (Name of Fractal)