An An Introduction Introduction

totoFractalsFractals

By: Brian FeuerBy: Brian Feuer

What is a Fractal?What is a Fractal?A word coined by A word coined by Benoit MandelbrotBenoit Mandelbrot

in 1975 to describe shapes that are in 1975 to describe shapes that are “self-similar”“self-similar” and are generated by a and are generated by a

repeating pattern, typically a repeating pattern, typically a “recursive”“recursive” or or “iterative”“iterative” process. process.

Fractals are independent of Fractals are independent of scale.scale.

Components of a FractalComponents of a Fractal Self SimilaritySelf Similarity

• Two or more objects having the same characteristics. In Two or more objects having the same characteristics. In fractals, the shapes of lines at different iterations look like fractals, the shapes of lines at different iterations look like smaller versions of the earlier shapes. smaller versions of the earlier shapes.

• Otherwise put, the shapes look the same at different Otherwise put, the shapes look the same at different magnifications.magnifications.

IterationIteration• Repeating a set or rules or steps over and over.Repeating a set or rules or steps over and over.• One single step is called an iterate.One single step is called an iterate.

RecursionRecursion

. . Given some starting information and a rule Given some starting information and a rule for howfor how to use it to get new information, the rule is to use it to get new information, the rule is thenthen repeated using the new information.repeated using the new information.

Fractals in NatureFractals in NatureCan you think of any fractals found Can you think of any fractals found

in nature?in nature?

More Fractals in NatureMore Fractals in Nature

Complex FractalsComplex Fractals

The Sierpinski Triangle The Sierpinski Triangle

Named after the Polish mathematician Waclaw Named after the Polish mathematician Waclaw

Sierpinski 1916.Sierpinski 1916. It is one of the simplest fractal shapes in It is one of the simplest fractal shapes in

existence existence It can be generated by infinitely repeating a It can be generated by infinitely repeating a

procedure of connecting the midpoints of each procedure of connecting the midpoints of each side of the triangle to form four separate side of the triangle to form four separate triangles, and cutting out the triangle in the triangles, and cutting out the triangle in the center. center.

ReferencesReferences http://www.dcet.k12.de.us/teach/quest/shari.htm by http://www.dcet.k12.de.us/teach/quest/shari.htm by

Shari GalganoShari Galgano http://math.rice.edu/~lanius/frac/ by Cynthia Laniushttp://math.rice.edu/~lanius/frac/ by Cynthia Lanius http://astronomy.swin.edu.au/~pbourke/fractals/http://astronomy.swin.edu.au/~pbourke/fractals/

fuzzylogic/ by Roger Bagulafuzzylogic/ by Roger Bagula http://occawlonline.pearsoned.com/bookbind/http://occawlonline.pearsoned.com/bookbind/

pubbooks/billstein_awl/medialib/shodor_site/lessons/pubbooks/billstein_awl/medialib/shodor_site/lessons/frac1.html frac1.html by The Shodor Education Foundation, Inc.by The Shodor Education Foundation, Inc.