S: How To Have Fun In Your Math Class By Creating Fractals
FC: How to Have Fun In Your Math Class By Creating Fractals Angie Durham
1: Daydreaming in your math class...does that ever happen to you? Not according to your teacher, maybe, but every once in a while, do you think, when can you ever learn anything fun in math anyway?’ Well, I am here to say that, among the very exciting things you can use math for, creating fractals is a fun way to use your math skills. Now, it may not help you settle world peace, and it may not make you run faster or jump higher, but they are pretty, and more importantly, may help you get an A in your math class. | So, grab paper and pencil (colored if you're creative!) and let's begin creating fractals.
2: What is a fractal? | Fractals are geometrical figures that are created by beginning with a very simple pattern and allowing it to grow through the application of rules. | Many times, these rules allow the figure to grow from one stage to the next, | by taking the original figure and modifying it or adding to it. This can be done an infinite number of times.
3: There are five common techniques for generating fractals: Iterated Function Systems | Escape-time fractals | Random fractals | Strange attractors | L-Systems -These are generated by string rewriting and are used to model the branching patterns of plants
4: Let us begin by visualizing a simple example, as the one above. We begin with a simple design, like the one on this page. We then encourage our image to ‘grow’ by adding very small duplicates of this image to it, a recursive (or infinite) amount of times. Begin by drawing your own simple design and adding very small duplicates to it. | Making a Fractal
5: Now, if you want to be able to see anything, you must stop adding to your picture, but eventually you can have the image above. On your own example, add as many parts as you'd like.
6: This is similar to many natural objects. Does it seem as if our plant might follow a regular mechanism to accomplish its growth? This tree seems to use the technique known as L-systems. L-Systems begin with a line, and then we decide what operation we would like to do with the line to make it 'grow'. What should happen to our picture now? | Let us look at the tree that we created before. It appears as though the branches branch out in very regular patterns. | The Math of Fractal Creation:
7: Most likely, this will result in a multi-lined figure. Now, do the same system to each of the new lines. Because we are repeating the same step, we are doing so recursively. Pretty simple? Follow the picture clockwise to see what this may look like. Then repeat this for your own picture.
9: More Fractal Math Using our 'tree' from our recursion steps, let's find a simple sequence of the number of line ends for each iteration. Let's look at the first figure first. At our 0th iteration (step), we have one line. Let's now figure out how many line ends are added with each iteration (recursion step). We will take any one line end to see how it gets transformed to go to the next iteration. What is happening? Let's remember that the exact same thing happens to each line end.!!How about a table that shows the number of line ends for each iteration? You do one, and I will have one on the next page filled out.
10: We can use the formula: Line Ends = 2^(Iteration) Did your table match mine? | Iteration 0 1 2 3 4 5 6 7 Line Ends 1 2 4 8 16 32 64 128
11: How are fractals useful? | Graphics - Fractals can be used to generate the appearance of natural terrain or the complexity of terrain | Fractals can be used for many different things in many different situations. Medicine - They can classify tissue slides using fractals | Armed Forces - Fractals are used to generate patterns or camouflage more efficiently | Music - Fractals can be used to generate new music | Photography - Digital photography can use fractals o enlarge photographs | Computer and video game design - Fractals are used to generate environments | Fashion - They're super awesome looking | There are tons of other ways they are used as well.
12: Fractals in nature | Can you see the fractals found on this page? | Fractals can be found in nature in mountains, galaxy clusters, clouds, lightning, leaves, shells, snowflakes, trees, and many other places.
14: Famous Fractals Sierpinski Triangle | Space-Filling Curve This was discovered in 1890 by Peano who wanted to construct a self-intersecting curve that passes through every point on the unit square.
15: Koch Snowflake This was created by the Swedish mathematician Niels Fabian Helge von Koch to show that it is possible to have figures that are continuous everywhere but differentiable nowhere. | Lyapunov Fractal These are constructed by mapping the regions of stability and chaotic behavior.
16: Dragon Curve This is a recursive nonintersecting curve that can be constructed by representing a left turn by 1 and a right turn by 0, and then using a string of digits to create.
17: There are an infinite amount of fractals you can create!
18: Fractals and Chaos Theory The term chaos in math, refers to whether or not it is possible to make a long-term prediction about how a system will act. Sometimes, a chaotic system can develop in a way that may appear to be very smooth and ordered. A famous example of chaos theory wwas exploredby Benoit Mandelbrot who wanted to know how long the coastline of Britain was. His colleagues simply told him that they were not going to waste their time on this, so he should just look it up. Well, Mandelbrot was not going to do this, because he already knew the answer: the coastline of Britain is infinite, and he wasn't going to waste his time looking at wrong answers. How you may ask???
19: Well, take a picture of Britain. The most simple shape that will circumscribe Britain, as closely as possible, will be a triangle, and this triangle will approximate the perimeter of Britain. Take that same triangle, and increase the amount of vertices going around the coastline. Our area will become closer to Britain. Therefore, the more vertices there are, the closer the circumscribing line will be able to conform to the dips and protrusions of Britain's coastline.
20: But, wait! When our number of vertices increases, our perimeter increases as well by triangle inequality, doesn't it? And, we never have a maximum number of verties. There is no point which we can say that our shape defines the coastline of Britai, since making the coast of Britain would include each and every rock, tide pool, pebble, everything that lies on the edge of Britain! By the time you think you've gotten somewhere on the project, something has moved! But, from our original map, the coastline looked so darn smooth...
21: Just like our coastline, fractals are geometric shapes that are very complex and infinitely detailed. You can zoom in on a section and it will have just as much detail as the whole fractal. They are recursively defined and small sections of them are similar to large ones. One way to think of fractals for a function f(x) is to consider x, f(x), f(f(x)), f(f(f(x))), f(f(f(f(x)))), etc. Fractals are related to chaos because they are complex systems that have definite properties.
22: Resources | The Mathematics that Every Secondary School Math Teacher Needs to Know By: Alan Sultan and Alice F. Artzt Copyright: 2011 Publisher: Routledge, New York, New York Math and Art: An Introduction to Visual Mathematics By: Sasho Kalajdzievski Copyright: 2008 Publisher: Taylor and Francis Group, Boca Raton, FL The Colours of Infinity: The Beauty and Power of Fractals By: Nigel Lesmoir Gordon Copyright: 2010 Publisher: Springer-Verlag London Limited, United Kingdom 101+ Great Ideas for Introducing Key Concepts in Mathematics By: Alfred S Posamentier and Herbert A Hauptman Copyright: 2006 Publisher: Corwin Press, Thousand Oaks, California Fractals: An Introduction http://ejad.best.vwh.net/java/fractals/intro.shtml By: Jacobo Bulaevsky Last updated: February 2, 2001
23: Angie Durham is a Michigan State graduate, currently obtaining her teaching certificate from Eastern Michigan University in Mathematics and Physics. She resides in Ypsilanti, Michigan, with her boyfriend, two dogs, and two cats. She believes everyone should read Mental Floss Magazine. | Published 2011