Fractals

Similar Figures Same shape Corresponding angles are congruent Corresponding sides are proportional

Fractal Geometry In high school – Euclidean geometry Fractal geometry is younger –Studies mainly over past 2 centuries –Easier with computers –Boom in the 60’s, 70’s, and 80’s

Fractal Characteristics Self-Similarity Formation through iteration Fractal Dimension

Self – Similar shapes The main characteristic of fractals is that they are self similar Smaller figures are similar to the large figure Not a necessity

Sierpinski Triangle One example of a self similar fractal is the Sierpinski Triangle. It can hold an infinite amount of smaller triangles inside the one larger triangle. It is strictly self similar, meaning the same figure is repeated.

Formation Through Iteration To repeat the same process, but each time make something more complicated. Koch Snowflake

Fractal Dimension In Euclidean Geometry we stay within 3 dimensions Involves logarithms in fractal geometry

Math Behind the Fractals There is math behind the pretty pictures Mandelbrot set – discovered by Mandelbrot in the 1960’s –Z=Z 2 +C –C is actually a complex number involving the imaginary number i

Where do we see fractals? Nature shows characteristics of fractals –Human circulatory system –Ferns –Broccoli

Art Landscapes in movies are computer generated using fractals Death Star – “Star Wars: Return of the Jedi” Genesis Effect – “Star Trek – Wrath of Kahn”

Landscapes "Clouds are not spheres, mountains are not cones, coastlines are not circles, and bark is not smooth, nor does lightning travel in a straight line." Benoit Mandelbrot Here are more examples of landscapes produced through fractals

Fractal Music Programs that allow you to create music based on fractals. Notes are assigned to numbers and as numbers are entered by the computer into an equation the output creates a sound

Other Uses for Fractals They help predict things that seem random Professions –Astronomers –Mathematicians –Scientists –Doctors –Stock Brokers

Bibliography ArtbyMath, Cynthia Lanius’ Lessons: A fractal lesson, Fractals : Useful Beauty, Model/Fractals-Useful-Beauty.htm Model/Fractals-Useful-Beauty.htm “Fractals,” World Book Encyclopedia, 70.htm 70.htm

More Fractal Designs