Fractals. Similar Figures Same shape Corresponding angles are congruent Corresponding sides are...
-
Upload
aubrey-carr -
Category
Documents
-
view
214 -
download
0
Transcript of Fractals. Similar Figures Same shape Corresponding angles are congruent Corresponding sides are...
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