Stata to Draw Fractals Seth Lirette MS Inspiration Types Of Fractals Escapetime Fractals Formula iteration in the complex plane Iterate many times If doesnt diverge to infinity it belongs in the set and you mark it ID: 746198
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Slide1
Structured Chaos:
Using Mata and
Stata
to Draw Fractals
Seth Lirette, MSSlide2
InspirationSlide3
Types
Of FractalsSlide4
Escape-time
FractalsFormula iteration in the complex plane
Iterate many timesIf doesn’t diverge to infinity, it belongs in the set and you mark it.Otherwise, color the point depending on how fast it escapes to infinity.
Mandelbrot Set
Julia Sets
Burning Ship FractalSlide5
Iterated Function Systems (IFS)
Draw a shapeReplace that shape with another shape, iteratively
Koch Snowflake
Peano
Curve
Barnsley
FernSlide6
Lindenmayer
Systems (L-systems)Different “Language”A form of string rewiringStarts with an axiom and has a set of production rules
Levy Curve
Dragon CurveSlide7
Strange Attractors
Solutions of intial-value differential equations that exhibit chaos
Lorenz Attractor
Rossler
Attractor
Double Scroll AttractorSlide8
mata
+
ExamplesSlide9
Mandelbrot Set
The set M of all points c such that the sequence z → z2 + c does not go to infinity.Slide10
Mandelbrot SetSlide11
Barnsley
FernCreated by Michael Barnsley in his book Fractals Everywhere.
Black Spleenwort
+
+
+
Defined by four transformations
w
ith assigned probabilities:Slide12
Barnsley
FernSlide13
Koch Snowflake
Based on the Koch curve, described in the 1904 paper “On a continuous curve without tangents, constructible from elementary geometry” by Helge von KochConstruction: (1) Draw an equilateral triangle;
(2) Replace the middle third of each line segment with an equilateral triangle; (3) Iterate Slide14
Koch SnowflakeSlide15
Dragon Curve
First investigated by NASA physicists John Heighway, Bruce Banks, and William Harter.Construction as an L-system: Start: FXRule: (X
X + YF), (Y FX – Y)Angle: 90oWhere:
F = “draw forward” - = “turn left 90o” + = “turn right 90o”Slide16
Dragon CurveSlide17
Lorenz Attractor
Plots the “Lorenz System” of ordinary differential equations
:Slide18
Lorenz AttractorSlide19
Finite Subdivisions
Random Fractals
Brownian Motion
Cantor Set
Sierpinski
Triangle
Levy FlightSlide20
Thank You