Introduction to Fractal Geometry FRACTAL GEOMETRY Fractal geometry is based on the idea of self-similar forms. To be selfsimilar, a shape must be able to be divided into parts that are smaller copies which are more or less similar to the whole. Because of the smaller similar divisions of fractals, they appear similar at all magnifications. However, while all fractals are self-similar, not all self-similar forms are fractals. (For example, a straight Euclidean line and a tessellation are self-similar, but are not fractals because they do not appear similar at all magnifications). Many times, fractals are defined by recursive formulas. Fractals often have a finite boundary that determines the area that it can take up, but the perimeter of the fractal continuously grows and is infinite. The Cantor Set, introduced by George Cantor, a German mathematician, in 1883, is one of the easiest ways to see the divisions similar to the whole after being magnified. The Cantor Set is a series of line segments in which the middle third is removed. After the middle third is removed, the first third and the last third remain. Of each of those segments, the middle third is removed leaving the first and last thirds. This goes on forever, removing the middle third of each new segment that is created. One of the most famous fractals is the Mandelbrot Set. French mathematician, Benoit Mandelbrot, began to study self-similarity in the 1960 s, and by 1980 was interested in graphing complex numbers. He used the recursive formula z z^2+c, where c is some real number and z is a complex number such as a+bi. Depending on the number put in, Mandelbrot discovered that some get larger and go off to infinity, while some get smaller and closer to zero. Mandelbrot then set the computer up to color the pixels for each number, or point on the complex plane. If the number got smaller and closer to zero, the computer colored it black. If it got larger it would get a different color. The colors depended on how quickly the number approached infinity. The picture he got turned out to be the most famous examples of fractal geometry. Fractal Geometry in Nature The basic definition of a fractal is something with a shape that gets smaller and repeats infinitely (i.e. if you were to zoom in on any one area of the object, you would be looking at the original picture). Fractals can be found in many forms. In nature and in the environment, approximate fractals are found everywhere, including some of the vegetables we eat. Below are some examples of fractals found in nature. http://en.wikipedia.org/wiki/image:animated_fractal_mountain.gif
This website shows an animated version of how fractal geometry can be found in mountains and mountain ranges. http://en.wikipedia.org/wiki/image:unknown_quartz_crystal_67.jpg The quartz stone is one example of a fractal image in nature. It has triangular points and would look similar if you were to zoom in on any part of the stone. http://www.flickr.com/photos/roddh/307187374/ The fern is a very good example of fractal geometry. It is very easy to demonstrate the zooming in idea.
This photo shows how a sea anemone and how even sea creatures can show fractal geometry. http://images.search.yahoo.com/search/images/view?back=http%3a%2f%2fimages.search.yahoo.com%2fsearch%2fimages%3f_adv_prop%3dimage%26va%3dfractal%26sz%3dall%26ei%3du TF-8%26ni%3D20%26fr%3Dyfp-t-501- s%26b%3d181&w=500&h=400&imgurl=static.flickr.com%2f226%2f474456954_0cfb8139e4_m.jp g&rurl=http%3a%2f%2fwww.flickr.com%2fphotos%2fseractal%2f474456954%2f&size=91.6kb &name=474456954_0cfb8139e4.jpg&p=fractal&type=jpeg&no=196&tt=357,067&oid=6de15e5d268 5f0c8&fusr=Seractal&tit=Fractal+anemone&hurl=http%3A%2F%2Fwww.flickr.com%2Fphotos%2Fs eractal%2f&ei=utf-8&src=p This picture shows a cross between broccoli and cauliflower that is a very good example of fractal geometry. http://www.flickr.com/photos/64011960@n00/1901012282/
Trees also demonstrate the concept of fractals. Tree branches usually grow and split and then grow and split and continue the pattern. Therefore, at any point, you can take a picture and zoom in and you will be looking at a very similar picture. http://rds.yahoo.com/_ylt=a9gnmiyqbvzhwkqaraejzbkf/sig=125bb0473/exp=1196906154/ **http%3a//www.flickr.com/photos/chickarun/432178870/ Queen Anne s Lace is also a very good example of fractal geometry. Each of the petals look like a smaller version of the flower.
http://rds.yahoo.com/_ylt=a0wtefvsblzhmmqauwmjzbkf/sig=1278ricor/exp =1196906450/**http%3A//www.flickr.com/photos/8589388@N04/521508634/ History of Fractal Geometry It is necessary to include some essential information about the history of fractal geometry, considering this is a history of mathematics course. Fractal geometry is very new area of mathematics. It allows us to show shapes and structures using formulas. Fractal geometry actually began in the 17 th century with philosopher Leibniz contemplating self-similarity. About a century later, Karl Weierstrass showed a function that had the property of being non-intuitive. This means it was continuous everywhere, but not differentiable. Today his graph would be considered fractal. Another very famous person in the world of fractal geometry is that of Helge Von Koch, who presented a more geometric definition. Following is the Koch snowflake: Also worth mentioning is the work of Waclaw Sierpinski, who in 1915 constructed what is now know as the Sierpinski Triangle and Sierpinski Carpet, both of which are pictured below. Both are 2D, even though once considered curves.
Sierpinski Triangle Sierpinski Carpet Furthermore, one individual, Paul Piere Levy, investigated self-similar curves even further and came up with what is today known as the Levy C curve. A picture can be viewed below or an animation version can be accessed at: http://moles.deviantart.com/art/fractal-levy-c-curve-38938489. Many others explored iterated functions in the complex plane. However, until computer graphics, they were unable to visualize the many objects they discovered. Levy C Curve Most notable in fractal geometry is Benoit B. Mandelbrot, mentioned earlier. He is responsible for the name, Fractal Geometry. He also discovered the Mandelbrot set in 1980 after working with Gaston Julia s theorems. These theorems were published in 1917 and now that we have super computers to
make the millions of calculation necessary, his theorems could be tested. In conclusion, fractal geometry is just being to take off and already we have found practical application, such as reducing file size of images and greatly enhancing their resolution. There is no doubt that the fractal geometry will continue to become more important in mathematics, science, and technology. Hands-On Fractal Demonstration Compact Disc Mirror Class Activity: Koch Snowflake Materials: straight edge, pencils Procedure: Make sure each student has a straight edge, pencil, and paper. Have them create an equilateral triangle. Then divide each side of the triangle in half and make a dot. Connect the dots to create a triangle within. Continue this process as many times as possible. Below are some pictures to help better understand the process:
WORKS CITED Scratching the Surface: What are Fractals and Fractal Geometry. http://www.sunleitz.com/whatarefractals.html Growth Factors. http://www.mathaware.org/mam/00/master/essays/dimension/dimen09.ht ml Sierpinski Carpet. http://www.answers.com/topic/sierpinskicarpet?cat=technology Fractal. http://commons.wikimedia.org/wiki/fractal?uselang=de Moles Fractal: levy c curve.http://moles.deviantart.com/art/fractal-levy-ccurve-38938489 Koch Snowflake. http://www.scientificweb.com/testreport/mathbench4/mathbench4_introduc tion.html Point Symmetry: 7.5 mm Square Paper. http://www.mav.vic.edu.au/pointsym/075sqdot.htm Cantor Set. Wikipedia. 19 November 2007. Wikimedia Foundation, Inc. 3 December 2007. <http://en.wikipedia.org/wiki/cantor_set>. The Math of Fractals. Cool Math. 1997-2007. CoolMath.com, Inc. 3 December 2007. <http://www.coolmath.com/fractals/fractals_lesson.html> Fractal. Wikipedia. 2 December 2007. Wikimedia Foundation, Inc. 3 December 2007. <http://en.wikipedia.org/wiki/fractal_geometry> DlimitR. Fractals Mandelbrot. YouTube. 17 June 2006. YouTube, LLC. 3 December 2007. <http://youtube.com/watch?v=ivrqdbaduom&feature=related>. Lanius, Cynthia. Sierpinski s Triangle. 1997-2007. 3 December 2007. <http://math.rice.edu/~lanius/fractals/selfsim.html>. The Sierpinski Triangle. http://math.rice.edu/~lanius/fractals/ Animated Mountain http://en.wikipedia.org/wiki/image:animated_fractal_mountain.gif Quartz Crystal Photo http://en.wikipedia.org/wiki/image:unknown_quartz_crystal_67.jpg Fern Photo http://www.flickr.com/photos/roddh/307187374/ Sea Anemone Photo http://images.search.yahoo.com/search/images/view?back=http%3a%2f %2Fimages.search.yahoo.com%2Fsearch%2Fimages%3F_adv_prop%3 Dimage%26va%3Dfractal%26sz%3Dall%26ei%3DUTF- 8%26ni%3D20%26fr%3Dyfp-t-501- s%26b%3d181&w=500&h=400&imgurl=static.flickr.com%2f226%2f4744 56954_0cfb8139e4_m.jpg&rurl=http%3A%2F%2Fwww.flickr.com%2Fphot os%2fseractal%2f474456954%2f&size=91.6kb&name=474456954_0cf b8139e4.jpg&p=fractal&type=jpeg&no=196&tt=357,067&oid=6de15e5d26
85f0c8&fusr=Seractal&tit=Fractal+anemone&hurl=http%3A%2F%2Fwww.f lickr.com%2fphotos%2fseractal%2f&ei=utf-8&src=p Broccoli/Cauliflower Photo http://www.flickr.com/photos/64011960@n00/1901012282/ Tree Branch Photo http://rds.yahoo.com/_ylt=a9gnmiyqbvzhwkqaraejzbkf/sig=125bb047 3/EXP=1196906154/**http%3A//www.flickr.com/photos/chickarun/432178 870/ Queen Anne s Lace Photo http://rds.yahoo.com/_ylt=a0wtefvsblzhmmqauwmjzbkf/sig=1278rico r/exp=1196906450/**http%3a//www.flickr.com/photos/8589388@n04/52 1508634/