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=Z2+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, www.artbymath.com

• Cynthia Lanius’ Lessons: A fractal lesson, http://math.rice.edu/~lanius/frac/

• Fractals : Useful Beauty, http://www.fractal.org/Bewustzijns-Besturings-Model/Fractals-Useful-Beauty.htm

• “Fractals,” World Book Encyclopedia, http://www.aolsvc.worldbook.aol.com/ar?/na/ar/co/ar208470.htm

More Fractal Designswww.Artbymath.com